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.
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.
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.
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.
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.
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.
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.
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.
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 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.
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.
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.
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.
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?
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.
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.
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... + ...))
It would get ugly pretty quick
If it makes you happy.
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.
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.
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.
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.
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?
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.
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?
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.
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.
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.
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.
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?
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.
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?
We can talk about abstract math next too.
We have been talking about abstract math. You don't seem to accept it.
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.
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?
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.
Knowing whether 0.9999... is the same as 1 is not useful in every day life.
Oh what a devastating argument. I'm humbled.
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.
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.
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.
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."
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.
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.