I cannot believe though, how a simple question as whether 0.999... is equal to 1 has elicited over 500 responses.

technically as i said earlier it is a finite set. but i can't think of the upper bound on it, who knows how long a mathematician's paper can be, so i just treat it as infinite. but what i'm saying is that including all those undefinable numbers does nothing. and they should never have been included in the real line since you can't describe them since they don't exist in our universe.

semck is the advanced math major here, he can do the arguments.

First of all, I don't even know what it means to say a number "exists in our universe."
Second, I've already pointed out that including the reals does a LOT, like get you calculus.

i am using the phrase exist in our universe to describe everything possible you could ever imagine. because when i said definable you wanted more. so i'm saying put in whatever else you want. put all that together. but obviously your numbers have to be distinguishable or at least that each number only has a finite number of copies of itself. if you include infinite copies of numbers you can get to a set of uncountable size, i guess. but then you wouldn't be describing a set of unique numbers.

OK, bolshoi. Well, none of this is precise to the level necessary when discussing mathematics carefully. I can imagine an uncountable set by defining it. Obviously, I can not then imagine either an uncountable or even a countable number of distinct individuals in the set. But I CAN prove that there are uncountably many distinct individuals.

And so on. Anyway, I have actual math to do, and this discussion is degenerating into complete non rigor, so I'm off for a good while.

you can claim a set is uncountable, but then you will only be able to use countably many of the elements of it. leaving 100% of the set unused.

What do you mean by "use"?

describe, for one thing.

the reason that i don't think you need uncountable reals is that we can't distinguish the difference in continuity between the real line and the rational line. so that increase in continuity people keep talking about is nonsense. all that extra continuity cannot be used.

Uh huh.
We can't "distinguish" it? What does that mean? We can't pick it up and run our hands over it? Guess what, Einstein, you can't touch numbers.

We sure as heck can distinguish it mathematically. For example, every Cauchy sequence of real numbers converges, whereas there are uncountably many distinct Cauchy sequences of rationals that do not converge. This property of the reals -- analytic completeness -- gets used _constantly_ in analysis. You say we never use them. Guess what? I use uncountably many of them all at once every time I prove almost anything in analysis, because without the property of completeness, my proofs and the proofs of most major theorems in analysis would fall completely apart.

So yeah, you have no idea what you're talking about, I'm afraid.

0.999999999999... < 1

so its not equal.

^ lolololololololololol, omg so funny and ironic

" there are uncountably many distinct Cauchy sequences of rationals that do not converge." no. what i'm saying is that there are only countably many.

if your cauchy sequences are defined with finite formulas, then you can only have countably many. you would need infinite-length formulas to get uncountably many.

The problem is, you have to worry about Cauchy sequences that might not be defined by a formula at all -- arbitrary Cauchy sequences.

it's what i said earlier, any cauchy sequence you will ever run into fits into this countable set. you can tell yourself that you're moving through a numberline and passing through all these undefinable numbers that must exist, but there's no evidence you are, since you can never see those numbers.

lol.
Any Cauchy sequence "I will ever run into"? I run into ARBITRARY Cauchy sequences whenever I use Cauchy sequences in an analytic proof, and I can PROVE that there are uncountably many Cauchy sequences that those sequences could be.

And again, there's no "evidence" that there are infinitely many integers, if you're just talking about what we could see in our lifetime.

i can prove that there are uncountably many cauchy sequences that those sequences COULD be.
yeah, could. well i can prove that there are uncountably many names i can use for bacon, so i guess there are uncountably many names in this world. just ignore the fact that infinite-length names don't exist.

i'm not sure if this link will be helpful to you or not. probably not though since i don't think anything will convince you that we only ever deal with countability in our universe. http://math.stackexchange.com/questions/86357/does-an-uncountable-discrete-subspace-of-the-reals-exist

How much "time" would it take to count all the Reals between 0...1 ?

even if you could count all the integers in one second, it would still take infinite amount of time to count those reals.

@bolshoi, The link is interesting, but completely irrelevant. Do you even know what a discrete subspace is? I have to wonder, since you think this is relevant.

Also, your last statement is absurd. You could NOT count all the integers in one second, however fast you went, so, the comparison is ridiculous.

... you could not count the integers in one second, so lets not figure out what would happen if you could? if you're against making assumptions about things that can't exist why do you even believe in uncountable numbers? if my statement has no merit, then neither does cantor's proof.

you need discrete numbers if you want to count them

you can only ever do anything with discrete numbers. even your continuous functions, you have to evaluate them somewhere. and there are only countably many places where you can evaluate them.

@bolshoi, Well, the thing is, if we're going to assume you could count the integers in a second, why not assume you could count the reals in a second, too? I mean, if we're making up fantastic worlds, where does it stop? One big difference between your little "thought" experiment and Cantor's proof was that Cantor's proof was a proof.

As I suspected, you don't understand the site you linked. It is about discrete subsets, not discrete numbers (whatever those are). Discrete subsets are subsets with no accumulation points, meaning every point in the set has a neighborhood whose intersection with the set is only itself.

The rationals, for example, do not form a discrete subset. In fact, they are the opposite, a dense subset. So, I'm not sure where that leaves your point about "discrete numbers," but anyway.

oh well... if rationals are not discrete then maybe i did confuse that. but cantor's proof is not different. because the size of the integers is not defined, is it? so by your genius logic, you can't say reals are larger than integers because they are both infinite in size, so have no size.

which, coincidentally, points to there being no such thing as uncountability.