This is a simple argument. (1/3 expressed as a decimal to n places) x 3 is a nice, clean 1.0 for any real number of n.
No, it isn't. 1/3 to n decimal places is 0.3...3, with n 3s. That times 3 gives you 0.9...9, with n 9s. Not 1.0.
My whole point is that infinity isn't a real number,
Nobody said it was...
therefore 1/3 x 3 expressed as .9r isn't real.
No, this doesn't follow. Infinity doesn't need to be a real number for 0.9r to be a real number. Here's another decimal expansion: 1.0r. Is that not a real number? If it is, then why doesn't your logic apply in this case? Even better, try it with the sequence 0.3, 0.33, 0.333,... Are you really going to claim that 1/3 isn't real?
Try it at n = 3. Try it at n = 12341. Try it at n = 2341234823469812634. It is always 1.0.
Except that it's
never 1! This is, in fact, the whole
point of completion: to also have the limits of such sequences in the set!
My whole point is the way that limits require you to assume that infinity is a number you actually get to.
I think you need to learn more about limits in a formal context, to be honest.
I'm starting to grasp the fundamental disconnect.
<snip>
Real doesn't mean possible to exist.
Ah, see, now I think we're getting to the heart of the matter. Unfortunately, AlphaNumeric (or Guest, I can't remember) was absolutely right: You need to learn what the word "real" means in mathematics. "Real" has a very specific meaning, and your intuitive (not intuitionist) idea of what it means is simply wrong. Furthermore, what you actually seem to be getting at is actually the
philosophy of mathematics, where one can ask such questions as: What connection does the mathematical concepts have with reality?
However, that has
absolutely no impact on the hard, provable fact that 0.9r = 1.
Oh, and you definitely need to think your examples through a bit more.
Thus, a logically impossible number, such as the largest member of an infinite set is "real" to a classical mathematician and its logical impossibility doesn't make it not real.
This is completely wrong, and shows that you have pretty much no idea what you're talking about.
First of all, it is entirely possible to have a largest member of an infinite set, in a logically possible manner that both intuitionist and "classical" mathematicians will agree on. Consider the set {0,-1,-2,-3,...}. This is an infinite set. However, under the standard ordering of the integers, is has a largest element, 0, but no least element. Or the set $$\{ x \in \mathbb{R} \,|\, 0 \leq x \leq 1\}$$, which has both, under the standard ordering of the reals.
Secondly, if you can show the logical impossibility of a largest element of an infinite set, then
no "classical" mathematician would argue that such an element existed. In fact, proof by contradiction is a central tool in almost all branches where mathematics is used.
Third, you have your schools completely reversed: It is the
intuistionists, not the "classical", mathematicians that reject the law of the excluded middle, reductio ad absurdam etc. Saying that "classical" mathematians will not accept logical impossibility as proof is simply, and embarrasingly, given the context, wrong.
Just a gentle (or not-so-gentle) reminder that a bit of modesty is a good idea before you go putting other, far more knowledgable, members down. Especially in fields you admit to being a novice in.