Baby Coin!!!!!

The tender member of your lady love?

Then be assured its texture is unmatched

Now softened as it is by womanhood

KING: Thou dares impugn the honor of thy queen?!

Were we not so close I would have thy head!

FELLATIO: Tis futile. You may have my head but once,

Take the real numbers. There are numbers that are uncomputable, such as Chaitin's constant.

Take a standard theory, like ZFC. Say that a number x is describable¹ if there is a predicate P that is only true for x. Each reasonable number (1983, sqrt(2), π) is described by a predicate, but even Chaitin's constant is describable. However, as there is only a countable number of predicates, most numbers are not describable in ZFC.

(And yeah, because of Lövenheim-Skolem theorem, there exists a countable model of the first-order theory of real arithmetic. Note that it doesn't prove that R is countable. And I don't completely understand this theorem so I will not spend too much time on that.)

I have the idea that what "exists" is what can be written. Whatever the language we use (first-order logic, bytes in memory, natural language), we only use finite descriptions in a finite alphabet, so there are only a countable number of "existing" objects. The concept of non-describable existing objects is paradoxical to me. And yeah, I'm probably influenced by my computer science background, I'm used to think in terms of computability.

I've heard about finitism in mathematics, but does there exists a "countabilism" philosophy? The point of view that all math is essentially countable?

If you generate a countable model of ZFC (eg. take any model of ZF, take it's L, take its Skolem model) it is still a model of ZF. Therefore Cantors Diagonalization still holds internally. So while there was originally a bijection from the countable model (and in particular its reals) to the naturals, there is none in this countable model. You forgot it, too.

Similarly, if you only take the computable reals, which there are countably many of then if you had a enumeration of those you could still compute the diagonal +1 real, which is not in the list. Therfore the enumeration of the computable reals is not computable.

You can never escape uncountable cardinalities unless you give up enough constructive strength so you can no longer execute diagonalization. And diagonalization does not need a lot of constructive strength. At this point what you end up with has no longer any claim of being a foundational model of any sort.

If you want to learn more about the constructive strength required for diagonalization look up Yanofsky's Theorem or check out my pinned post.