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

Posted: Tue Mar 14, 2017 12:55 am

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?
2+2=4 is less controversial. But if I put up a thread that read "Is 2+2 really the same as 4", we can plainly see that I'm not asking whether current mathematics teaches that or what calculators output as an answer to that question. And 0.9999... deals with infinity and limits making it more complex. But when I programmed mathematics into a turning machine, I had to hard-wire certain mathematical operations into it like xor, or, and. Maybe our brains are hard-wired to believe certain self-evident axioms that are just not true. How we perceive the world is based off of what's best for our survival as a species, not necessarily what's true. In a "2+2" thread I would break down the thought process that goes into convincing someone that it is the same as 4. The law of identity and the law of non-contradiction would come up.
wtf wrote: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.
Hardly solved unless you use that term loosely. There's a difference between having one solution and having it solved. They have one particular answer which is 0.9999... = 1. Other possible answers are 0.9999... < 1 or maybe 0.9999... is undefined like 1/0. Mathematics returns to us whatever assumptions we feed to it. What's the smallest number greater than 0? Well there is no such number n the real numbers, which has no holes in it, has no smallest number greater than 0. Does that make sense to you? If I walk from point A to point B, did I walk across an infinite number of points or a finite number of points? Real numbers provides one way to comprehend it but no real way to verify it outside of itself.
wtf wrote: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?"
And maybe they'll decided that it's better that a knight move a different way if there's a good justification for doing it. Like rules change in many games such as NFL football. http://operations.nfl.com/the-rules/evo ... nfl-rules/
wtf wrote: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.
extremefinitism did make some good points at the end though. Mathematics uses some misleading language. Like when they say x approaches infinity. Any finite number is no closer to infinity than any other, whether it be 1 or 10^googleplex, so how can x approach infinity? It's probably best to say 0.9999... has a variable number of 9's instead of infinite 9's when we talk about the reals. A variable number of 9's is less than 1. They don't have infinitesimals or "infinity 9's" in the real number system.

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

Posted: Tue Mar 14, 2017 1:35 am
marsh8472 wrote:... In a "2+2" thread I would break down the thought process that goes into convincing someone that it is the same as 4. The law of identity and the law of non-contradiction would come up. ...
Sorry, I'm loving this thread as whilst I can understand very little of it I'm enjoying the little philosophy of mathematics that pops up that I can understand and I'm really scared to be breaking in here but I'm at a bit of a loss as to what you mean here about convincing someone as my remedial mathematics has the symbol "+" as something like "and" and "2" as shorthand for "1+1" and "4" as shorthand "1+1+1+1" so I'm at a loss as to how "2+2=4" can ever be 'wrong' or needs to be explained as it says "1+1+1+1=1+1+1+1"? Are you trying to ground numbers in a logical base? If so I thought this had already been done by Russell and Whitehead? And before that someone called Peano had sorted the numbers?
p.s.
Ooo! Just wikkied and seen that C.S.Peirce had axiomatised them as well! Wow!! As I love that guy even tho' he was also way above my weight of understanding.

Sorry again.

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

Posted: Tue Mar 14, 2017 2:16 am
Arising_uk wrote:Sorry, I'm loving this thread as whilst I can understand very little of it I'm enjoying the little philosophy of mathematics that pops up that I can understand and I'm really scared to be breaking in here
Oh no please by all means jump in. We need fresh meat. I'd be happy to explain any points of curiosity or confusion. Nobody on math-oriented websites discusses whether .999... = 1. It's not a serious dispute. If I have any purpose in being here at all it's to try to explain the viewpoint of modern math to anyone who's interested. Modern in this case meaning after 1850 or so when Augustin-Louis Cauchy invented the epsilon business that's bedeviled students ever since.
Arising_uk wrote: but I'm at a bit of a loss as to what you mean here about convincing someone as my remedial mathematics has the symbol "+" as something like "and" and "2" as shorthand for "1+1" and "4" as shorthand "1+1+1+1" so I'm at a loss as to how "2+2=4" can ever be 'wrong' or needs to be explained as it says "1+1+1+1=1+1+1+1"? Are you trying to ground numbers in a logical base? If so I thought this had already been done by Russell and Whitehead?
R&W are no longer relevant. Their entire thesis that math could be reduced to logic (this is called logicism) was totally destroyed by Gödel.

Today we model the natural numbers (and everything else that depends on them, including the real numbers) on set theory. In parts of contemporary math, set theory has been replaced by category theory, which can be called a structuralist approach. (I know more about category theory than I do about structuralism). There are also new approaches to foundations like type theory, which actually goes back to Russell. So perhaps he'll have the last laugh. Type theory II, the Revenge of Russell. It's a thing in modern math but I don't know much about it. It's big in computer science.
Arising_uk wrote: And before that someone called Peano had sorted the numbers?
A very interesting point, the difference between Peano and set theory. Peano set down some axioms for the natural numbers. He based everything on the idea of a successor function that inputs 0 and outputs 1, inputs 1 and outputs 2, etc.

So with Peano we have all the natural numbers 0, 1, 2, 3, 4, ... However we do NOT have a completed set of them. We have no container that encompasses them all. Peano's axioms describe Aristotle's potential infinity.

In set theory, one of the most important axioms is the axiom of infinity. The axiom of infinity says that there is a set that models the natural numbers. That's the von Neumann definition. Zero is the empty set, 1 is the set containing 0, and so forth.

A set of natural numbers is Aristotle's actual infinity. It was Cantor's bold and brilliant breakthrough to assume he had a completed set of the natural numbers, and then derive further logical consequences from that. It is precisely the axiom of infinity that distinguishes math from physics. We have no evidence in physics that there are any infinite collections. In math, infinite sets are taken as given.

Essentially all .999... = 1 opposition eventually comes down to objecting to the axiom of infinity. The Extremefinitism site (which I label a crank site, but I love cranks so I'm perfectly willing to read them for entertainment value) explicitly objects to the mathematical assumption of infinity.

It's very relevant that the Extremefinitism site rejects the axiom of infinity and .999... = 1. Those two positions go hand in hand.

I note for the record that finitism itself is a perfectly valid philosophical stance. The Extremefinitism guy is a crank because he thinks he's a finitist but doesn't actually know what he's talking about. If he actually studied the subject he claims to be blogging about he'd have a much better site.

There is an intermediate viewpoint called intuitionism or constructivism. As I understand it, intuitionism is a little mystical (points come into existence through the active selection of an observer); and constructivism is computational: a point exists if and only if it can be generated by an algorithm. So (as I understand it, which isn't much) an intuitionist accepts pi = 3.14159... because we can write a computer program to crank out the digits. But they do not accept the so-called noncomputable numbers, which are strings of digits that can not possibly be generated by any algorithm because they are essentially random.

So the intuitionist/constructivist real line is shot full of holes, but it's still infinite.

I should mention that earlier I noted that the hyperreal line is shot full of holes too. But the hyperreal line is much richer than the standard line. The hyperreal line has all the reals PLUS the infinitesimals. The constructivist real line is restrictive. It has the computable real numbers but not the noncomputable ones.

It's perhaps philosophically interesting that the standard real line is complete (every nonempty set bounded above has a least upper bound); and both the restrictive (constructivist/intuitionist) and the expansive (hyperreal) variants are not complete. Like Goldilock's porridge, the standard real line is "just right."

As I mentioned earlier there are in fact philosophies of math called finitism and ultrafinitism that deny the axiom of infinity. It's a perfectly legitimate position. In fact it is logically consistent to deny the axiom of infinity. When you do, you get the exact same system as described by Peano. You get most of number theory, including mathematical induction. You just don't get any infinite sets, no Cantorian set theory, no transfinite numbers.

The biggest problem with finitism is that it's very difficult to get the theory of the real numbers off the ground. But a hundred years from now who's to say what intellectual ideas will be in vogue. Anything's possible.

Arising_uk wrote:
Ooo! Just wikkied and seen that C.S.Peirce had axiomatised them as well! Wow!! As I love that guy even tho' he was also way above my weight of understanding.
Ah C.S. Peirce. I've heard a little about him. He apparently anticipated the structuralist approach to mathematics. His point is that the continuum, if it means anything at all, can not possibly be made of discrete points as it is in set theory. And as it turns out, the category theory revolution in contemporary math is evidently a rediscovery of the ideas of Peirce.

In fact in the categorical account of set theory, sets don't have any internal structure. We don't ask what is "inside" of sets. Rather we only care what relations a given set has with other sets. There are no more points.

Evidently Peirce was a century ahead of his time and modern math is just starting to catch up. I wish I knew more about him too.
Arising_uk wrote: Sorry again.
You raise great points. Everyone is welcome here as far as I'm concerned. Especially devotees of the Flying Spaghetti Monster.

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

Posted: Tue Mar 14, 2017 11:43 am
Arising_uk wrote:
marsh8472 wrote:... In a "2+2" thread I would break down the thought process that goes into convincing someone that it is the same as 4. The law of identity and the law of non-contradiction would come up. ...
Sorry, I'm loving this thread as whilst I can understand very little of it I'm enjoying the little philosophy of mathematics that pops up that I can understand and I'm really scared to be breaking in here but I'm at a bit of a loss as to what you mean here about convincing someone as my remedial mathematics has the symbol "+" as something like "and" and "2" as shorthand for "1+1" and "4" as shorthand "1+1+1+1" so I'm at a loss as to how "2+2=4" can ever be 'wrong' or needs to be explained as it says "1+1+1+1=1+1+1+1"? Are you trying to ground numbers in a logical base? If so I thought this had already been done by Russell and Whitehead? And before that someone called Peano had sorted the numbers?
p.s.
Ooo! Just wikkied and seen that C.S.Peirce had axiomatised them as well! Wow!! As I love that guy even tho' he was also way above my weight of understanding.

Sorry again.
2+2=4 is as right as right can be I'm sure. It's more clear if we're using a unary number system where there are less symbols to work with, define, and justify definitions of
1=1
2=11
3=111
4=1111
etc..

The idea is that 2+2 (base 10)= 4 (base 10) is asking whether 11+11 (base 1)=1111 (base 1)
Now we just have to figure out how to justify 11+11=1111
The "+" means we combine them, one way to do that is manipulate the left side by removing the "+" which results in the "11" being put with the other "11".
Like this 1111=1111. Then the next step is justifying 1111=1111. They look the same but what if they're not the same? It's assumed that any "1" is the same as any other "1" I suppose.

But if we had something like
11+11=1111
Can we still say 1111=1111? It depends on the definition of "=". Even if they look the same in every way detectable that does not prove that there's not something that makes them different in some undetectable way. Claiming they are the same because no way can be found to distinguish them or even no way exists to distinguish them is committing an ignorance fallacy.

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

Posted: Tue Mar 14, 2017 6:52 pm
I found another video about 0.999... = 1. I'm guessing wtf will like this better since they say it equals 1 at the end https://www.youtube.com/watch?v=rT1sIVqonE8

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

Posted: Wed Mar 15, 2017 4:33 pm
marsh8472 wrote:...
But if we had something like
11+11=1111
Can we still say 1111=1111? It depends on the definition of "=". Even if they look the same in every way detectable that does not prove that there's not something that makes them different in some undetectable way. Claiming they are the same because no way can be found to distinguish them or even no way exists to distinguish them is committing an ignorance fallacy.

I'd have thought in your example it'd depend upon what you mean by the colours rather than the equals symbol? But does it really matter as you appear to be asking what the symbol "1" stands for in the real world and I'd have thought if it stands for anything it stands for an object(I understand that there could be metaphysical objections about whether there are any objects but phenomenologically this appears undoubtable), so at base 1 is one object, 11 two objects and so forth, so 1+1+1+1=1+1+1+1 just says there are four objects and they are equal in number to four objects? So 2+2=4 is the same as two objects added to two objects is equal in number to four objects, as I thought in the main Mathematics was originally the language of number and shape. I thought symbolic numbers probably arose from writing and was used to help or improve the counting of objects so before a shepherd might well have just picked up different stones and for each sheep that went through to pasture a stone was assigned and on the way back a stone discarded(a primitive set method?) to count that all the sheep were there and it doesn't matter what size, shape or colour the stones are, number symbols are just a refinement of this process.

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

Posted: Wed Mar 15, 2017 6:24 pm
Arising_uk wrote:a shepherd might well have just picked up different stones and for each sheep that went through to pasture a stone was assigned and on the way back a stone discarded(a primitive method?) to count that all the sheep were there
A beautiful illustration of the idea of one-to-one correspondence, the basis of Cantor's transfinite set theory!

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

Posted: Fri Mar 17, 2017 5:15 am
This looks promising https://www.youtube.com/watch?v=aRUABAUcTiI http://mathworld.wolfram.com/SurrealNumber.html Like with the hyper-reals I have not looked into the surreal numbers before much either. But according to this, 0.9999... and 1 are different values within the Surreal number system.

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

Posted: Fri Mar 17, 2017 6:01 pm
Am I required to weigh in here? Surreals, ok.
marsh8472 wrote:This looks promising
Promising what? I confess I'm at a loss to follow your point. Promising a dramatic refutation of the previous two centuries of math? I'd say you've done no such thing. Perhaps if you stated a clear thesis or argument, we could analyze and discuss it. Posting links to red herrings doesn't help anyone. Not you, not the readers.
marsh8472 wrote: Like with the hyper-reals I have not looked into the surreal numbers before much either. But according to this, 0.9999... and 1 are different values within the Surreal number system.
The hyperreals are a nonstandard model of the reals that have been suggested as bringing pedagogical benefits in the teaching of calculus. That's a perfectly reasonable point.

But the surreals have no such nice properties. For one thing they are a proper class, not a set. They are "too big" to be a set. http://math.stackexchange.com/questions ... ned-in-zfc

So we can't even talk about them in standard (ZFC) set theory. We need different set-theoretic axioms to even have a mathematical conversation about them.

The expression .999... is not even defined in the surreals. http://math.stackexchange.com/questions ... equal-to-1

And, you can't do calculus with them since there's no theory of integration in the surreals. http://math.stackexchange.com/questions ... al-numbers

So as I say, I'm at a loss to understand their relevance to this discussion except as a further instance of the random linking of decontexualized factoids that don't actually bear on the topic at hand.

Here's the math.stackexchange search page for the surreals. Have fun. http://math.stackexchange.com/search?q=surreal

But hey, weird number systems? Check out the p-adic numbers, in which the expression ...999 (infinitely to the left) is defined and is equal to -1. The p-adics have relevance in computer science. The 2-adics for example are an infinitary version of 2-'s complement arithmetic. https://en.wikipedia.org/wiki/Two's_com ... ic_numbers

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

Posted: Sat Mar 18, 2017 3:20 am
wtf wrote:Am I required to weigh in here? Surreals, ok.
marsh8472 wrote:This looks promising
Promising what? I confess I'm at a loss to follow your point. Promising a dramatic refutation of the previous two centuries of math? I'd say you've done no such thing. Perhaps if you stated a clear thesis or argument, we could analyze and discuss it. Posting links to red herrings doesn't help anyone. Not you, not the readers.
marsh8472 wrote: Like with the hyper-reals I have not looked into the surreal numbers before much either. But according to this, 0.9999... and 1 are different values within the Surreal number system.
The hyperreals are a nonstandard model of the reals that have been suggested as bringing pedagogical benefits in the teaching of calculus. That's a perfectly reasonable point.

But the surreals have no such nice properties. For one thing they are a proper class, not a set. They are "too big" to be a set. http://math.stackexchange.com/questions ... ned-in-zfc

So we can't even talk about them in standard (ZFC) set theory. We need different set-theoretic axioms to even have a mathematical conversation about them.

The expression .999... is not even defined in the surreals. http://math.stackexchange.com/questions ... equal-to-1

And, you can't do calculus with them since there's no theory of integration in the surreals. http://math.stackexchange.com/questions ... al-numbers

So as I say, I'm at a loss to understand their relevance to this discussion except as a further instance of the random linking of decontexualized factoids that don't actually bear on the topic at hand.

Here's the math.stackexchange search page for the surreals. Have fun. http://math.stackexchange.com/search?q=surreal

But hey, weird number systems? Check out the p-adic numbers, in which the expression ...999 (infinitely to the left) is defined and is equal to -1. The p-adics have relevance in computer science. The 2-adics for example are an infinitary version of 2-'s complement arithmetic. https://en.wikipedia.org/wiki/Two's_com ... ic_numbers
0.1 < 0.2 < 0.3 < 0.4 < 0.5 < 0.6 < 0.7 < 0.8 < 0.9 < 0.99 < 0.999 < 0.9999 all have a units digit of 0, all are less than one. All real numbers with a units digit of 0 are less than one. 0.9999... < 1 It's over

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

Posted: Sat Mar 18, 2017 4:30 am
marsh8472 wrote: 0.1 < 0.2 < 0.3 < 0.4 < 0.5 < 0.6 < 0.7 < 0.8 < 0.9 < 0.99 < 0.999 < 0.9999 all have a units digit of 0, all are less than one. All real numbers with a units digit of 0 are less than one. 0.9999... < 1 It's over
'<' changes to '<=' in the limit. You need to get your money back from whoever taught you real analysis.

If you say it's over, is that your concession?

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

Posted: Sat Mar 18, 2017 12:49 pm
wtf wrote:
marsh8472 wrote: 0.1 < 0.2 < 0.3 < 0.4 < 0.5 < 0.6 < 0.7 < 0.8 < 0.9 < 0.99 < 0.999 < 0.9999 all have a units digit of 0, all are less than one. All real numbers with a units digit of 0 are less than one. 0.9999... < 1 It's over
'<' changes to '<=' in the limit. You need to get your money back from whoever taught you real analysis.

If you say it's over, is that your concession?
It's 99.9999...% over

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

Posted: Sat Mar 18, 2017 2:16 pm
This seems to cover the general discussion and points to where the confusions may lie.

https://en.wikipedia.org/wiki/0.999...

The skepticism part seems apt but it appears that one can have 0.999... less then 1 if one is prepared to give up some very useful parts of mathematics and even then this approach appears to have its difficulties.

All very interesting philosophically even with my remedial maths.

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

Posted: Sat Mar 18, 2017 5:34 pm
Arising_uk wrote:This seems to cover the general discussion and points to where the confusions may lie.

https://en.wikipedia.org/wiki/0.999...

The skepticism part seems apt but it appears that one can have 0.999... less then 1 if one is prepared to give up some very useful parts of mathematics and even then this approach appears to have its difficulties.
Could you quote the exact part you're asking about? We've been over the Wiki page and in particular Professor Katz's paper. Prof Katz is correct as far as he goes but his title is misleading and he admits in his paper that his point is pedagogical and not mathematical. .999... = 1 is a theorem in the hyperreals.

If there's something else in the Wiki article perhaps you can say what you are meaning to refer to.
Arising_uk wrote: All very interesting philosophically even with my remedial maths.
I don't think so (that it's interesting philosophically). It's interesting pedagogically. We don't teach math very well. And it's interesting psychologically. People want symbols to mean something beyond what they mean. Like burning a flag. It's a piece of cloth. People add meaning that's not inherent in the thing itself.

Philosophically I think it's a category error to say that .999 = 1 "means" something outside of whatever the symbols say it means. If you define the symbols as they are defined in standard math, then the statement is true. If you change the definition then you can make it say anything you want. The problem comes in when people think it's a meaningful statement about something other than the way the symbols are defined in math.

In other words, say we define the symbol ".999..." to mean the number 47. Since 47 isn't 1, the statement .999... = 1 is false.

If we define the symbols ".999...", "1", and "=" as they are defined in math, then .999... = 1 is a theorem provable from the axioms of math.

So in the end it's nothing more than a formal symbolic process. It doesn't necessarily mean anything at all. It can't be proved in a physics lab, for example, not with current technology and theory.

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

Posted: Sun Mar 19, 2017 9:22 pm
wtf wrote:...

Could you quote the exact part you're asking about? We've been over the Wiki page and in particular Professor Katz's paper. Prof Katz is correct as far as he goes but his title is misleading and he admits in his paper that his point is pedagogical and not mathematical. .999... = 1 is a theorem in the hyperreals.

If there's something else in the Wiki article perhaps you can say what you are meaning to refer to. ...
My apologies, I didn't see where you had alredy discussed this page. I was interested in the later bits on skepticism and the alternative number systems where it doesn't equal 1.
I don't think so (that it's interesting philosophically). ...
Sorry unclear, I meant I found the discussion between the two of you interesting.