Having a "different definition" of the reals doesn't make it correct, even if you tend to say it.

I would expect that most people with a passing knowledge of basic calculus would be able to eventually understand the argument that not only does 0.999... = 1, but that the real numbers are uncountable. It might take some convincing, but the truths are provable and very well understood across the world.

It's a definitional issue. "God created the integers. All else is the work of man." The theory of reals is a convenient abstraction defined by axioms. It's not created by construction.

There are not countably many real numbers. How can you claim your definition is correct?

There's nothing special about the definition. Use any construction you like, but do it over a countable model of set theory[1]. Now you have a countable set of real numbers, although there is no correspondence between it and the natural numbers within the model. The statement that 'there is an uncountable set' is also provable, although the set is countable. It's one of those Goedelian tricks, like the statement that is proven true but is unprovable.

That said, I prefer to say that only real numbers that can be emitted by computer programs exist.

[1] Use the Downward Lowenheim-Skolem theorem to get a countable set theory, and choose something like ZFC as your set theory.

I think you are confusing the object-level and the meta-level (and incidentally confusing everyone who doesn't know advanced set theory).

In any case, if you want to talk about the computable reals, then call them computable reals. Don't say "reals" for "computable reals", even if you think the former don't exist. Unicorns don't exist either, but that doesn't justify defining "unicorn" as "a horse".

Thanks for the interesting read.

I didn't understand anything on that page.

This is not the easiest thing to explain briefly, but let's give it a shot anyway.

There are several ways of defining real numbers, and one of them is the axiomatic definition. Real numbers are defined by a list of axioms they must satisfy. These would include, among others:

(1) If x and y are reals, then x + y = y + x. (2) If S is a nonempty subset of reals with an upper bound, then S has a least upper bound.

There is a crucial difference between these two. In (1) the variables x and y only quantify over reals, but in (2) the variable S quantifies over subsets of reals. We say that (1) is a first-order axiom and (2) is a second-order axiom. Actually, among all of the axioms of real numbers, (2) is the only one that is second-order. Therefore, it is natural to ask: can we rid of it?

No, we cannot. Löwenheim-Skolem theorem says that if we only have first-order axioms, then it is impossible to distinguish between countable and uncountable sets - even if we have an infinite number of first-order axioms. In particular, this means that if we try to define real numbers using only first-order axioms, then the definition cannot even capture the basic fact that there is an uncountable number of reals.

From here on, there are two roads you could take. If you're like me, then you just accept that real numbers cannot be defined using first-order axioms. By my standards, any definition that only uses first-order axioms cannot be a satisfactory definition of the real numbers.

But some people don't want to accept definitions that are not based on first-order axioms. And this is not as crazy as it might sound. First-order axioms are very nice from a theoretic point of view. For example, with first-order axioms it is absolutely clear what it means to prove something based on those axioms. With second-order axioms, the situation is a lot hairier.