Is 0.9999... really the same as 1?

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Is 0.9999... really the same as 1?

Yes 0.9999... = 1
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31%
No 0.9999... is slightly less than 1
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44%
Other
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25%

marsh8472
Posts: 54
Joined: Sun Oct 19, 2014 3:06 pm

Re: Is 0.9999... really the same as 1?

wtf wrote:I skimmed the Better Explained piece. It seems like a compendium of all the wrong and misleading things people say on these .999... discussion forum threads.

In particular, the author claims that .999... < 1 in the hyperreals. This is simply not only false, but provably impossible. Professor Amit explained why in the very Quora article that you yourself linked.

Let me explain how this works, although this goes beyond what Prof. Amit said.

Consider the first-order theory of the real numbers. This is the formal syntactic list of axioms involving addition, multiplication, order, etc. It does not included the completeness axiom, which is second order (we quantify over subsets and not just over elements).

The standard reals are a model of this theory. So are the hyperreals. It's a theorem of model theory (the theory of formal systems and their models) that all models must satisfy the same first-order theorems.

.999... = 1 is a first-order theorem of the theory of the real numbers. That means it's provably true (ie there is a logical proof from the axioms) in the reals and also in the hyperreals.

It can not possibly be otherwise because both the reals and the hyperreals are models of the same set of axioms. So any proof that works in the standard reals also works in the hyperreals.

Now I am NOT saying that infinitesimals aren't a useful intuition; or that ultimately the physicists may say some day 200 years from now that we can detect infinitesimals in a particle accelerator.

What I am saying is that .999... = 1 is a theorem in the hyperreals; and that this MUST be so.

Finally, what about second-order properties such as completeness? Well, the hyperreasls are not complete. They are shot full of holes. In my opinion that's a very poor model for the intuitive idea of a continuum. The hyperreals are a tremendous distraction in the .999... discussion but a very common one.
Without getting too involved in the semantics of hyperreals, they do allow us to demonstrate a difference between 0.99999... and 1. The distinction is between limit and st(limit) if I read it correctly

the limit of the sequence 0.9, 0.99, 0.999, ... is less than 1 in the hyperreals whereas the st(limit of the sequence 0.9, 0.99, 0.999, ... ) = 0. The st operation acts as a rounding up to the nearest real number operation similar to the ceiling function in conventional math.

The real analysis book I have says this about the decimal representation:
1) decimal representation represents an infinite series
2) an infinite series represents an infinite sequence
3) the least upper bound of the sequence represents the limit of the infinite sequence
4) the limit of the sequence represents the value or sum of the sequence
5) the "=" sign represents the value

Therefore 0.9999... = 1 according to that

http://imagizer.imageshack.us/a/img921/1559/gQBd4b.jpg "If lim S exists, we say that this series is convergent and call this limit the sum or the value of this series." That's about as deep as I saw them go with it, I'm assuming this is the axiom.

Pluto used to be defined as a planet, now it's not. If I were to have asked "Is Pluto really a planet?" back then I would not be inquiring about what it is currently defined as or what I was taught in school and pounded into my head all my life. I would be asking whether or not it is justified that Pluto is classified as a planet. The real question is whether or not 0.9999... should be classified as the limit of a sequence.

And found this definition of "<", ">" and "=" in the book

http://imagizer.imageshack.us/a/img924/8806/Ksl4AO.jpg They use the subtraction logic and if the difference is positive then the "a>b" or "a<b" is used according to that. But they did say that when defining "a>b" that a and b must both be elements of the real numbers. This would require a proof that 0.9999... is a real number and then determining whether the axioms of that proof are justified or not as well.

The argument for that is 0.9999... is 1 because there are no real numbers between 0.9999... and 1 therefore 0.9999... and 1 are the same because they do not differ by a real number. All real numbers differ by other real numbers where it's assumed that 0.9999... is a real number. But if 0.9999... is not a real number because 1 and 0.9999... do not differ by a real number but do differ by something then 0.9999... is not a real number could make sense too to explain why no real number exists between them rather than just assuming they're equal. Claiming 0.9999... = 1 because a real number cannot be produced between them commits an argument from ignorance fallacy https://www.logicallyfallacious.com/too ... -Ignorance
Argument from Ignorance

(also known as: appeal to ignorance, absence of evidence, argument from personal astonishment, argument from Incredulity)

Description: The assumption of a conclusion or fact based primarily on lack of evidence to the contrary. Usually best described by, “absence of evidence is not evidence of absence.”

Logical Form:
X is true because you cannot prove that X is false.
X is false because you cannot prove that X is true.

wtf
Posts: 805
Joined: Tue Sep 08, 2015 11:36 pm

Re: Is 0.9999... really the same as 1?

marsh8472 wrote: Without getting too involved in the semantics of hyperreals, they do allow us to demonstrate a difference between 0.99999... and 1. The distinction is between limit and st(limit) if I read it correctly

the limit of the sequence 0.9, 0.99, 0.999, ... is less than 1 in the hyperreals whereas the st(limit of the sequence 0.9, 0.99, 0.999, ... ) = 0. The st operation acts as a rounding up to the nearest real number operation similar to the ceiling function in conventional math.
No. .999... = 1 in the hyperreals. It's a theorem that follows from the first-order axioms of the real numbers; therefore it must be true in every model of those axioms, of which the hyperreals are one.

Note that in professor Katz's paper, he does show that the hyperreal .999...;...999 < 1. This is true. I'm using the Lightstone notation where Katz uses a variant involving underbraces. But the point is that in this notation for hyperreals, the thing to the left of the semicolon is not the same as the real number .999... And more importantly, the expression to the right of the semicolon terminates. It doesn't go on forever. Because in Lightstone notation, .999...;...999... IS exactly 1. (Note the trailing dots). But again, the thing to the left of the semicolon no longer has the same meaning it does in the reals.

Why are there dots directly to the right of the semicolon? Because there is no smallest infinite integer in the hyperreals. This alone should make you realize that naive intuitions about the hyperreals are wrong. The hyperreals are a highly technical construction in set theory, sort of like Cauchy sequences on steroids, and they are very strange and hard to understand. They are no panacea for the educational difficulties in teaching the real numbers in my opinion.

You say you don't want to get "too involved" in the semantics of the hyperreals, but if you're going to bring them up then you have to. So the question is how deeply DO you want to get involved in this subject, which as I say is a tremendous distraction from just trying to understand the reals?

I found an article by professor Katz here. http://u.cs.biu.ac.il/~katzmik/999.html [In the first sentence he refers to himself in the third person, which makes it look like it's someone else's article. But it's by prof Katz, it's on his website].

Let me make a couple of points about it. The article is a question/answer type format. From the article:
Professor Katz wrote: Question 1. Aren't there many standard proofs that 0.999...=1? Since we can't have that and also 0.999...≠1 at the same time, if mathematics is consistent, then isn't there necessarily a flaw in the proof given in the text "strict non-standard inequality"?

Answer. The standard proofs are of course correct, in the context of the standard real numbers.
As you see, professor Katz is perfectly well aware that .999... = 1 is true in the real numbers. He then goes on to point out that "students are confused." You will see if you read through that article that time and time again Katz says, "Students are confused," and "How can we explain this to highschoolers," and so forth.

Katz is grinding a pedagogical axe. He perfectly well knows that .999... = 1.

Now to say any more I would have to start diving deeper into the hyperreals, and there would be no end to it. I went back and reread Lightstone's paper on his semicolon notation, and he points out for example that 0;...999... is not a hyperreal number. It's not .1;...000... as you might expect. In the hyperreals, some (Lightstone) decimal expressions do not represent hyperreal numbers at all. This is related to the fact that the hyperreals fail to be complete.

See https://en.wikipedia.org/wiki/A._H._Lightstone and https://www.jstor.org/stable/2316619

That latter link is to Lightstone's short and somewhat brilliant paper describing his notation for hyperreals. If you are at all serious about hyperreals you might as well read it. It's a soft paywall, you register a free account and you can put the paper on your "shelf" as they call it, and read it there.

Now I'm completely aware that I'm talking over everyone's head, and that I am just barely familiar with this material myself, having dived into it last year due to that other .999... thread, and having forgotten most of what I learned back then.

But the real point is that this is not a productive exercise. You do not have any correct intuitions about the hyperreals. Nobody does. They're far stranger and more murky than the "Better Explained" level of misunderstanding and confusion.

And the bottom line is that .999... = 1 in the hyperreals by simple virtue of the fact that it's a theorem provable from the first-order axioms for the real numbers, which are satisfied by both the standard reals and the hyperreals.

If you want me to say more about all this I'll be happy to do so, but it's not going to add any clarity to the .999... discussion. The real bottom line is that Professor Katz is making a pedagogical point and he clearly admits that whenever you read his work closely instead of relying on Better Explained types of distortions.

Honestly, that's a lot of words for nothing. .999... = 1 in the reals and also in the hyperreals, if we give those symbols their standard meaning. And professor Katz knows that.

'Nuff o' that.
marsh8472 wrote: The real analysis book I have says this about the decimal representation:
1) decimal representation represents an infinite series
2) an infinite series represents an infinite sequence
3) the least upper bound of the sequence represents the limit of the infinite sequence
4) the limit of the sequence represents the value or sum of the sequence
5) the "=" sign represents the value

Therefore 0.9999... = 1 according to that

"If lim S exists, we say that this series is convergent and call this limit the sum or the value of this series." That's about as deep as I saw them go with it, I'm assuming this is the axiom.
It's a definition, not an axiom. And please, may I ask, haven't I said exactly the same thing several times? The sum of an infinite series is defined as the limit of the partial sums. They tell you that in calculus class but to be fair nobody learns anything in calculus class beyond pulling down the exponent and subtracting one. Calculus pedagogy is a disaster. I don't blame professor Katz for wanting it to be better.
marsh8472 wrote: Pluto used to be defined as a planet, now it's not. If I were to have asked "Is Pluto really a planet?" back then I would not be inquiring about what it is currently defined as or what I was taught in school and pounded into my head all my life. I would be asking whether or not it is justified that Pluto is classified as a planet. The real question is whether or not 0.9999... should be classified as the limit of a sequence.
Perfectly valid point, if one is a nihilist. I wish to demonstrate that things fall down if I'm standing on the surface of the earth. To that end I explain to you Newtonian gravity, and after you have mastered that I explain special and then general relativity, and the Higgs boson, and gravity waves and loop quantum gravity. And after I've done all that, you say to me: "Yes, but isn't it true that someday physicists might redefine down to be up?"

Well yeah they might. I haven't got an answer for that. And the International Astronomical Union should be condemned to purgatory for putting this lame idea into people's heads. The damage they did to the public's understanding of science is incalculable. If Pluto's a planet one day and not the next, how can we rely on science?

Damn good question. Take it up with the IAU.
marsh8472 wrote: And found this definition of "<", ">" and "=" in the book

They use the subtraction logic and if the difference is positive then the "a>b" or "a<b" is used according to that. But they did say that when defining "a>b" that a and b must both be elements of the real numbers. This would require a proof that 0.9999... is a real number and then determining whether the axioms of that proof are justified or not as well.

The argument for that is 0.9999... is 1 because there are no real numbers between 0.9999... and 1 therefore 0.9999... and 1 are the same because they do not differ by a real number. All real numbers differ by other real numbers where it's assumed that 0.9999... is a real number. But if 0.9999... is not a real number because 1 and 0.9999... do not differ by a real number but do differ by something then 0.9999... is not a real number could make sense too to explain why no real number exists between them rather than just assuming they're equal. Claiming 0.9999... = 1 because a real number cannot be produced between them commits an argument from ignorance fallacy
First, tell me this. Do you believe that the sequence .9, .99, .999, ... has the limit 1, based on the epsilon/N definition of a limit? If yes, then we're done, since that is exactly the sequence of partial sums of the series 9/10 + 9/100 + ...

If no, then let's talk about the limit of the sequence, since that's where the problem lies. If you don't think .999... represents a real number, you must at some level not believe that the limit of the sequence .9, .99 etc is 1, and that's the core issue we need to address.

marsh8472
Posts: 54
Joined: Sun Oct 19, 2014 3:06 pm

Re: Is 0.9999... really the same as 1?

wtf wrote:
marsh8472 wrote: Without getting too involved in the semantics of hyperreals, they do allow us to demonstrate a difference between 0.99999... and 1. The distinction is between limit and st(limit) if I read it correctly

the limit of the sequence 0.9, 0.99, 0.999, ... is less than 1 in the hyperreals whereas the st(limit of the sequence 0.9, 0.99, 0.999, ... ) = 0. The st operation acts as a rounding up to the nearest real number operation similar to the ceiling function in conventional math.
No. .999... = 1 in the hyperreals. It's a theorem that follows from the first-order axioms of the real numbers; therefore it must be true in every model of those axioms, of which the hyperreals are one.
When I said "they do allow us to demonstrate a difference between 0.9999... and 1", "0.9999..." refers to an open question about what "0.9999..." should be defined as, not how it's currently classified in standard reals.
wtf wrote: Note that in professor Katz's paper, he does show that the hyperreal .999...;...999 < 1. This is true. I'm using the Lightstone notation where Katz uses a variant involving underbraces. But the point is that in this notation for hyperreals, the thing to the left of the semicolon is not the same as the real number .999... And more importantly, the expression to the right of the semicolon terminates. It doesn't go on forever. Because in Lightstone notation, .999...;...999... IS exactly 1. (Note the trailing dots). But again, the thing to the left of the semicolon no longer has the same meaning it does in the reals.

Why are there dots directly to the right of the semicolon? Because there is no smallest infinite integer in the hyperreals. This alone should make you realize that naive intuitions about the hyperreals are wrong. The hyperreals are a highly technical construction in set theory, sort of like Cauchy sequences on steroids, and they are very strange and hard to understand. They are no panacea for the educational difficulties in teaching the real numbers in my opinion.
Maybe you didn't read this link https://www.quora.com/Is-0-999-dots-1-in-the-hyperreals You're using the phrase "the hyperreals" which implies there's a standard for them. Maybe there is, maybe there's not.
wtf wrote: You say you don't want to get "too involved" in the semantics of the hyperreals, but if you're going to bring them up then you have to. So the question is how deeply DO you want to get involved in this subject, which as I say is a tremendous distraction from just trying to understand the reals?
I don't really want to get involved with hyperreals. If you understood what I said about the difference between limit and st(limit) then they're useful, otherwise we don't need to talk about them. What I've been saying is if 1 and 0.9999... differ by "something" then it may be inappropriate to say 0.9999... = 1. Hyperreals are just one element in the set of "something" or the universal set. Hyperreals is something you bought up actually, in the your very first post on the thread.
wtf wrote: I found an article by professor Katz here. http://u.cs.biu.ac.il/~katzmik/999.html [In the first sentence he refers to himself in the third person, which makes it look like it's someone else's article. But it's by prof Katz, it's on his website].

Let me make a couple of points about it. The article is a question/answer type format. From the article:
Professor Katz wrote: Question 1. Aren't there many standard proofs that 0.999...=1? Since we can't have that and also 0.999...≠1 at the same time, if mathematics is consistent, then isn't there necessarily a flaw in the proof given in the text "strict non-standard inequality"?

Answer. The standard proofs are of course correct, in the context of the standard real numbers.
As you see, professor Katz is perfectly well aware that .999... = 1 is true in the real numbers. He then goes on to point out that "students are confused." You will see if you read through that article that time and time again Katz says, "Students are confused," and "How can we explain this to highschoolers," and so forth.

Katz is grinding a pedagogical axe. He perfectly well knows that .999... = 1.
I don't know Katz personally. Maybe he's being a troll. I just threw his proof out there since you rejected mine I thought something more formal looking would satisfy whatever it was that caused you to reject mine. He said the standard proofs are correct in the context of the standard real numbers. Santa Clause exists and lives at the north pole in the context of some storybooks. This does not mean Santa Clause actually exists and lives in the north pole. Saying he "knows" .999... = 1, is a claim that doesn't really say much. You would have to define what knowledge means when you say that if I were to evaluate that claim. There's an entire branch of philosophy on that topic. Claiming 0.9999... = 1 because Katz agrees with you, standard teachings of reals agrees with you, or even hyperreals agrees with you is committing a genetic fallacy in case you're trying to do that. https://www.logicallyfallacious.com/too ... ic-Fallacy
Description: Basing the truth claim of an argument on the origin of its claims or premises.

Logical Form:
The origin of the claim is presented.
Therefore, the claim is true/false.

Example #1:
Lisa was brainwashed as a child into thinking that people are generally good. Therefore, people are not generally good.
wtf wrote: Now to say any more I would have to start diving deeper into the hyperreals, and there would be no end to it. I went back and reread Lightstone's paper on his semicolon notation, and he points out for example that 0;...999... is not a hyperreal number. It's not .1;...000... as you might expect. In the hyperreals, some (Lightstone) decimal expressions do not represent hyperreal numbers at all. This is related to the fact that the hyperreals fail to be complete.

See https://en.wikipedia.org/wiki/A._H._Lightstone and https://www.jstor.org/stable/2316619

That latter link is to Lightstone's short and somewhat brilliant paper describing his notation for hyperreals. If you are at all serious about hyperreals you might as well read it. It's a soft paywall, you register a free account and you can put the paper on your "shelf" as they call it, and read it.

Now I'm completely (no pun) aware that I'm talking over everyone's head, and that I myself am just barely familiar with this material myself, having dived into it last year due to that other .999... thread, and having forgotten most of what I learned back then.

But the real point is that this is not a productive exercise. You do not have any correct intuitions about the hyperreals. Nobody does. They're far stranger and more murky than the "Better Explained" level of misunderstanding and confusion.
No need to dive deeper into that, as I said in the previous post. If the lim and st(lim) distinction was useful than fine, otherwise it's not useful.
wtf wrote: And the bottom line is that .999... = 1 in the hyperreals by simple virtue of the fact that it's a theorem provable from the first-order axioms for the real numbers, which are satisfied by both the standard reals and the hyperreals.
There's no such thing as "The hyperreals" according to that article I posted earlier in this post. I've already challenged whether 0.9999... is a real number. If 0.9999... is not a real number then it would not be a hyperreal either according to that logic.
wtf wrote: If you want me to say more about all this I'll be happy to do so, but it's not going to add any clarity to the .999... discussion. The real bottom line is that Professor Katz is making a pedagogical point and he clearly admits that whenever you read his work closely instead of relying on Better Explained types of distortions.

Honestly, that's a lot of words for nothing. .999... = 1 in the reals and also in the hyperreals, if we give those symbols their standard meaning. And even professor Katz knows that.
Standard meanings are determined by discussions like these. Saying "Honestly, that's a lot of words for nothing. .999... = 1 in the reals and also in the hyperreals, if we give those symbols their standard meaning." is like saying "Pluto is a planet" a few decades ago. If it's generally accepted that 0.9999... is a real number, should it be? Saying 0.9999... is considered a real number over and over again is not helping you either. This is another fallacy in case that's supposed to prove something
(also known as: argument from nagging, proof by assertion)

Description: Repeating an argument or a premise over and over again in place of better supporting evidence.
Logical Form:

X is true. X is true. X is true. X is true. X is true. X is true... etc.
wtf wrote:
marsh8472 wrote: The real analysis book I have says this about the decimal representation:
1) decimal representation represents an infinite series
2) an infinite series represents an infinite sequence
3) the least upper bound of the sequence represents the limit of the infinite sequence
4) the limit of the sequence represents the value or sum of the sequence
5) the "=" sign represents the value

Therefore 0.9999... = 1 according to that

"If lim S exists, we say that this series is convergent and call this limit the sum or the value of this series." That's about as deep as I saw them go with it, I'm assuming this is the axiom.
It's a definition, not an axiom. And please, may I ask, haven't I said exactly the same thing several times? The sum of an infinite series is defined as the limit of the partial sums. They tell you that in calculus class but to be fair nobody learns anything in calculus class beyond pulling down the exponent and subtracting one. Calculus pedagogy is a disaster. I don't blame professor Katz for wanting it to be better.
Definition of axiom...

http://www.dictionary.com/browse/axiom
Logic, Mathematics. a proposition that is assumed without proof for the sake of studying the consequences that follow from it.
It's a bit nitpicky. I thought of using a different word but for me to accept 0.9999... = 1 I would need to accept some definition or premise as an axiom and treat it as an axiom.
wtf wrote:
marsh8472 wrote: Pluto used to be defined as a planet, now it's not. If I were to have asked "Is Pluto really a planet?" back then I would not be inquiring about what it is currently defined as or what I was taught in school and pounded into my head all my life. I would be asking whether or not it is justified that Pluto is classified as a planet. The real question is whether or not 0.9999... should be classified as the limit of a sequence.
Perfectly valid point, if one is a nihilist. I wish to demonstrate that things fall down if I'm standing on the surface of the earth. To that end I explain to you Newtonian gravity, and after you have mastered that I explain special and then general relativity, and the Higgs boson, and gravity waves and loop quantum gravity. And after I've done all that, you say to me: "Yes, but isn't it true that someday physicists might redefine down to be up?"

Well yeah they might. I haven't got an answer for that. And the International Astronomical Union should be condemned to purgatory for putting this lame idea into people's heads. The damage they did to the public's understanding of science is incalculable. If Pluto's a planet one day and not the next, how can we rely on science?

Damn good question. Take it up with the IAC.
I think a more appropriate analogy is whether "string theory" or "atomic theory" are correct or whether the multi-verse theory is correct. Claiming 0.9999... is a real number or represents a limit of an infinite sequence. 0.9999... is funky as a representation of a real number and it is not necessary to keep it there if there's already a decimal representation for 1, we don't need 2 representations of the same number especially if it's going to be used to make an epistemic claim like 0.9999... = 1.
wtf wrote:
marsh8472 wrote: And found this definition of "<", ">" and "=" in the book

They use the subtraction logic and if the difference is positive then the "a>b" or "a<b" is used according to that. But they did say that when defining "a>b" that a and b must both be elements of the real numbers. This would require a proof that 0.9999... is a real number and then determining whether the axioms of that proof are justified or not as well.

The argument for that is 0.9999... is 1 because there are no real numbers between 0.9999... and 1 therefore 0.9999... and 1 are the same because they do not differ by a real number. All real numbers differ by other real numbers where it's assumed that 0.9999... is a real number. But if 0.9999... is not a real number because 1 and 0.9999... do not differ by a real number but do differ by something then 0.9999... is not a real number could make sense too to explain why no real number exists between them rather than just assuming they're equal. Claiming 0.9999... = 1 because a real number cannot be produced between them commits an argument from ignorance fallacy
First, tell me this. Do you believe that the sequence .9, .99, .999, ... has the limit 1, based on the epsilon/N definition of a limit? If yes, then we're done, since that is exactly the sequence of partial sums of the series 9/10 + 9/100 + ...
Asking me that question suffers the same problem as asking whether "0.9999... = 1". I showed you in the real analysis book that an infinite series is defined as an infinite sequence to represent it's value. You've already said before that an we do not add an infinite amount of numbers when computing the value of an infinite series. That's the problem with defining an infinite series as an infinite sequence. 0.9, 0.99, 0.999, appears like something that approaches 1. A limit is defined as something that determines a value of something as that something is approached. That similarity does not mean the computation of a limit should be considered the same thing as the value of an infinite series.

An example of a limit not being the same as a value is If we say f(x) = 5 for all x not equal to 2 and 10 for all x = 2, the limit of f(x) as x approaches 2 equals 5. But this does not prove f(2) = 5. In fact we know from the function that f(2) = 10. The problem is that they're using infinity in the summation and artificially defined how to handle it.
wtf wrote: If no, then let's talk about the limit of the sequence, since that's where the problem lies. If you don't think .999... represents a real number, you must at some level not believe that the limit of the sequence .9, .99 etc is 1, and that's the core issue we need to address.
I don't know if 0.9999... is a real number. If 0.9999... is not an element of the set 0.9, 0.99, 0.999, 0.9999 then I don't see how it can be proved that it is a real number. I think I already addressed the core issues. They come from these definitions in that real analysis book:

1) decimal representation represents an infinite series
2) an infinite series represents an infinite sequence
3) the least upper bound of the sequence represents the limit of the infinite sequence
4) the limit of the sequence represents the value or sum of the sequence
5) the "=" sign represents the value

Something that approaches 1 is not the same thing as something that equals one which I demonstrated with the f(x) example. It is pretty much the core issue as you say.

wtf
Posts: 805
Joined: Tue Sep 08, 2015 11:36 pm

Re: Is 0.9999... really the same as 1?

marsh8472 wrote:
I don't know if 0.9999... is a real number.
If you don't believe in the real numbers, even as a meaningless formal axiomatic system as described in your real analysis book, there's nothing to discuss.

I am sure your book contains a list of the axioms for the real numbers. Addition/subtraction, multiplication/division, distributivity of multiplication over addition, total order (trichotomy), and the completeness axiom.

From those principles we can prove as theorem (not an axiom or assumption, but a logical step-by-step proof from the axioms) that .999... = 1.

If you don't believe in the axioms to start with -- even as a purely formal system -- then why would you even care whether .999 = 1? If somebody doesn't care about chess, why would they want to replay and find fault with the immortal games of Capablanca?

Why do you bother to argue about .999...? It's clear at this point that you reject the axioms of the real numbers as a formal system. That's your right, but then why waste time arguing with their logical consequences?

If you accept the axioms, then .999... = 1 is an easy consequence. A machine could crank out the proof. It's all right there in your real analysis book. You are rejecting the game altogether. So why the passionate disagreement with a legal position in the game?
marsh8472 wrote: You're using the phrase "the hyperreals" which implies there's a standard for them. Maybe there is, maybe there's not.
I was trying to avoid talking about nonprincipal ultrafilters and the fact that the question of whether there are non-isomorphic models of the hyperreals relates, amazingly enough, to the Continuum hypothesis.

Here are a couple of Mathoverflow threads on the topic. These are serious professional mathematicians, some of them well-known in the math world, kicking around the question of whether there are multiple non-isomorphic copies of the hyperreals. This is why I say the hyperreals are a tremendous distraction. They are very complicated mathematical objects, far more so than the standard reals, and nothing at all like the simplified and distorted accounts on Wikipedia and Better Explained.

http://mathoverflow.net/questions/88292 ... lem-solved

http://mathoverflow.net/questions/13672 ... of-mathbbn

[Nobody (myself included) is expected to understand those links, MO is for professional mathematicians. My point is only to note that the uniqueness of the hyperreal field is a question that is considered interesting by contemporary mathematicians. It's way beyond anything we're discussing here.]

For a Wiki-lite intro to ultrafilters, see this. https://en.wikipedia.org/wiki/Ultrafilter

The existence of the hyperreals (using "the" loosely as noted) depends on a weak form of the Axiom of Choice. From a philosophical standpoint the hyperreals require a far stronger ontological commitment than do the standard reals.

http://math.stackexchange.com/questions ... -of-choice

marsh8472
Posts: 54
Joined: Sun Oct 19, 2014 3:06 pm

Re: Is 0.9999... really the same as 1?

wtf wrote:
marsh8472 wrote:
I don't know if 0.9999... is a real number.
If you don't believe in the real numbers, even as a meaningless formal axiomatic system as described in your real analysis book, there's nothing to discuss.
It's not a real numbers or nothing problem. Rules are subject to change. Decimals are a representation of real numbers, but how they represent real numbers can be changed.

This probably explains it best https://en.wikipedia.org/wiki/0.999...
Proofs of this equality have been formulated with varying degrees of mathematical rigor, taking into account preferred development of the real numbers, background assumptions, historical context, and target audience.
The reason I am skeptical about the rules is because you can lead them to a conclusions like 0.9999... = 1. Applying Reductio ad Absurdum

https://www.logicallyfallacious.com/too ... d_Absurdum

Description: A mode of argumentation or a form of argument in which a proposition is disproven by following its implications logically to an absurd conclusion. Arguments which use universals such as, “always”, “never”, “everyone”, “nobody”, etc., are prone to being reduced to absurd conclusions. The fallacy is in the argument that could be reduced to absurdity -- so in essence, reductio ad absurdum is a technique to expose the fallacy.

Logical Form:
Assume P is true.
From this assumption, deduce that Q is true.
Also deduce that Q is false.
Thus, P implies both Q and not Q (a contradiction, which is necessarily false).
Therefore, P itself must be false.
With a conclusion like 0.9999... = 1, it leads to the implication is that an infinite number of 9's after a decimal point implies that this value becomes 1. From a natural numbers or units digit point of view, that's just not true. That conclusion is counter-intuitive. This would mean that either the counter-intuitive conclusion is the correct conclusion anyway or one of the premises that lead to this conclusion is false. Decimal representation is part of the problem which I'm sure you've heard said before

Here's how they defined decimal representation
http://imageshack.com/a/img923/4900/XCjMKR.jpg

(thinking outloud here in case you've seen this before)

They do talk about the infinite 9 issue on that page. It's a consequence from creating the decimal number from the real number between two possible sub-intervals. Then generating a series which then is changed to decimal representation at the end based on the numerators of that series.

There's no problem here with a number like Pi, the steps for creating the decimal are deterministic
3 + 1/10 + 4/100 + 1/1000 + 5/10000 <= Pi <= 3 + 1/10 + 4/100 + 1/1000 + (5+1)/10000

But the procedure for converting terminating decimals creates ambiguity on one step
0 + 4/10 <= 1/2 <= 0 + (4+1)/10

When converting 1/2 to a decimal it allows the tenths digit to be 4 or 5 while still being able to capture what the value of 1/2 is by using that decimal notation.

If we pick 4 as the tenths digit for the next step we get this
0 + 4/10 + 9/100 <= 1/2 <= 0 + 4/10 + (9+1)/100 <--- still a true statement that 0.49 <= 1/2 <= 0.5

If we pick 5 as the tenths digit for the next step we get this
0 + 5/10 + 0/100 <= 1/2 <= 0 + 5/10 + (0+1)/100 <--- still a true statement that 0.5 <= 1/2 <= 0.6

The 0.9999... = 1.0000... = 1 problem would then arise because the number 1 can fall within either of these two sub-intervals: 1.0 <= 1 <= 1.1 and 0.9 <= 1 <= 1.0 allowing for the tenths digit of 1 to be either 0 or 9 using this conversion procedure. Given people don't prefer to say 0.99999... over 1 anyway, this procedure could be changed to make the ambiguous step a deterministic step and just exclude the infinite 9 decimals as representations of real numbers altogether while still being able to represent every real number as a decimal.

Then the "..." represents the pattern continues where people see it as an infinite number of digits with an infinitesimal precision which the archimedean property negates the possibility of immediately.

The conversion back from decimal to a value of a real number involves assuming an infinite series has the same value as upper bound which I mentioned my grievances about that on the previous post already. The least upper bound undoes the flaws of this real number to decimal number conversion by having the effect of rounding up even if they are not equal, making them appear equal anyway. That makes the support of the conclusion 0.9999... = 1 misleading.
wtf wrote: I am sure your book contains a list of the axioms for the real numbers. Addition/subtraction, multiplication/division, distributivity of multiplication over addition, total order (trichotomy), and the completeness axiom.

From those principles we can prove as theorem (not an axiom or assumption, but a logical step-by-step proof from the axioms) that .999... = 1.
Yeah I looked at all that already. The 0.999... can be shown to equal 1 but it looks more like proof by equivocation to me. We can see that any decimal in general is not a real number but representation of a real number. Having 0.9999... represent a real number looks more like a preference on their part rather than a necessity. 0.9999... can represent a number outside of the real number system too where 0.9999... is an infinitesimal amount less than 1. If 0.9999... is less than 1 by any amount, even by an infinitesimal amount, then this would lead one to conclude 0.9999... < 1.
wtf wrote: If you don't believe in the axioms to start with -- even as a purely formal system -- then why would you even care whether .999 = 1? If somebody doesn't care about chess, why would they want to replay and find fault with the immortal games of Capablanca?

Why do you bother to argue about .999...? It's clear at this point that you reject the axioms of the real numbers as a formal system. That's your right, but then why waste time arguing with their logical consequences?
You made a knowledge claim that 0.999... = 1. I'm seeing how you can know that and I can't.
wtf wrote: I say you reject the axioms because if you accept them, then .999... = 1 is an easy consequence. A machine could crank it out. It's all right there in your real analysis book. You are rejecting the game altogether. So why the passionate disagreement with a legal position in the game?
It's a language problem and philosophical problem too. Accepting the real number system and accepting 0.9999... is a real number is not required for this "game" called philosophy when determining whether 0.9999... is the same as 1. That objection would be more appropriate on a forum strictly about real analysis.

wtf
Posts: 805
Joined: Tue Sep 08, 2015 11:36 pm

Re: Is 0.9999... really the same as 1?

marsh8472 wrote:
It's not a real numbers or nothing problem.
Perhaps, but as a divide-and-conquer strategy it's vital to nail this down for the real numbers before going off to other interpretations. You are clear that you don't accept the real number interpretation.
marsh8472 wrote: Rules are subject to change. Decimals are a representation of real numbers, but how they represent real numbers can be changed.

Sophistry.
marsh8472 wrote: This probably explains it best https://en.wikipedia.org/wiki/0.999...
It explains it worse. It's marginally less worse than the Better Explained page.

I keep trying to talk math, and you keep posting popularized and highly misleading Wiki pages. I confess I'm pretty much done here. If I haven't yet ripped that page to shreds it's because I already wore out my keyboard on everything else. What's the point of arguing with bad Wiki pages? "So many bad Wiki pages, so little time."

marsh8472 wrote: The reason I am skeptical about the rules is because you can lead them to a conclusions like 0.9999... = 1. Applying Reductio ad Absurdum
If I've said anything at all of interest, that's to the good. If you remain unconverted, go in peace and sin no more. You can cos all you like.
marsh8472 wrote: That conclusion is counter-intuitive.
But math is full of counterintuitive things. That's why we have formal proof. That's exactly why we have formal proof. So as not to be led astray by intuition.

Surely the profound example of general relativity and quantum theory show that intuition is an unreliable guide.
marsh8472 wrote: This would mean that either the counter-intuitive conclusion is the correct conclusion anyway or one of the premises that lead to this conclusion is false. Decimal representation is part of the problem which I'm sure you've heard said before
I'm glad we can find a point of agreement. Decimal representation is not a fundamental attribute of the real numbers. It's secondary, in the sense that we first define what the real numbers are, and later prove that each real has one or two decimal representations. Decimal representation does have flaws.
marsh8472 wrote: (thinking outloud here in case you've seen this before)
Hey let's talk photography! (1) Lighting; (2) Focus.
marsh8472 wrote: There's no problem here with a number like Pi, the steps for creating the decimal are deterministic
3 + 1/10 + 4/100 + 1/1000 + 5/10000 <= Pi <= 3 + 1/10 + 4/100 + 1/1000 + (5+1)/10000
You believe in pi but not .999...? That's logically inconsistent. You can believe in both or neither. What makes you think the digits of pi converge to their least upper bound? What makes you think there's any point on the real number line at all that represents pi and to which the infinite series 3 + 1/10 + etc converges? The only reason anyone believes that is because of the completeness axiom. If we're in the rational numbers, which are much more ontologically reasonable than the reals, then the digits of pi don't converge to anything at all.
marsh8472 wrote: When converting 1/2 to a decimal it allows the tenths digit to be 4 or 5 while still being able to capture what the value of 1/2 is by using that decimal notation.
Nobody is unduly troubled by this but I've noted this difficulty as a point of agreement between us.
marsh8472 wrote: Then the "..." represents the pattern continues where people see it as an infinite number of digits with an infinitesimal precision which the archimedean property negates the possibility of immediately.
You're still talking about infinitesimals. There is no purpose to that other than obfuscation and confusion.
marsh8472 wrote: The conversion back from decimal to a value of a real number involves assuming an infinite series has the same value as upper bound which I mentioned my grievances about that on the previous post already.
I am not personally responsible for this state of affairs. If there's any one person to blame, I'd suggest Augustin-Louis_Cauchy. He gave us epsilonics. Some consider that one of humanity's great intellectual achievements. If you dissent, that's your right and truly it has nothing to do with me. Even if you converted me to your point of view, nothing would change in math because I don't get invited to the secret meetings.

Hell I don't even do a very good job of spreading the gospel, let alone deciding what's in it.

marsh8472 wrote: The least upper bound undoes the flaws of this real number to decimal number conversion by having the effect of rounding up even if they are not equal, making them appear equal anyway. That makes the support of the conclusion 0.9999... = 1 misleading.
I actually understand exactly what you're saying. And I agree that mathematical formalism does not precisely capture our ancient intuitions, nor the ultimate truth, if there be such a thing, of the world.

But it's the formalism we've got. When we do math, we accept the formalisms and if we're geniuses we create new and better formalisms. That's how we crawled out of caves and built all this. Reductionism, Aristotelian logic, Western values. All under attack in these postmodern days of ours. I quite understand where you're coming from.
marsh8472 wrote: Yeah I looked at all that already. The 0.999... can be shown to equal 1 but it looks more like proof by equivocation to me.
As it happens, and this is entirely an accident of history, I had a marvelous real analysis professor and had a profound intellectual experience the summer I took that class. I wish I could impart my sense of the beauty and intellectual achievement of this subject, which took humanity thousands of years to create.
marsh8472 wrote: We can see that any decimal in general is not a real number but representation of a real number.
I note this as another point of absolute agreement between us.

marsh8472 wrote: Having 0.9999... represent a real number looks more like a preference on their part rather than a necessity.
I agree that on some other planet they probably have the real numbers, but a very different notation for them. And on that planet Zork and Frisbee are having more or less the same discussion we are.
marsh8472 wrote: 0.9999... can represent a number outside of the real number system too where 0.9999... is an infinitesimal amount less than 1.
Statement without supporting evidence. The hyperreals don't help you and you have not presented any other model of the real numbers.
marsh8472 wrote: If 0.9999... is less than 1 by any amount, even by an infinitesimal amount, then this would lead one to conclude 0.9999... < 1.
If 2 + 2 = 5 then I am the pope. Principle of explosion. False implies true in logic. Your argument is valid but not sound. That is, your conclusion follows from the premise; but your premise is not true.

marsh8472 wrote: You made a knowledge claim that 0.999... = 1. I'm seeing how you can know that and I can't.
I had a kickass class in real analysis, followed by a very classic education in pure math where I had these concepts beaten into me by professors at some of the finest universities.

marsh8472 wrote: It's a language problem and philosophical problem too.
Where you and I diverge is that I fully agree that the mathematical solution is not by any means the philosophical or physical solution. You occasionally give lip service to that point of view, but most of the time you refuse to accept modern math in its entirety.

I see you as being a little disingenuous on that point. If we agreed on the standard math interpretation then we could have a great conversation about the philosophy and the physics. But you don't accept the math either. That leaves me no place to stand. Even Descartes realized he had to stand somewhere. I think therefore I am. But you will accept nothing. "Oh what if they change the definition of convergence?" Do you regard that as a serious way to discuss the nature of the real numbers?
marsh8472 wrote: Accepting the real number system and accepting 0.9999... is a real number is not required for this "game" called philosophy when determining whether 0.9999... is the same as 1.
But I've said that many times. And YOU say it, only to pull the rug out by then refusing to accept the math. In so doing we've had to leave the really interesting parts of this discussion untouched. The intuitionistic and constructivist real line. The modern structuralist revolution in math in terms of category theory, that's basically supplanted set theory as the modern foundation for much of advanced math. All the physical speculation about the discrete or continuous nature of spacetime. All that stuff and more.

So you SAY you want to talk about that, but you can't get yourself to grok what's in your real analysis text, and you won't accept what explanations I've done my best to provide.

marsh8472 wrote: That objection would be more appropriate on a forum strictly about real analysis.
But you'd object there too. We haven't been talking philosophy. We've been talking real analysis and you won't accept the basics of the subject even on their own terms.

If you would simply say, "Real analysis is bullshit but it's interesting bullshit," then we could talk about it like we'd marvel over a well-played game of chess. Or tennis for that matter. But you won't grant me even that. I say real analysis is interesting but in no way necessarily true. You won't even grant me that. You want to reject what's in your dark and fuzzy real analysis book. (Don't you have a lamp or something?)

That's how it seems to me.

marsh8472
Posts: 54
Joined: Sun Oct 19, 2014 3:06 pm

Re: Is 0.9999... really the same as 1?

wtf wrote:
marsh8472 wrote:
It's not a real numbers or nothing problem.
Perhaps, but as a divide-and-conquer strategy it's vital to nail this down for the real numbers before going off to other interpretations. You are clear that you don't accept the real number interpretation.
That's fine, if you have more points to make about real numbers you can make them. I accept conventional math enough for most things but believing 0.9999... = 1 is too theoretical and pointy of a question to make assumptions on for me. Conflicting answers are at odds with the fundamentals of real numbers. I'd like to see a line by line indisputable proof of why 0.9999... = 1, if you want to make one of those up. I thought what I wrote earlier was sufficient enough though.
wtf wrote:
marsh8472 wrote: Rules are subject to change. Decimals are a representation of real numbers, but how they represent real numbers can be changed.

Sophistry.
I think I looked at it under the hood pretty closely already. What's deeper than what I've already looked at? It feels like a con job that the justification for 0.9999... = 1 is rather baseless from all the attempts to prove it that I've seen.
wtf wrote:
marsh8472 wrote: This probably explains it best https://en.wikipedia.org/wiki/0.999...
It explains it worse. It's marginally less worse than the Better Explained page.

I keep trying to talk math, and you keep posting popularized and highly misleading Wiki pages. I confess I'm pretty much done here. If I haven't yet ripped that page to shreds it's because I already wore out my keyboard on everything else. What's the point of arguing with bad Wiki pages? "So many bad Wiki pages, so little time."
I like the quote from the wiki page though, no need to rebut it.
wtf wrote:
marsh8472 wrote: The reason I am skeptical about the rules is because you can lead them to a conclusions like 0.9999... = 1. Applying Reductio ad Absurdum
If I've said anything at all of interest, that's to the good. If you remain unconverted, go in peace and sin no more. You can cos all you like.
I don't reject the idea. I just do not have maximal certainty that 0.9999... = 1. It's reasonable that it is, but also reasonable that it is not. I wouldn't even know where to begin to compute the likelihood that it is true or false. We are talking about whether a number that's infinitesimally close to 1 is less than 1. You apparently think they are equal and I'm undecided. It's not that big of a deal.
wtf wrote:
marsh8472 wrote: That conclusion is counter-intuitive.
But math is full of counterintuitive things. That's why we have formal proof. That's exactly why we have formal proof. So as not to be led astray by intuition.

Surely the profound example of general relativity and quantum theory show that intuition is an unreliable guide.
I don't use intuition as a guide but it's useful when you don't know all the facts. Something that's counter-intuitive provides a reason to doubt it. We probably wouldn't exist without that instinct. If I doubt it, then I don't fully accept it unless I see some strong proof. Like the birthday problem might seem counter-intuitive but the proof is pretty solid. I'm not feeling that with this real analysis logic.
wtf wrote:
marsh8472 wrote: This would mean that either the counter-intuitive conclusion is the correct conclusion anyway or one of the premises that lead to this conclusion is false. Decimal representation is part of the problem which I'm sure you've heard said before
I'm glad we can find a point of agreement. Decimal representation is not a fundamental attribute of the real numbers. It's secondary, in the sense that we first define what the real numbers are, and later prove that each real has one or two decimal representations. Decimal representation does have flaws.
It's almost like even though the reals do not have holes in them as an axiom, the decimal notation shows holes which are covered up by saying 0.9999... = 1.0000... . When dividing a number into 10 equal parts, then those parts into 10 equal parts, etc.. the problems arise when trying to place decimals into the right bucket when they are divided by lines that are infinitesimal in length or no length at all. If the repeated 9's were excluded from the decimal system then there would be no ambiguity and everyone would just know 0.9999... is either undefined like infinity or a non-real infinitesimal number where 0.9999... < 1.
wtf wrote:
marsh8472 wrote: (thinking outloud here in case you've seen this before)
Hey let's talk photography! (1) Lighting; (2) Focus.
marsh8472 wrote: There's no problem here with a number like Pi, the steps for creating the decimal are deterministic
3 + 1/10 + 4/100 + 1/1000 + 5/10000 <= Pi <= 3 + 1/10 + 4/100 + 1/1000 + (5+1)/10000
You believe in pi but not .999...? That's logically inconsistent. You can believe in both or neither. What makes you think the digits of pi converge to their least upper bound? What makes you think there's any point on the real number line at all that represents pi and to which the infinite series 3 + 1/10 + etc converges? The only reason anyone believes that is because of the completeness axiom. If we're in the rational numbers, which are much more ontologically reasonable than the reals, then the digits of pi don't converge to anything at all.
I meant that there are no issues as far as multiple decimal notations in the decimal generating algorithm. It's only with the terminating decimals like 0.5 can also be 0.4999, or 0.05 can also be 0.04999... That's a good question for you too about points. Do points have an infinitesimal length or 0 length? Typically 2 points are required for a length to be computable, then the length would be whatever the absolute value of their difference is.

I was thinking about writing a program that would break a decimal series apart into alternative decimals over and over just for the heck of it.
Like this
0.999... = 0.9 + 0.09 + 0.009 + ...
0.999... = (0.8999..) + (0.08999...) + (0.008999...) + ...
0.999... = (0.8 + 0.09 + 0.009 + 0.0009 + ... ) + (0.08 + 0.009 + 0.0009 + 0.00009) + (0.008 + 0.0009 + 0.00009 + 0.000009) + ...
0.999... = ((0.7999... + 0.08999... + 0.008999... + 0.0008999... + ...) + (0.07999... + 0.008999... + 0.0008999... + 0.00008999... + ...) + (0.007999... + 0.0008999... + 0.00008999... + 0.000008999... + ...))
ect...
It would get ugly pretty quick
wtf wrote:
marsh8472 wrote: When converting 1/2 to a decimal it allows the tenths digit to be 4 or 5 while still being able to capture what the value of 1/2 is by using that decimal notation.
Nobody is unduly troubled by this but I've noted this difficulty as a point of agreement between us.
ok good we agree moving on from that
wtf wrote:
marsh8472 wrote: Then the "..." represents the pattern continues where people see it as an infinite number of digits with an infinitesimal precision which the archimedean property negates the possibility of immediately.
You're still talking about infinitesimals. There is no purpose to that other than obfuscation and confusion.
Here's the definition of infinitesimal though
an indefinitely small quantity; a value approaching zero.
That word is describing the difference between 0.9999... and 1 or this if you'd like lim x->infinity of 1 - (1 - 10^-x) = lim x->infinity of 10^-x <--- it's a value that approaches 0. A value of difference between two numbers implies that one is greater than or less than the other.
wtf wrote:
marsh8472 wrote: The conversion back from decimal to a value of a real number involves assuming an infinite series has the same value as upper bound which I mentioned my grievances about that on the previous post already.
I am not personally responsible for this state of affairs. If there's any one person to blame, I'd suggest Augustin-Louis_Cauchy. He gave us epsilonics. Some consider that one of humanity's great intellectual achievements. If you dissent, that's your right and truly it has nothing to do with me. Even if you converted me to your point of view, nothing would change in math because I don't get invited to the secret meetings.

Hell I don't even do a very good job of spreading the gospel, let alone deciding what's in it.
I don't have a point of view except I don't accept either completely. If you were arguing 0.9999... < 1, I may have argued 0.9999... = 1 instead. I've done that before. I don't see how either can be rejected, both sound reasonable.
wtf wrote:
marsh8472 wrote: The least upper bound undoes the flaws of this real number to decimal number conversion by having the effect of rounding up even if they are not equal, making them appear equal anyway. That makes the support of the conclusion 0.9999... = 1 misleading.
I actually understand exactly what you're saying. And I agree that mathematical formalism does not precisely capture our ancient intuitions, nor the ultimate truth, if there be such a thing, of the world.

But it's the formalism we've got. When we do math, we accept the formalisms and if we're geniuses we create new and better formalisms. That's how we crawled out of caves and built all this. Reductionism, Aristotelian logic, Western values. All under attack in these postmodern days of ours. I quite understand where you're coming from.
Okay no problem here we agree
wtf wrote:
marsh8472 wrote: Yeah I looked at all that already. The 0.999... can be shown to equal 1 but it looks more like proof by equivocation to me.
As it happens, and this is entirely an accident of history, I had a marvelous real analysis professor and had a profound intellectual experience the summer I took that class. I wish I could impart my sense of the beauty and intellectual achievement of this subject, which took humanity thousands of years to create.
It was one of my more difficult proving math courses. Other proving courses I remember taking are calc-based statistic course, calc-based physics, numerical analysis, linear numerical analysis, abstract math, and number theory.
wtf wrote:
marsh8472 wrote: We can see that any decimal in general is not a real number but representation of a real number.
I note this as another point of absolute agreement between us.
marsh8472 wrote: Having 0.9999... represent a real number looks more like a preference on their part rather than a necessity.
I agree that on some other planet they probably have the real numbers, but a very different notation for them. And on that planet Zork and Frisbee are having more or less the same discussion we are.
ok no problem there
wtf wrote:
marsh8472 wrote: 0.9999... can represent a number outside of the real number system too where 0.9999... is an infinitesimal amount less than 1.
Statement without supporting evidence. The hyperreals don't help you and you have not presented any other model of the real numbers.
I kind of made up a model in that proof I made up that you rejected. I just tweaked the definition of the comparison operators to allow the 0.999... numbers to be less than their alternative decimal notations. But aside from that, I'm talking about a system that can exist that we don't know about. It's epistemically possible which means it's possible for all I know due to my lack of omniscience.
wtf wrote:
marsh8472 wrote:If 0.9999... is less than 1 by any amount, even by an infinitesimal amount, then this would lead one to conclude 0.9999... < 1.
If 2 + 2 = 5 then I am the pope. Principle of explosion. False implies true in logic. Your argument is valid but not sound. That is, your conclusion follows from the premise; but your premise is not true.
Except my conclusion is more like a tautology really actually isn't it? In the event 0.9999... = 1 like you believe, people still exist that are led to believe 0.9999... < 1 regardless of the reality of whether it's true or not. It's like asking whether the earth resolves the sun or vice versa or whether particles move randomly or deterministicly. A modal of either possibility can explain what we observe.
wtf wrote:
marsh8472 wrote: You made a knowledge claim that 0.999... = 1. I'm seeing how you can know that and I can't.
I had a kickass class in real analysis, followed by a very classic education in pure math where I had these concepts beaten into me by professors at some of the finest universities.
That sounds like indoctrination. I prefer to be open minded then I'm not as blind-sided when I'm wrong.
wtf wrote:
marsh8472 wrote: It's a language problem and philosophical problem too.
Where you and I diverge is that I fully agree that the mathematical solution is not by any means the philosophical or physical solution. You occasionally give lip service to that point of view, but most of the time you refuse to accept modern math in its entirety.

I see you as being a little disingenuous on that point. If we agreed on the standard math interpretation then we could have a great conversation about the philosophy and the physics. But you don't accept the math either. That leaves me no place to stand. Even Descartes realized he had to stand somewhere. I think therefore I am. But you will accept nothing. "Oh what if they change the definition of convergence?" Do you regard that as a serious way to discuss the nature of the real numbers?
0.9999... = 1 is too pointy of a question and hits at the heart of the fundamentals of reals where you have no choice but to critically analyze them. We agree that 0.9999... is a representation of a real number but not a real number. In that sense because real numbers are real numbers and decimal numbers are representations of real numbers, the law of identity would say this proves that 0.9999... is not equal to 1 if we include the definition of "=" to distinguish between the set of representation of real numbers and real numbers rather than just "the value". Since logic is more fundamental than math, we can say 0.9999... is not the same as 1 right there based on that.

https://en.wikipedia.org/wiki/Law_of_identity
In logic, the law of identity is the first of the three classical laws of thought. It states that each thing is identical with itself. By this it is meant that each thing (be it a universal or a particular) is composed of its own unique set of characteristic qualities or features, which the ancient Greeks called its essence.

In its symbolic representation, "a = a" or "For all x: x = x".

In logical discourse, violations of the Law of Identity (LOI) result in the informal logical fallacy known as equivocation. That is to say, we cannot use the same term in the same discourse while having it signify different senses or meanings – even though the different meanings are conventionally prescribed to that term. In everyday language, violations of the LOI introduce ambiguity into the discourse, making it difficult to form an interpretation at the desired level of specificity. The LOI also allows for substitution.
I'd agree with them on the law of identity wiki page that there's equivocation going on with the equal sign.
wtf wrote:
marsh8472 wrote: Accepting the real number system and accepting 0.9999... is a real number is not required for this "game" called philosophy when determining whether 0.9999... is the same as 1.
But I've said that many times. And YOU say it, only to pull the rug out by then refusing to accept the math. In so doing we've had to leave the really interesting parts of this discussion untouched. The intuitionistic and constructivist real line. The modern structuralist revolution in math in terms of category theory, that's basically supplanted set theory as the modern foundation for much of advanced math. All the physical speculation about the discrete or continuous nature of spacetime. All that stuff and more.

So you SAY you want to talk about that, but you can't get yourself to grok what's in your real analysis text, and you won't accept what explanations I've done my best to provide.
We can talk about abstract math next too. Rather than these rigid rules in the reals. I went there a little bit when I invented that compare function that you rejected.
wtf wrote:
marsh8472 wrote: That objection would be more appropriate on a forum strictly about real analysis.
But you'd object there too. We haven't been talking philosophy. We've been talking real analysis and you won't accept the basics of the subject even on their own terms.

If you would simply say, "Real analysis is bullshit but it's interesting bullshit," then we could talk about it like we'd marvel over a well-played game of chess. Or tennis for that matter. But you won't grant me even that. I say real analysis is interesting but in no way necessarily true. You won't even grant me that. You want to reject what's in your dark and fuzzy real analysis book. (Don't you have a lamp or something?)

That's how it seems to me.
Not accepting is not the same thing as rejecting. I reject proofs as actually proving things. They only maybe prove something if the assumptions and definitions are true and there's no mistakes in it. It's useful except when asking questions like these then it's either you accept their assumptions and definitions or you don't. Then it's all just theory. Knowing whether 0.9999... is the same as 1 is not useful in every day life. If you say it's not necessarily true then why don't you agree that you don't know whether 0.9999... is the same as 1? I took the real analysis course and have been looking at the book but nothing's jumping out at me as a convincing proof for 0.9999... = 1. We actually covered the book and went beyond the book and proved things with balls in addition to real numbers.

These 0.9999... = 1 proofs are useful for people who accept the current math education as gospel. The first time I questioned 0.9999... = 1 was in high school actually. I just noticed that 1/3 and 2/3 added produced infinite 9's and asked about it after class. I was told they were the same thing. I'm not in high school anymore and don't just accept things that I'm told.

In general I don't believe in absolute certainty just maximum certainty. I went as far as being unable to completely justify basic arithmetic before too. I realized that arithmetic like adding is built off of counting, counting is based off the idea of recognizing different values are actually different values like 1 and 2 are different so we know what more and less are. This is built off of the law of non-contradiction which I've come to conclude that even that cannot be completely proven or disproven because it's a necessary tool in order to prove or disprove anything. But it's the most obvious thing that's true so I say I have maximum certainty that the law of non-contradiction is true. But we went on and on about whether it was true or not for quite a while in this "anything is possible" thread http://www.onlinephilosophyclub.com/for ... dd4ebe8fa1

wtf
Posts: 805
Joined: Tue Sep 08, 2015 11:36 pm

Re: Is 0.9999... really the same as 1?

marsh8472 wrote: That's fine, if you have more points to make about real numbers you can make them.
I have nothing to add. Everything I have to say, I've said. This post is idle chatter. We're at the end.

marsh8472 wrote: I accept conventional math enough for most things but believing 0.9999... = 1 is too theoretical and pointy of a question to make assumptions on for me.
So be it.

marsh8472 wrote: Conflicting answers are at odds with the fundamentals of real numbers. I'd like to see a line by line indisputable proof of why 0.9999... = 1, if you want to make one of those up.
I've done this several times. If you'll tell me what principles or axioms you accept, I can do the proof. But if you say things like, "Oh what if tomorrow they change the definition of convergence?" then there is no conversation to be had.
marsh8472 wrote: I thought what I wrote earlier was sufficient enough though.
Are you saying you proved to your own satisfaction that .999... = 1? I don't know what your remark refers to.
marsh8472 wrote: Rules are subject to change. Decimals are a representation of real numbers, but how they represent real numbers can be changed.

Like I say. This is the position of either a nihilist or a sophist. Nothing can be known because the rules can be changed.
marsh8472 wrote: I think I looked at it under the hood pretty closely already. What's deeper than what I've already looked at? It feels like a con job that the justification for 0.9999... = 1 is rather baseless from all the attempts to prove it that I've seen.
Fine with me. You own a real analysis textbook and it contains the accumuulated intellectual achievement of the 200 years of thought post-Newton. I can't say anything new that I haven't said many times already.
marsh8472 wrote: I don't reject the idea. I just do not have maximal certainty that 0.9999... = 1.
What do you want me to say? If this were 20 posts ago I'd say something like, "Well, we can prove it from first principles." But having done so, and having outlined the proof several times, I'd be foolish to say the same things again.
marsh8472 wrote: It's reasonable that it is, but also reasonable that it is not. I wouldn't even know where to begin to compute the likelihood that it is true or false.
Your real analysis text seems like a good start. Except that it's so dark and fuzzy.
marsh8472 wrote: We are talking about whether a number that's infinitesimally close to 1 is less than 1. You apparently think they are equal and I'm undecided. It's not that big of a deal.
If you still think there are infinitesimals in the real numbers, why would I say anything at all at this point?
marsh8472 wrote: I don't use intuition as a guide but it's useful when you don't know all the facts. Something that's counter-intuitive provides a reason to doubt it. We probably wouldn't exist without that instinct. If I doubt it, then I don't fully accept it unless I see some strong proof. Like the birthday problem might seem counter-intuitive but the proof is pretty solid. I'm not feeling that with this real analysis logic.
Fine with me. There's a diminishing return on arguing with .999... doubters and I'm way past that point here.

marsh8472 wrote: It's almost like even though the reals do not have holes in them as an axiom, the decimal notation shows holes which are covered up by saying 0.9999... = 1.0000... . When dividing a number into 10 equal parts, then those parts into 10 equal parts, etc.. the problems arise when trying to place decimals into the right bucket when they are divided by lines that are infinitesimal in length
If I say to you: "But there are no infinitesimals in the real numbers," I would be an idiot. I prefer to say nothing. It's certainly frustrating that after all this, you didn't read a word I wrote about infinitesimals.

marsh8472 wrote: or no length at all. If the repeated 9's were excluded from the decimal system then there would be no ambiguity and everyone would just know 0.9999... is either undefined like infinity or a non-real infinitesimal number where 0.9999... < 1.
Fine. You can't bait me anymore. Surely it must be clear to you that you are simply repeating the same points over and over now.
marsh8472 wrote:
There's no problem here with a number like Pi, the steps for creating the decimal are deterministic
3 + 1/10 + 4/100 + 1/1000 + 5/10000 <= Pi <= 3 + 1/10 + 4/100 + 1/1000 + (5+1)/10000
Whatever. I already made a perfectly sensible response to that point, which of course you completely ignored, just to repeat the same point.
marsh8472 wrote: I meant that there are no issues as far as multiple decimal notations in the decimal generating algorithm. It's only with the terminating decimals like 0.5 can also be 0.4999, or 0.05 can also be 0.04999... That's a good question for you too about points. Do points have an infinitesimal length or 0 length?
Having interacted with me for a week, and having read what I've written, how do you think I would answer that last question?

marsh8472 wrote: Typically 2 points are required for a length to be computable, then the length would be whatever the absolute value of their difference is.
Computabiility is a completely different subject. As a CS major I'm sure you know that.
marsh8472 wrote: I was thinking about writing a program that would break a decimal series apart into alternative decimals over and over just for the heck of it.
Like this
0.999... = 0.9 + 0.09 + 0.009 + ...
0.999... = (0.8999..) + (0.08999...) + (0.008999...) + ...
0.999... = (0.8 + 0.09 + 0.009 + 0.0009 + ... ) + (0.08 + 0.009 + 0.0009 + 0.00009) + (0.008 + 0.0009 + 0.00009 + 0.000009) + ...
0.999... = ((0.7999... + 0.08999... + 0.008999... + 0.0008999... + ...) + (0.07999... + 0.008999... + 0.0008999... + 0.00008999... + ...) + (0.007999... + 0.0008999... + 0.00008999... + 0.000008999... + ...))
ect...
It would get ugly pretty quick
If it makes you happy.

marsh8472 wrote: Then the "..." represents the pattern continues where people see it as an infinite number of digits with an infinitesimal precision which the archimedean property negates the possibility of immediately.
Surely you can step back, pretend you are not the person who wrote those words, and see that it's incoherent.
marsh8472 wrote: Here's the definition of infinitesimal though
an indefinitely small quantity; a value approaching zero.
That is not the definition of an infinitesimal. An infinitesimal is a quantity x such that for every positive integer n, 0 < x < 1/n. That is the mathematical definition. The infinitesimals in the hyperreals satisfy that definition. It's perfectly simple to show that no real number does.
marsh8472 wrote: The conversion back from decimal to a value of a real number involves assuming an infinite series has the same value as upper bound which I mentioned my grievances about that on the previous post already.
Grievance so noted.
marsh8472 wrote: I don't have a point of view except I don't accept either completely. If you were arguing 0.9999... < 1, I may have argued 0.9999... = 1 instead.
Oh no I don't believe that. Your initial post innocently asked, "What do people think?" And of course your actual agenda is to dig in and argue your point of view. This is a common pattern.
marsh8472 wrote: The least upper bound undoes the flaws of this real number to decimal number conversion by having the effect of rounding up even if they are not equal, making them appear equal anyway. That makes the support of the conclusion 0.9999... = 1 misleading.
Have you got anything new to say? Isn't it clear that we're done?
marsh8472 wrote: Yeah I looked at all that already. The 0.999... can be shown to equal 1 but it looks more like proof by equivocation to me.
What happens if I don't take the bait? I happen to live in a rural area and every once in a while a cute little field mouse gets in the house. I set traps. I know what happens to mice who take the bait.

marsh8472 wrote: It was one of my more difficult proving math courses. Other proving courses I remember taking are calc-based statistic course, calc-based physics, numerical analysis, linear numerical analysis, abstract math, and number theory.
Did you argue with your number theory professor that (1) We have no idea what a number is; and (2) We have no evidence that there can be infinitely many of them? And then refuse to learn the formalisms? When the prof said that we have a binary relation called "divides," according to which 3 divides 9 and 5 divides 10 but 3 doesn't divide 7, did you say, "Oh but what if they change the definition? We can't really be certain that 3 divides 9, can we?"

Did you so argue in number theory class? Or is this obfuscatory stance reserved for me?
marsh8472 wrote: I kind of made up a model in that proof I made up that you rejected. I just tweaked the definition of the comparison operators to allow the 0.999... numbers to be less than their alternative decimal notations. But aside from that, I'm talking about a system that can exist that we don't know about.
As I say, that's nihilsm or sophistry. Professor so-and-so proves that the electron has such and such energy. And you say, "Well, there could be other models of the electron where that's not true." Surely you can't expect me to take that point of view seriously. We'd still be living in caves. We do the best we can. Knowlege is always changing, as are historically contingent attitudes and beliefs.
marsh8472 wrote: It's epistemically possible which means it's possible for all I know due to my lack of omniscience.
Sophistry or nihilsm. You reject scientific progress because "anything is possible." After all, the Flying Spaghetti Monster might have created us five minutes ago, along with our memories of the past. Woo hoo, you have solved everything.
marsh8472 wrote: Except my conclusion is more like a tautology really actually isn't it? In the event 0.9999... = 1 like you believe, people still exist that are led to believe 0.9999... < 1 regardless of the reality of whether it's true or not. It's like asking whether the earth resolves the sun or vice versa or whether particles move randomly or deterministicly. A modal of either possibility can explain what we observe.
Sure. But you have presented no alternative model. The hyperreals don't help you. Your only alternative is that "anything is possible" and "maybe they'll change the definition of convergence." That is not intellectually satisfying as a model. The contents of your real analysis book IS an intellectual model.

After we're done here, why don't you spend some time actually putting together a model. Start with, say, 17th century calculus, and see how you would try to formalize a logically coherent theory of limits. Nobody will take you seriously if your theory of limit is, "Maybe they'll change the definition of convergence." You can't expect me to continue to engage on a point like that.
marsh8472 wrote: That sounds like indoctrination. I prefer to be open minded then I'm not as blind-sided when I'm wrong.
Taking an obviously tongue-in-cheek remark as serious so you can score a debating point isn't very impressive. It marks you as a humorless scold with an agenda to push at all costs.

marsh8472 wrote: 0.9999... = 1 is too pointy of a question and hits at the heart of the fundamentals of reals where you have no choice but to critically analyze them. We agree that 0.9999... is a representation of a real number but not a real number.
Yes. And 3 is a representation of a positive integer but is not a positive integer. What of it? Until we develop telepathy we're stuck with symbols. Words aren't always very good representations of thoughts, and even thoughts are only mere representations or shadows of the things they represent. Is that really the extent of your mathematical ontology?
marsh8472 wrote: In that sense because real numbers are real numbers and decimal numbers are representations of real numbers, the law of identity would say this proves that 0.9999... is not equal to 1 if we include the definition of "=" to distinguish between the set of representation of real numbers and real numbers rather than just "the value". Since logic is more fundamental than math, we can say 0.9999... is not the same as 1 right there based on that.
If I say 2 + 2 = 4, aren't those two different representations of the same abstract concept? So that by your logic, 2 + 2 = 4 is not true. Did I understand your point correctly?
Wiki surfing is fun. But if that's all you've got, you haven't got anything.
marsh8472 wrote: In its symbolic representation, "a = a" or "For all x: x = x".

In logical discourse, violations of the Law of Identity (LOI) result in the informal logical fallacy known as equivocation. That is to say, we cannot use the same term in the same discourse while having it signify different senses or meanings – even though the different meanings are conventionally prescribed to that term. In everyday language, violations of the LOI introduce ambiguity into the discourse, making it difficult to form an interpretation at the desired level of specificity. The LOI also allows for substitution.
Can I ask you a question? Why do you think copy/pasting completely irrelevant quotes from Wikipedia constitutes an actual argument?
marsh8472 wrote: We can talk about abstract math next too.
We have been talking about abstract math. You don't seem to accept it.

marsh8472 wrote: Rather than these rigid rules in the reals. I went there a little bit when I invented that compare function that you rejected.
Now that is a flat out lie. Go back to that post. You presented your compare function. I responded that "< becomes <= in the limit."

And now you say I "rejected" your compare function, as if you either

* Didn't read what I wrote; or

* Didn't understand what I wrote; or

* Pretended not to have read or understood it, so you could lie about it a few posts later.

Disingenuous. Does not reflect well on you. I wonder -- I genuinely and truly wonder -- why you never engage with what I say? When I said that < becomes <= in the limit, why didn't you respond with, "How do we know that," or "I object because ..." But you don't do that. Instead you ignore what I said and later pretend I said something else.

marsh8472 wrote:I reject proofs as actually proving things.
Nihilism and sophistry. Look at what you just wrote.

marsh8472 wrote: They only maybe prove something if the assumptions and definitions are true and there's no mistakes in it.
Well duh, yeah. So you want to live in a cave because nothing is absolutely certain? Why are you making this childish argument?

marsh8472 wrote: It's useful except when asking questions like these then it's either you accept their assumptions and definitions or you don't. Then it's all just theory.
Yes but you won't even accept the logical argument from premises. That's a deeper problem than just saying that soundness depends on the truth of the premises.

marsh8472 wrote: Knowing whether 0.9999... is the same as 1 is not useful in every day life.
Oh what a devastating argument. I'm humbled.
marsh8472 wrote: If you say it's not necessarily true then why don't you agree that you don't know whether 0.9999... is the same as 1?
I do know it. I know it with more certainty that I know the sun will rise tomorrow.

marsh8472 wrote: I took the real analysis course and have been looking at the book but nothing's jumping out at me as a convincing proof for 0.9999... = 1.
I will not repeat the same argument again. If you can't follow the proof, whose fault is that? You know the definition of a convergent sequence in terms of epsilon and N. You know the definition of a convergent series as the limit of the sequence of partial sums. That's the proof. If you don't get it you don't get it. I can do no more because you say you don't believe it but you won't ask any specific questions about it.

You won't engage on the math, you just retreat into "Oh what if they change the definition," and "But nothing is certain." That's nihilism or sophistry, depending on whether you are sincere or trolling, respectively. If you ever engaged on the math, we'd make some progress on the math.

marsh8472 wrote:
These 0.9999... = 1 proofs are useful for people who accept the current math education as gospel.
Oh no, I am a huge critic of contemporary math education. We have not been talking about math education at all. The state of which, since you asked, is dismal.

marsh8472 wrote: The first time I questioned 0.9999... = 1 was in high school actually. I just noticed that 1/3 and 2/3 added produced infinite 9's and asked about it after class. I was told they were the same thing. I'm not in high school anymore and don't just accept things that I'm told.
You argue like you're in elementary school. "We can't know anything so nothing we know is true, nyah nyah nyah."
marsh8472 wrote: In general I don't believe in absolute certainty just maximum certainty. I went as far as being unable to completely justify basic arithmetic before too.
Start with a book on set theory then. Or homotopy type theory, or category theory. Plenty of good starting points these days, you can literally choose your own foundations and end up at the same place as everyone else.
marsh8472 wrote: I realized that arithmetic like adding is built off of counting, counting is based off the idea of recognizing different values are actually different values like 1 and 2 are different so we know what more and less are. This is built off of the law of non-contradiction which I've come to conclude that even that cannot be completely proven or disproven because it's a necessary tool in order to prove or disprove anything. But it's the most obvious thing that's true so I say I have maximum certainty that the law of non-contradiction is true. But we went on and on about whether it was true or not for quite a while in this "anything is possible" thread http://www.onlinephilosophyclub.com/for ... dd4ebe8fa1
"Anything is possible." Well, you sure have presented a compelling alternate theory of math.

Is it not painfully clear that we're done? Nothing new has been said the last few posts by either of us.

Arising_uk
Posts: 11919
Joined: Wed Oct 17, 2007 2:31 am

Re: Is 0.9999... really the same as 1?

wtf wrote:... After all, the Flying Spaghetti Monster might have created us five minutes ago, along with our memories of the past. Woo hoo, you have solved everything. ...
BLASPHEMY!! There's no might about it and it solves everything. Ramen.

marsh8472
Posts: 54
Joined: Sun Oct 19, 2014 3:06 pm

Re: Is 0.9999... really the same as 1?

Based on this earlier statement you made in a previous post I'm assuming you agree with me on some level. I will accept your concession if you are done here, here's what you said again:
Please note that I made no claim that .999... = 1 is true. I only say that it's a valid derivation from a set of axioms.
I won't respond to the replies that involve name calling (nihilist/sophist stuff) or anything that represents a defeatist mindset of your position. But there's a few things worth responding to.
wtf wrote:
marsh8472 wrote: Conflicting answers are at odds with the fundamentals of real numbers. I'd like to see a line by line indisputable proof of why 0.9999... = 1, if you want to make one of those up.
I've done this several times. If you'll tell me what principles or axioms you accept, I can do the proof. But if you say things like, "Oh what if tomorrow they change the definition of convergence?" then there is no conversation to be had.
Actually you haven't once done it. You talk about it in sentences informally with nuggets of knowledge here and there where you tell me to look it up rather than explain it yourself but nothing formal. Whereas I explain it myself like with how decimal notation is constructed. I scanned pages of the book to prove that I was looking at it.

Here's what I mean for line-by-line proof

The real analysis book I have says this about the decimal representation:
1) decimal representation represents an infinite series
2) an infinite series represents an infinite sequence
3) the least upper bound of the sequence represents the limit of the infinite sequence
4) the limit of the sequence represents the value or sum of the sequence
5) the "=" sign represents the value

Therefore 0.9999... = 1 according to that

That was the post I made to you that I expected you would either get it or you don't why that's not good enough. It's not my fault if you do not understand it.
wtf wrote:
marsh8472 wrote: Conflicting answers are at odds with the fundamentals of real numbers. I'd like to see a line by line indisputable proof of why 0.9999... = 1, if you want to make one of those up.
I've done this several times. If you'll tell me what principles or axioms you accept, I can do the proof. But if you say things like, "Oh what if tomorrow they change the definition of convergence?" then there is no conversation to be had.
ok I looked them over, these all look acceptable
Synthetic approach

The synthetic approach axiomatically defines the real number system as a complete ordered field. Precisely, this means the following. A model for the real number system consists of a set R, two distinct elements 0 and 1 of R, two binary operations + and × on R (called addition and multiplication, respectively), and a binary relation ≤ on R, satisfying the following properties.

(R, +, ×) forms a field. In other words,
For all x, y, and z in R, x + (y + z) = (x + y) + z and x(yz) = (xy)z. (associativity of addition and multiplication)
For all x and y in R, x + y = y + x and xy = yx. (commutativity of addition and multiplication)
For all x, y, and z in R, x(y + z) = (xy) + (xz). (distributivity of multiplication over addition)
For all x in R, x + 0 = x. (existence of additive identity)
0 is not equal to 1, and for all x in R, x*1 = x. (existence of multiplicative identity)
For every x in R, there exists an element −x in R, such that x + (−x) = 0. (existence of additive inverses)
For every x ≠ 0 in R, there exists an element x^−1 in R, such that x*x^−1 = 1. (existence of multiplicative inverses)
(R, ≤) forms a totally ordered set. In other words,
For all x in R, x ≤ x. (reflexivity)
For all x and y in R, if x ≤ y and y ≤ x, then x = y. (antisymmetry)
For all x, y, and z in R, if x ≤ y and y ≤ z, then x ≤ z. (transitivity)
For all x and y in R, x ≤ y or y ≤ x. (totalness)
The field operations + and × on R are compatible with the order ≤. In other words,
For all x, y and z in R, if x ≤ y, then x + z ≤ y + z. (preservation of order under addition)
For all x and y in R, if 0 ≤ x and 0 ≤ y, then 0 ≤ xy (preservation of order under multiplication)
The order ≤ is complete in the following sense: every non-empty subset of R bounded above has a least upper bound. In other words,
If A is a non-empty subset of R, and if A has an upper bound, then A has a least upper bound u, such that for every upper bound v of A, u ≤ v.

The rational numbers Q satisfy the first three axioms (i.e. Q is totally ordered field) but Q does not satisfy axiom 4. So axiom 4, which requires the order to be Dedekind-complete, is crucial. Axiom 4 implies the Archimedean property. Several models for axioms 1-4 are given below. Any two models for axioms 1-4 are isomorphic, and so up to isomorphism, there is only one complete ordered Archimedean field.
Given I don't have an objection with anything that makes a real number a real number, I don't think it's a disagreement with the real numbers so much as a disagreement with definitions of some notations like the infinite summation, how the equal sign is being used, and whether 0.9999... should represent a real number in the context of asking "Is 0.9999... really the same as 1?". You'll notice for all of these axioms x, y, and z are required to be real numbers. If 0.9999... should not represent a real number in this context then these axioms are not much help.
wtf wrote:
marsh8472 wrote: I thought what I wrote earlier was sufficient enough though.
Are you saying you proved to your own satisfaction that .999... = 1? I don't know what your remark refers to.
No I'm saying I produced the line by line proof with this
The real analysis book I have says this about the decimal representation:
1) decimal representation represents an infinite series
2) an infinite series represents an infinite sequence
3) the least upper bound of the sequence represents the limit of the infinite sequence
4) the limit of the sequence represents the value or sum of the sequence
5) the "=" sign represents the value

Therefore 0.9999... = 1 according to that
It's not satisfying. We could add this one too "0) decimal notation represents a real number". That's at least 6 levels of rigid definitions in order to get to 0.9999... = 1. People who try to compare 0.9999... to 1 are thinking no matter how many 9's are added, this number will always be less than 1. So as 0.9, 0.99, 0.999, 0.9999, approaches infinity, does it suddenly become 1?

This would be the answer I would expect from your side of it:

1) 0.99999... represents an infinite series 9/10 + 9/100 + 9/1000 + 9/1000 +...
2) The infinite series 9/10 + 9/100 + 9/1000 + ... represents the sequence 0.9, 0.99, 0.999, ...
3) the limit of the sequence is the least upper bound of the sequence which is 1
4) the limit of the sequence is called the value of the infinite series
5) therefore 0.99999... is called 1

You said something similar when you stated this:
No, = means exactly equal. The reason is that we define the symbol .999... as a particular limit, and we define a limit to be an equality. In other words the limit of .9, .99, .999, .9999, ... is EXACTLY 1. It's part of the definition of a limit, which is that the sequence eventually gets as close as we want to 1. If that's true (which it is in this case) then the limit is exactly 1. It's because we define the limit to make this come out. is that clear or should I try to explain it better?
You also asked this question:
Then you are giving a nonstandard meaning of the notation. Shouldn't mathematical expressions be interpreted the way they are defined in math?
The answer to that question is not necessarily.

This proof answers the question with another question. Artificially defining the limit of the sequence as the value of the sequence allows the real numbers to remain complete with other real numbers but whether this definition is accurate is the question that answers whether 0.9999... really is the same thing as 1. That's another way of saying, as x approaches the value it is the value. That's the question being asked, but in this book they answered it with a definition and not a proof.

You asked this question too
It's much more philosophical than that I think. What is "infinite," really? Must we solve the riddle of the universe to know what this means?

No. Instead, in math we simply define the symbols and give formal definitions of the real numbers and limits in terms of the axioms of set theory; and then I can formally deduce .999... = 1.
That's the answer most people are looking for when they ask whether 0.9999... is really the same thing as 1. You need to solve the riddle to the universe to know what it means, not just arbitrarily define it and break out the champagne.

You stated here before that if 9/10 + 9/100 + 9/1000 + .... is seen as an never ending process we will never get a real value back when you said this
The sum of an infinite series is the limit of the sequence of partial sums. That's actually a clever trick. There really is no such thing as an infinite sum 9/10 + 9/100 + 9/1000 + ... Rather, that notation is defined as the limit of the SEQUENCE 9/10, 99/100, 999/1000, ... and you yourself provided the definition that the limit is 1.
and you were right, it is a clever trick. It's so clever it's managed to fool you into thinking you can really prove 0.9999... = 1 with it.

The real analysis book I have defines the infinite series x1 + x2 + x3 + ... xn + ...
as a sequence where s1 = x1, s2 = s1 + x2, s3 = s4 + x3 etc...

so if we consider this function = infinite_geometric_series( x= first term of the series, y=rate of change between terms, z=term number)

Written as a function 0.9999... = infinite_geometric_series(0.9, 0.1, 1)
where infinite_geometric_series(x, y, z) = x*0.1^(z-1) + infinite_geometric_series(x, y, z+1)

This infinite_geometric_series would not have a value because there would be no terminating condition. No value implies no real value, no real value implies it's not a real number.

The most obvious objection to 0.9999... being a real number I think is the closure property of addition which states that if you perform an add operation on two members of the same set of numbers you would produce another member of the same set. An infinite series as I defined it above and as a lot of people interpret it as does not produce a real value because 0.9999... is not an element of the set {0.9, 0.99, 0.999, ... } because there is no last element of the set by definition of infinity in math which is "without bound".

Another thing that you also mentioned is Cauchy. From another dreaded wiki page regarding construction of real numbers from cauchy sequences
https://en.wikipedia.org/wiki/Construct ... _sequences
Two Cauchy sequences are called equivalent if and only if the difference between them tends to zero. This defines an equivalence relation that is compatible with the operations defined above, and the set R of all equivalence classes can be shown to satisfy all axioms of the real numbers.
In other words (1 + 0.0 + 0.00 + 0.000 + ...) - (0.9 + 0.99 + 0.999 + ...) tends to 0 therefore they're considered equivalent. That does not answer the question about whether 9/10 + 9/100 + 9/1000 + ... approaches infinity is the same as 1. Rather they just define that they're the same. It's just more of the same type of circular proofs.
wtf wrote:
marsh8472 wrote: We are talking about whether a number that's infinitesimally close to 1 is less than 1. You apparently think they are equal and I'm undecided. It's not that big of a deal.
If you still think there are infinitesimals in the real numbers, why would I say anything at all at this point?
I don't think that. Are real numbers defined as "the only numbers that exist"? Or the union of real numbers and hyper reals "the only numbers that exist"?
wtf wrote:
marsh8472 wrote: It's almost like even though the reals do not have holes in them as an axiom, the decimal notation shows holes which are covered up by saying 0.9999... = 1.0000... . When dividing a number into 10 equal parts, then those parts into 10 equal parts, etc.. the problems arise when trying to place decimals into the right bucket when they are divided by lines that are infinitesimal in length
If I say to you: "But there are no infinitesimals in the real numbers," I would be an idiot. I prefer to say nothing. It's certainly frustrating that after all this, you didn't read a word I wrote about infinitesimals.
I would say fine, but that doesn't prove 0.9999.. = 1
wtf wrote:
marsh8472 wrote: Here's the definition of infinitesimal though
an indefinitely small quantity; a value approaching zero.
That is not the definition of an infinitesimal. An infinitesimal is a quantity x such that for every positive integer n, 0 < x < 1/n. That is the mathematical definition. The infinitesimals in the hyperreals satisfy that definition. It's perfectly simple to show that no real number does.
I grabbed that definition from the dictionary. That definition I showed you is by definition the definition of infinitesimal. But even with this definition you provided we can say x>0 and if 0.999... and 1 differ by an amount greater than 0 then they are not the same.
wtf wrote:
marsh8472 wrote: I don't have a point of view except I don't accept either completely. If you were arguing 0.9999... < 1, I may have argued 0.9999... = 1 instead.
Oh no I don't believe that. Your initial post innocently asked, "What do people think?" And of course your actual agenda is to dig in and argue your point of view. This is a common pattern.
I've done it before on another forum where I argued that 0.9999... = 1 and the other side argued endlessly until I realized their points were equally valid to my own.
wtf wrote:
marsh8472 wrote: Yeah I looked at all that already. The 0.999... can be shown to equal 1 but it looks more like proof by equivocation to me.
What happens if I don't take the bait? I happen to live in a rural area and every once in a while a cute little field mouse gets in the house. I set traps. I know what happens to mice who take the bait.
Like I didn't take your bait with hyperreals. It is a false dichotomy to claim that numbers are either reals or hyperreals. Then just disprove both and think it's proven.
wtf wrote:
marsh8472 wrote: It was one of my more difficult proving math courses. Other proving courses I remember taking are calc-based statistic course, calc-based physics, numerical analysis, linear numerical analysis, abstract math, and number theory.
Did you argue with your number theory professor that (1) We have no idea what a number is; and (2) We have no evidence that there can be infinitely many of them? And then refuse to learn the formalisms? When the prof said that we have a binary relation called "divides," according to which 3 divides 9 and 5 divides 10 but 3 doesn't divide 7, did you say, "Oh but what if they change the definition? We can't really be certain that 3 divided 9, can we?"
If I asked whether artificial intelligence was alive my answer wouldn't depend on whether artificial life follows the 8 life processes found in science text books. Similarly, if I'm asking a question relating to infinity I wouldn't rely solely on the interpretations of real numbers.
wtf wrote:
marsh8472 wrote: It's epistemically possible which means it's possible for all I know due to my lack of omniscience.
Sophistry or nihilsm. You reject scientific progress because "anything is possible." After all, the Flying Spaghetti Monster might have created us five minutes ago, along with our memories of the past. Woo hoo, you have solved everything.
There's no such thing as proved in science. Everything is provisionally true. The attitude of not accepting a conclusion is what keeps science going instead of becoming a dogmatic.
wtf wrote:
marsh8472 wrote: Except my conclusion is more like a tautology really actually isn't it? In the event 0.9999... = 1 like you believe, people still exist that are led to believe 0.9999... < 1 regardless of the reality of whether it's true or not. It's like asking whether the earth resolves the sun or vice versa or whether particles move randomly or deterministicly. A modal of either possibility can explain what we observe.
Sure. But you have presented no alternative model. The hyperreals don't help you. Your only alternative is that "anything is possible" and "maybe they'll change the definition of convergence." That is not intellectually satisfying as a model. The contents of your real analysis book IS an intellectual model.

After we're done here, why don't you spend some time actually putting together a model. Start with, say, 17th century calculus, and see how you would try to formalize a logically coherent theory of limits. Nobody will take you seriously if your theory of limit is, "Maybe they'll change the definition of convergence." You can't expect me to continue to engage on a point like that.
I don't have to if I wanted to justify an answer of "don't know" rather than pick "0.9999... = 1" or "0.9999... < 1". Even if 0.9999... is more correct to place 0.9999... in an inconceivable number set rather than in the real number set. I just say I don't know. I can prove I don't know by stateing I am not aware of inconceivable number sets by definition of inconceivable.

wtf
Posts: 805
Joined: Tue Sep 08, 2015 11:36 pm

Re: Is 0.9999... really the same as 1?

marsh8472 wrote:I will accept your concession
And you wonder why I call you disingenuous and other words that you don't like.

I concede nothing. I see that we are near the end of our conversation because we are just repeating the same points over and over. Abandoning an unproductive discussion is not conceding, except by virtue of the online principle that whoever keeps going wins.

You did make some substantive mathematical points in this post that I'd like to address later on, after I've stopped being infuriated at a couple of things you said, the use of the word "concession" being among the most prominent.

marsh8472 wrote: I won't respond to the replies that involve name calling (nihilist/sophist stuff
Someone who responds to a mathematical argument by saying:

* For all we know they might change the definitions someday; and

* We can be certain of nothing, and therefore -- your words -- "a proof doesn't prove anything";

may legitimately be accused of nihilism or sophistry, depending on whether they are sincere or trolling.

I simply can't find it in my heart to retract those words. Your repeated insistence that since the definition of convergence might change, you can't accept a mathematical proof, makes it pointless for me to present a mathematical argument. And my abandonment of trying to converse intelligently with someone who takes that stance is not concession, it's abandonment in recognition of futility.

If I'm grossly misunderstanding your position, this would be the time to clarify it. If you reject standard mathematical definitions, we can't talk math. That would seem obvious.

If you ask me to explain or clarify a definition, I'll do so to the best of my ability.

And if you ask me if I think definitions are historically contingent and subject to change, I will agree.

But given that, once we have a definition, it's fair to consider the logical consequences of that definition. You have to drive a stake somewhere. You can't just say that nothing is certain and therefore you disagree with anything I might possibly say. I'm sure you can see why that would be frustrating to me. Yes?
marsh8472 wrote: or anything that represents a defeatist mindset of your position.
I'm not sure what you mean by that. But it has been my experience that arguing with people about .999... never results in them changing their minds. In that respect I acknowledge defeat even before I start. I don't expect to convert you and I haven't ever expected to convert you. But you must understand that by explicitly, over and over, rejecting standard definitions, it makes it pointless for me to even try.

Now before I go further, I would like you to respond to a point I made in my previous post. You pointed out that .999... is not a real number, but is rather a representation of a real number; a point with which I agree.

You then said that since .999... and 1 are different representations (of something, but not necessarily of the same thing) then by the law of identity the equality .999... = 1 is false.

If this is an unfair paraphrase of your argument please clarify it.

Now I pointed out that by the exact same argument, 2 + 2 = 4 may be doubted. Because as you say, these are two different representations of something, not necessarily the same something, and they are clearly not the same.

So I want to know if you argue against 2 + 2 = 4 on the same grounds that you argue against .999... = 1.

Frankly from a formalist position, I believe the same argument DOES apply to both equally.

But you did something that you do a lot. I make a substantive point and you simply ignore it. The accumulated weight of these adds to my feeling of futility in continuing here.

So humor me. Talk about the law of identity with respect to 2 + 2 = 4, and how it does or does not fall victim to the exact same criticism you leveled at .999... = 1.
marsh8472 wrote: I've done it before on another forum where I argued that 0.9999... = 1 and the other side argued endlessly until I realized their points were equally valid to my own.
It was probably me with one of my other handles. Do you have a link?

I have been an amateur crankologist for years, and I have never once seen a .999... conversation enlighten the doubter. I participate in these conversations because it's a bad habit. It certainly doesn't surprise me that you got no satisfaction from that other thread. Your own unwillingness to accept a mathematical argument on its own terms gets in your way.
marsh8472 wrote: Like I didn't take your bait with hyperreals. It is a false dichotomy to claim that numbers are either reals or hyperreals.
If you can quote anywhere in this thread where I said any such thing, or remotely said anything that could be interpreted as saying that ... then I will "concede."

Now as I say you did actually make an attempt in your most recent post to engage on the mathematics. I do appreciate that and I will respond after I calm down from your claiming that I have given a "concession," and that I claimed that only reals and hyperreals exist (a complete absurdity), while ignoring the substantive remark I made about 2 +2. These commissions and omissions in your most recent post have not made me feel any more charitable towards your style of discourse than I did before.

But like I say, I appreciate the substantive mathematical points you brought up. So I'd welcome de-escalation rather than escalation. And direct evidence or a retraction, whichever is appropriate, of your claim that I said that reals and hyperreals are the only numbers.

wtf
Posts: 805
Joined: Tue Sep 08, 2015 11:36 pm

Re: Is 0.9999... really the same as 1?

Ok. Now that I got that out of my system ... I forgive your assuming I conceded anything. And I'm perfectly happy to discuss the math, and also the meaning and philosophy of the math. You owe me the 2 + 2 discussion since I believe one of your objections to .999... = 1 applies with equal force to 2 + 2 = 4, and I want to know if you object to 2 + 2 = 4 as well.

Now you brought up a lot of mathematical topics in your post. I'm going to respond one at a time in separate posts, because I think the excessive length of both of our posts is contributing to confusion.

To begin, you perfectly expressed everything there is to know about the subject.
marsh8472 wrote:
This would be the answer I would expect from your side of it:

1) 0.99999... represents an infinite series 9/10 + 9/100 + 9/1000 + 9/1000 +...
2) The infinite series 9/10 + 9/100 + 9/1000 + ... represents the sequence 0.9, 0.99, 0.999, ...
3) the limit of the sequence is the least upper bound of the sequence which is 1
4) the limit of the sequence is called the value of the infinite series
5) therefore 0.99999... is called 1
Yes. Yes yes yes yes and yes.

Let me outline how this works. Let me note that despite your claiming I never gave a complete argument, in fact you understand the complete argument perfectly well. Now you want this to "mean" something and frankly that's hopeless. Nobody knows whether the universe is finite or infinite, discrete or continuous. Nobody knows if we're in a simulation or a multiverse or the mind of the Flying Spaghetti Monster or something else. Nobody knows these things. Math is a historically contingent creation of human beings. It is good to put on our formalist hat when studying math, even if we are not actually formalists. If you insist that math should "mean" something, that's fine, let's talk about that later. For now let's just try to understand the math itself.

Now as you point out, there are two approaches to the real numbers, the synthetic and the various set-theoretic constructions.

You listed the axioms for the real numbers. Now in this case "axiom" does not actually mean a statement accepted without proof. Rather, in this case axiom just means "defining property." So if I say that a mammal is an animal that gives birth to live young and so forth, those defining characteristics are what we call axioms. I agree that this overloading of the word "axiom" is a point of confusion and when they put me in charge of math education I will use the phrase "defining characteristics" rather than "axioms" to describe the real numbers.

Ok so far? Ok.

Now the axioms for the real numbers -- that is, how we can tell, if we stumble on a mathematical object, whether or not it is the real numbers -- are as you point out:

* Multiplication, multiplicative identity (one), multiplicative inverses (except for 0);

* Multiplication distributes over addition; that is, a(b + c) = ab + ac.

* Total order called <= that satisfies trichotomy. Either x < y or x = y or y < x.

So far we have defined what we call a field. The rational numbers have all these properties but the rationals are not the reals. So there is one more defining property:

* Completeness. There are no holes. Every nonempty set of reals that is bounded above has a least upper bound, or LUB. In the literature the LUB is also called the "sup."

For example are the rationals complete? No, because for example the set of rationals whose square is less than 2 is bounded above (by 47, for example) but there is no least upper bound. .

But the real numbers are complete. Every nonempty set of reals that's bounded above has a LUB or sup.

Now any sensible person asks at this point: Well yeah, but how do we know there is such a thing? After all I can define a flying elephant as an elephant that flies, but there isn't one so I'm making a vacuous definition.

Now it does turn out that if we assume the axioms of set theory (in this case, statements assumed without proof!) then using the axioms, we can construct a particular set that satisfies all the axioms (defining properties!) of the real numbers, including completeness. One such construction is via equivalence classes of Cauchy sequences; another is called Dedekind cuts.

[Yes you are right, the dual-meaning of axiom is a great point of confusion. But ultimately I can argue that the two meanings are the same. For example the modern point of view is that set theory is the study of mathematical objects that satisfy the defining properties of sets. This is a profound philosophical point that has only become clear in recent years].

Now these are technical constructions (of the reals) that have absolutely no purpose other than to demonstrate that we can construct something that fits the definition of the real numbers within set theory. Every math major sees such a construction once and never uses it again. So you can take my word for it or look it up online that there are valid constructions (lots of them, all different) in set theory that give us a model of the axioms (defnining properties) of the real numbers.

From now on we will simply assume that there exists a mathematical object that satisfies the axioms of the real numbers.

Philosophic aside: Of course I make no claime whatsoever that the real numbers have physical existence. Personally I don't think so. I only claim that an object can be built using the rules of set theory that satisfies the defining properties of the reals. It has no more "meaning" than my telling you how the knight moves in chess.

Now we make the following definitions, which I'll just list.

* Within set theory I can define a thing called a function which has all the usual properties of functions as taught in high school. You put something in, you get something out. You put the same thing in twice, you get the same thing out both times. That's what makes it a function. So when we are in high school math class and I say "Let f(x) = x squared," a set theorist can tell you exactly what f is as a particular set.

* Within set theory we can model the natural numbers 0, 1, 2, 3, ...

* A sequence of real numbers is a function whose domain is the natural numbers.

Note this definition. There is no "magic infinite thingie" called an infinite sequence. There's nothing more than a function whose domain is the natural numbers. It's customary to notate the elements of a sequence using subscripts such as x_1, x_2, x_3, ... rather than f(1), f(2), f(3), ... but a sequence is nothing more than a function that inputs a natural number 'n' and outputs some real number that we think of as the n-th element of the sequence. [It's convenient to start sequences from 1 rather than 0, no harm is done since we only care about order and not arithmetic on the subscripts].

* Now we make the following definition, which as I say took 200 freaking years to get right. [And you say maybe it's not right? Maybe it's not. But let's talk about that later!]

If we have a sequence of real numbers x_1, x_2, x_3, ...; and we have some real number L; and it happens to be that case that for every positive real number epsilon, there exists a natural number N such that if n > N, then |x_n - L| < epsilon; in that case we say that L is the limit of the sequence.

That's it. A limit means nothing more and nothing less. And the development of this definition is profound. Newton didn't have it. Leibniz didn't have it. Nobody had it till the middle of the 19th century, 200 years after Newton used calculus to describe how gravity worked and thereby revolutionized the way we look at the world.

So what this means is that we have defined the limit of a sequence of real numbers as a number L such that the sequence of x's gets arbitrarily close and stays arbitrarily close. If you give me epsilon I'll give you back N.

Example. Consider the sequence 1/2, 1/4, 1/8, 1/16, ... I claim this sequence has the limit 0. Why? Say you give me epsilon = 1/100. Then every element of the sequence after 1/128 has distance from 0 of less than epsilon. Say you give me epsilon = 1/1000. Then I just go out to 1/1024. Any epsilon you give me, I can make the entire tail of the sequence within epsilon of 0. So the limit is 0 by definition.

Now the beauty of this definition is that we have abolished the idea of infinitesimals. There are no infinitesimals in the real numbers, and we have figure out how to define limits without them. That's an important philosophical point.

* Ok now, suppose that we have an "expression of the form" x_1 + x_2 + x+3 + ... How shall we deal with it?

Note that the axioms (defning properties) of the real numbers only say that if we have TWO real numbers x and y we can form their sum x + y. You mentioned this in your post. Nothing in the rules allows us to add up infinitely many real numbers. So we are going to have to do something clever.

Well, we can form the SEQUENCE x_1, x_1 + x_2, x_1 + x_2 + x_3, and so forth. And we already know how to take the limit of a sequence, it's just the epsilon/N/L business. So we define the "sum" of an infinite series as the limit of the SEQUENCE of partial sums. Done!!

* We can now make logical sense of an expression such as 9/10 + 9/100 + 9/1000 + ... It's just the limit of the sequwnce 9/10, 99/100, 999/1000, ... And that limit, by the epsilon/N definition is 1.

* Finally, we define the notation .999... to mean the sum of the series 9/10 + 9/100 + 9/1000 + ..., which is 1. So .999... = 1.

That is it. That is all there is to it. There are no claims that it's "true in the real world." There aren't even any claims that this is the perfect or ultimate defininition of convergence of a sequence or series. For all we now these will change in a hundred years just as our conception of limits changed radically from the time of Newton.

But today, this is the definition. Today, .999... = 1 is provable from first princples in math.

And this is ALL I claim. You want this to mean something. Let's discuss that later. For right now I just want to make sure that this derivation is clear to you, because that's everything you need to know about why .999... = 1.

You say it's not necesarily true in the physical world? I say that makes as much sense as asking if the way the knight moves in chess is true in the physical world. It's a category error to even ask the question.

You ask if this is the "right" definition of limit, and if perhaps this definition might be different in a hundred years. I freely admit it might be. After all, we can trace the development of the limit concept from the time of Archimedes and Eudoxus. It's changed a lot. Math is a historically contingent activity of humans.

But it's the definition we've got. So before we attack it, let's first understand it. And let's try to appreciate that in the 2000+ years since Archimedes, this is what we've come up with. Let's accept it on its own terms.

So this has been a long-assed piece of mathematical exposition, quite out of place on a forum like this. You asked for it, you got it.

So I suggest that if you have questions about this exposition, ask them in a reply to this post.

If you want to make a point or ask questions about the physical meaning (I claim there isn't any at all); or the philosophical meaning (is this the right model of the continuum? Many such question abound and they are good questions); or whether this is even the right or the ultimate definition within the field of math; I suggest asking them in separate posts. That way perhaps we can get multiple topics going but separately, so as to reduce confusion.

Let me tl;dr this shaggy dog of a post.

* I've presented the modern mathematical argument that .999.. = 1.

* This argument is historically contingent. Archimedes, Newton, Gauss, and Euler worked brilliantly with limits but would not recognize set theory or the modern theory of limits. By analogy, the theory might be different in a hundred years. That does not prevent us from learning and understanding and appreciating the current theory.

* I don't for a moment believe the real numbers as currently understood are necessarily the right model of the universe, which might be discrete or finite or a simulation or an idle thought in the mind of the Flying Spaghetti Monster. Those are interesting questions but they are not mathematical questions.

* I propose keeping each further post in this thread on one particular topic. Math or philsophy or physics or the historical development (past and future) of the limit concept. For sanity's sake let's just keep all these things separate.

Ok tell me what you think. About the math! If you think something about the philosophy, put it in a separate post. Maybe title it "philosophy." Or "real world." Or "Historical contingency of the limit concept." So that we can be on the same page instead of talking past each other.

And I still want to hear about whether you obect to 2 + 2 = 4. Because I say to you truly: The set-theoretical proof that 2 + 2 = 4 is every bit as doubtful as the proof that .999... = 1. For example what makes anyone think there are infinite sets? Once you accept infinite sets everything else follows. Ask me about the Axiom of Infinity. That's where all the trouble starts.

marsh8472
Posts: 54
Joined: Sun Oct 19, 2014 3:06 pm

Re: Is 0.9999... really the same as 1?

wtf wrote:So humor me. Talk about the law of identity with respect to 2 + 2 = 4, and how it does or does not fall victim to the exact same criticism you leveled at .999... = 1.
Correct, I was thinking 1 was not also a representation of a real number when I said that. The point though is if any property distinguishes the left side from the right side and we include that property as part of the criteria for the equal sign then they are not the same. Such as "0.999..." has ellipsis in it, has 9's in it, is 8 characters in length, represents an sum of an infinite series.

Suppose we were working in the natural numbers where a definition exists that the ceiling function of a real number represents the value of the number. Using this definition I can prove that Pi = 4 by showing ceiling(Pi) = 4. Does that really mean it's a true statement that 3.141592... = 4? The objection would be that 3.141592... < 4 and 3.141592 should really be treated as a real number. The same problem occurs here.

Given whatever people mean when they say "0.9999..." could disqualify that number as a real number within the context of what they meant by "0.9999..." . We can still make a claim that there could exist a set such that 0.9999... < 1. We could make up a set right here that states 0.9999... < 1 as an axiom and that 0.9999... added with a real number produces another element of that set like 1.9999..., 2.9999, etc... Then just keep making up other rules from there the same way they did in real analysis. That's just as powerful as the proof I've seen of 0.9999... = 1 with the exception of it not being widely accepted since I just made it up.
wtf wrote:
marsh8472 wrote: I've done it before on another forum where I argued that 0.9999... = 1 and the other side argued endlessly until I realized their points were equally valid to my own.
It was probably me with one of my other handles. Do you have a link?

I have been an amateur crankologist for years, and I have never once seen a .999... conversation enlighten the doubter. I participate in these conversations because it's a bad habit. It certainly doesn't surprise me that you got no satisfaction from that other thread. Your own unwillingness to accept a mathematical argument on its own terms gets in your way.
I tried looking for it, but can't find it. I either used another alias or the site is gone.
wtf wrote:
marsh8472 wrote: Like I didn't take your bait with hyperreals. It is a false dichotomy to claim that numbers are either reals or hyperreals.
If you can quote anywhere in this thread where I said any such thing, or remotely said anything that could be interpreted as saying that ... then I will "concede."

Now as I say you did actually make an attempt in your most recent post to engage on the mathematics. I do appreciate that and I will respond after I calm down from your claiming that I have given a "concession," and that I claimed that only reals and hyperreals exist (a complete absurdity), while ignoring the substantive remark I made about 2 +2. These commissions and omissions in your most recent post have not made me feel any more charitable towards your style of discourse than I did before.
Here:
But the claim that .999... = 1 is true "because" the number of 9's is infinite, can be falsified.

The Quora link marsh8472 gave is a clear and simple technical overview of the question of whether .999... = 1 in the hyperreals. I'm going to talk about this a little when I respond to marsh8472's post. Normally I don't like Quora, but in this case there is an answer by Alon Amit, who in my opinion is God when it comes to expositing math on Quora. In this case he did a beautiful job of showing an example where we can interpret a particular hyperreal notation as containing infinitely many 9's, yet the number so expressed would not be 1. The problem is that it's not "ALL" nines, which turns out to not be a number in the hyperreals. I will say something about all this later, it's not actually that difficult.

But the bottom line here is that we have to make the case that .999... = 1 in the hyperreals. It's not given just because "there are infinitely many 9's."
But if you're not saying it's hyperreals/reals or nothing then we can just forgot that part. I was saying in case that's what you're trying to say it's a false dichotomy.

[quote="wtf]Let me outline how this works. Let me note that despite your claiming I never gave a complete argument, in fact you understand the complete argument perfectly well. Now you want this to "mean" something and frankly that's hopeless. Nobody knows whether the universe is finite or infinite, discrete or continuous. Nobody knows if we're in a simulation or a multiverse or the mind of the Flying Spaghetti Monster or something else. Nobody knows these things. Math is a historically contingent creation of human beings. It is good to put on our formalist hat when studying math, even if we are not actually formalists. If you insist that math should "mean" something, that's fine, let's talk about that later. For now let's just try to understand the math itself.[/quote]

Right even if the universe is not infinite we can still create a model to represent the universe that says it is infinite. Whether 0.9999... = 1 or not does not have to be contingent on how the universe really behaves if we're speaking in terms of models of reality rather than reality itself.
wtf wrote:You listed the axioms for the real numbers. Now in this case "axiom" does not actually mean a statement accepted without proof. Rather, in this case axiom just means "defining property." So if I say that a mammal is an animal that gives birth to live young and so forth, those defining characteristics are what we call axioms. I agree that this overloading of the word "axiom" is a point of confusion and when they put me in charge of math education I will use the phrase "defining characteristics" rather than "axioms" to describe the real numbers.

Ok so far? Ok.
yes
wtf wrote:Well, we can form the SEQUENCE x_1, x_1 + x_2, x_1 + x_2 + x_3, and so forth. And we already know how to take the limit of a sequence, it's just the epsilon/N/L business. So we define the "sum" of an infinite series as the limit of the SEQUENCE of partial sums. Done!!

* We can now make logical sense of an expression such as 9/10 + 9/100 + 9/1000 + ... It's just the limit of the sequwnce 9/10, 99/100, 999/1000, ... And that limit, by the epsilon/N definition is 1.
I agree that's what they do. But that's sweeping the issue under the carpet too. Determining the value of an infinite sum is what makes it a challenging question.
wtf wrote:That is it. That is all there is to it. There are no claims that it's "true in the real world." There aren't even any claims that this is the perfect or ultimate defininition of convergence of a sequence or series. For all we now these will change in a hundred years just as our conception of limits changed radically from the time of Newton.

But today, this is the definition. Today, .999... = 1 is provable from first princples in math.
Right, if we were in a math class that would be correct. But asking a question like whether 0.999... is 1 is like asking whether Pluto is a planet 20 years ago. The two ways to interpret this question are: they're asking how it's currently classified as or they're asking whether it should be considered a planet regardless of whether it currently is considered one or not. On a philosophy board there is potential that the latter interpretation is more appropriate.

I found this demonstration from extremefinitism that says 0.999... is not equal to 1 using math rigor https://www.youtube.com/watch?v=xSlS2xE8rH8 and this video from Math Association of America https://www.youtube.com/watch?v=x-fUDqXlmHM that analyzes an informal proof of 0.9999... = 1 comes to the conclusion that the answer is contingent on what we choose to assume about the number in case you want to look at either of those to see if it is enlightening or not.

wtf
Posts: 805
Joined: Tue Sep 08, 2015 11:36 pm

Re: Is 0.9999... really the same as 1?

Excerpt from a conversation I recently had in my head:

Imaginary You: Thank you so much, wtf, for taking the time and trouble to write out that clear explanation of the mathematical argument that .999... = 1. It's better than anything in my dark and fuzzy real analysis book, and way more understandable than anything my real analysis professor ever said.

Me: You're very welcome.
marsh8472 wrote:
wtf wrote:So humor me. Talk about the law of identity with respect to 2 + 2 = 4, and how it does or does not fall victim to the exact same criticism you leveled at .999... = 1.
Correct, I was thinking 1 was not also a representation of a real number when I said that. The point though is if any property distinguishes the left side from the right side and we include that property as part of the criteria for the equal sign then they are not the same. Such as "0.999..." has ellipsis in it, has 9's in it, is 8 characters in length, represents an sum of an infinite series..
I wish to talk about one thing at a time to focus this conversation.

I ask you directly, are you questioning 2 + 2 = 4 on the same grounds that you question .999... = 1? Because the same argument applies. Different symbols, a plus sign, etc. So they are not the same.

Let me argue as you might that 2 + 2 is not necessarily 4.

* Isn't it the case that in math, "2", "+", "=", and "4" are symbols that have historically contingent definitions?

* And isn't it the case that the modern set-theoretic definitions are arguably nonsense? For example we are told in set theory that ∅ is the empty set; and that we define:

0 = ∅
1 = {1}
2 = {0, 1}
3 = {0, 1, 2}
4 = {0, 1, 2, 3}

etc. This definition is due to John von Neumann. https://en.wikipedia.org/wiki/Set-theor ... n_ordinals

* And as I say, isn't this definition manifestly nonsense that any schoolboy would recognize as such? For one thing, there is no such thing as the empty set. It's a mathematical fiction whose purpose is to make everything else work out; but any philosopher worth his or her salt could rip the concept to shreds with ease.

* Isn't it completely wrong to claim that the counting numbers are "sets," when in fact mathematical sets are not actually common sense collections as we're told in school, but that are actually very weird hypothetical abstract entities that, when you drill down into the technical bits, aren't anything like any collections we ever saw?

* Didn't philosopher Paul Benacerraf write a brilliant paper called What Numbers Could Not Be, in which he did in fact rip to shreds the notion that numbers are sets? http://isites.harvard.edu/fs/docs/icb.t ... cerraf.pdf

* And while we're on the subject, isn't it therefore the case that the set-theoretic statement, "3 is an element of 4" true is set theory, and obviously and profoundly false in the real world, in common sense, and in fact even in mathematics, outside of the field of set theory?

* And aren't these set-theoretic definitions used to prove in modern math that 2 + 2 = 4?

* Therefore I ask you: Isn't it entirely possible that someday mathematicians will throw off this nonsense, provide a far better definition of the natural numbers, and then it will be taught that 2 + 2 ≠ 4? After all, Pluto.

What say you to that argument? I say it's no different than .999... = 1, and that the only reason .999... gets so much attention is because it involves mysterious limits, whereas 2 + 2 = 4 seems to be an everyday fact of nature.

But arguing from "everyday facts of nature" can be misleading. It was considered an everyday fact of nature for thousands of years that the geometry of space was Euclidean. Kant even claimed that humans have an a priori knowledge of this fact. In the 1840's when Riemann and others demonstrated the logical consistency of non-Euclidean geometry, his discovery was met with at first shock, and then with the dismissal that it was no more than a mathematical curiosity that had nothing to do with the real world. And the some guy named Einstein came along and showed that the weird mathematics of non-Euclidean geometry was in fact true about the world.

You made some other points in your post that I'll get to but let's do one thing at a time. The 2 + 2 = 4 example has all the features of .999... without the counterintuitive baggage, so it's instructive.

tl;dr: Do you deny 2 + 2 = 4 on exactly the same grounds that you deny .999... = 1?

wtf
Posts: 805
Joined: Tue Sep 08, 2015 11:36 pm

Re: Is 0.9999... really the same as 1?

My comments on the rest of your most recent post.
marsh8472 wrote: I tried looking for it, but can't find it. I either used another alias ...
Perhaps you and I are the only two people on the Internet and everyone else is one of our many aliases.
marsh8472 wrote: But if you're not saying it's hyperreals/reals or nothing then we can just forgot that part. I was saying in case that's what you're trying to say it's a false dichotomy.
Well I better make sure you're not saying a brick is a tuna fish in case that's what you're trying to say. I really would like to know what I wrote that led you to think I was making that claim.
marsh8472 wrote: I agree that's what they do. But that's sweeping the issue under the carpet too. Determining the value of an infinite sum is what makes it a challenging question.
Yes, it was a challenging question for two thousand years before Newton and for another 200 years after. And determining the value of infinite sums is certainly a challenging problem for freshman calculus students.

But it's a solved problem. It's no objection to a scientific or mathematical fact that it used to be unsolved. If it's solved today, it's solved. It's sophistry (there's that word again but what other word would you use?) to claim that because a solved problem used to be unsolved that this somehow casts doubt on the solution. Fire used to be an unsolved problem. The wheel used to be an unsolved problem. The internal combustion engine, heavier-than-air flight, and television used to be unsolved problems. Today they're solved. Philosophers don't say, "Oh well, fire might not actually be hot, because after all it used to be an unsolved problem." Come on, man.
marsh8472 wrote: Right, if we were in a math class that would be correct. But asking a question like whether 0.999... is 1 is like asking whether Pluto is a planet 20 years ago.
But outside of math class it's not clear what the question even means. It's far from clear that it means anything at all.

If I'm teaching you chess and I say the knight moves in such-and-so manner, isn't it a category error to say, "Well yes, in chess class the knight moves that way. But how does it REALLY move?"
marsh8472 wrote: I found this demonstration from extremefinitism that says 0.999... is not equal to 1 using math rigor
I'm painfully familiar with that site. He's a complete crank. He doesn't even know what finitism is.

It so happens that finitism and its even more extreme cousin ultra-finitism are two extremely interesting philosophies of math. I myself am of the opinion that ultra-finitism is the only philosophy of math that makes sense in the real world and has even a potential claim to be true. The problem with it is that it's not axiomatizable. But neither is physics, and that doesn't stop anyone from doing physics.

If you want to read something intelligentabout finitism and ultra-finitism, have a look at this. https://plato.stanford.edu/entries/phil ... thematics/. Also this is pretty good. http://math.stackexchange.com/questions ... believe-it. And Wiki's not half bad. https://en.wikipedia.org/wiki/Ultrafinitism

I'd love to talk about ultra-finitism, it's super interesting and actually very technical. There was a serious, accomplished professional mathematician at Princeton named Ed Nelson who was the world's leading advocate of ultra-finitism. He passed away just a couple of years ago.

But listen man you have simply got to stop randomly linking crank sites. I have no idea what your point is in doing that. It makes me think you're not intellectually serious about any of this.

ps -- Serious alternative philosophies of math: Intuitionism/constructivism. Finitism/ultrafinitism. Structuralism (today actually winning the war and supplanting set theory. Benacerraf's article I linked got that party started).

Not serious: Internet crank sites.

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