Real numbers are defined as an equivalence class such that if the differences of two infinite sequences of rationals tend toward zero, then they are equal. The difference between 0.999... and 1.000... clearly tends towards zero as it heads of to infinity, and so they are equal.
If you want to argue that it doesn't then you have to come up with some other definition for numbers which have an infinite decimal expansion.
(Technically, of course, 1 is a rational number, but if you're using 0.9999... to represent it, you're using a real number representation, so you're bound by the definition)
> Real numbers are defined as an equivalence class such that if the differences of two infinite sequences of rationals tend toward zero, then they are equal. The difference between 0.999... and 1.000... clearly tends towards zero as it heads of to infinity, and so they are equal.
> If you want to argue that it doesn't then you have to come up with some other definition for numbers which have an infinite decimal expansion.
> (Technically, of course, 1 is a rational number, but if you're using 0.9999... to represent it, you're using a real number representation, so you're bound by the definition)
I'm not sure why you think I don't know the difference between rational and real numbers, but I assure you I do. What I said was I don't see how a proof involving the standard arithmetic operations found within the rational numbers, but not including any concepts of limits, completeness, etc. is invalid. Let me know if you still don't understand my point.
I think you have mis-stated yourself. Either that, or I don't understand what you're saying.
Trying to rearrange it and remove as many negatives as possible, I started with your statement:
> I don't see how a proof involving the standard arithmetic operations found within the rational numbers, but not including any concepts of limits, completeness, etc. is invalid.
I think what you mean is that any proof that does not use the concepts of limits and completeness is going to be invalid.
That seems clear to me, the reason being that one needs to define what one means by the sequence of symbols "0.9999...".
You can say "It's infinitely many 9s stretching off to the right", but that doesn't tell me what it means.
People seem to think it does, but when I dig deeper, they usually don't have any sense of what it means. And therein lies the problem (as I see it). People blithely write the glyphs, but don't have a concrete interpretation.
My personal experience is that people want to argue from intuition about what 0.9999... means, and when you try to make it precise they say that it's obvious. Then they derive all sorts of nonsense and conclude that mathematics is all rubbish.
If someone really wants to understand it then I'll explain current mathematical thinking, including non-standard analysis and the surreals. But most people don't want to put in the work to understand how these issues have been resolved, and just want to argue from their intuition.
Someone else here brought up the surreal numbers and my intuition says that it's right to do that. The various arithmetic proofs thrown around here don't explicitly make use of completeness. As such they should be correct proofs in the surreal numbers as well. But they basically are not. Here is a blog post about it:
I don't quite know how to formalize it, but I'm pretty certain that if these proofs logically worked (in the "theory of proofs sense"), then they should work in the surreals as well.
Anyway it's just intuition. My main point in this thread is that I don't really accept the proofs of this that don't use completeness as a step. Though I do suspect that proofs not making use of it are actually incorrect proofs in their own right. If I were curious enough I'd think back about formal proofs and models and all that jazz, but I probably already have spent more time in this thread than I should. :)
edit: The more I think about it I feel like someone actually explained to me this (i.e. why this proof is wrong using surreals as reasoning) a long time ago and I'm just remembering echos of it in my mind. Wish I could remember something more useful...or that I were a logician...
Real numbers are defined as an equivalence class such that if the differences of two infinite sequences of rationals tend toward zero, then they are equal. The difference between 0.999... and 1.000... clearly tends towards zero as it heads of to infinity, and so they are equal.
If you want to argue that it doesn't then you have to come up with some other definition for numbers which have an infinite decimal expansion.
(Technically, of course, 1 is a rational number, but if you're using 0.9999... to represent it, you're using a real number representation, so you're bound by the definition)