Beyond that, you say: "all real numbers that you're likely to hear of" - this is a fallacy twice over.

First, since my lifetime is finite, it's easy to count all the real numbers that I will hear in my life. But I don't think that's the argument you want to make.

Second, there are people who professionally construct unlikely real numbers; just because you choose to ignore those numbers doesn't mean that your countable sequence of all real numbers can. There's a difference between "things that cannot be written down" and "things you can't write down".

It seems you didn't need to redefine the reals as programs to make this argument--the reals are classically defined as Cauchy sequences of rationals and (if the argument were valid) you could make the same claim using these sequences instead of programs.

I don't think the actual philosophical question is whether the reals are countable, it's whether the reals "exist". You can't define the reals such that they're countable because if you did, they wouldn't be the same structure.

I am fully aware of the constructionist approach to mathematics.

But your argument goes somewhat like this:

The reals are real

In computer programs however, we can only do so much

Every computer program represents a real number

For every bounded amount of bits, there is only a countable set of computer programs

Therefore, the reals are countable.

This is not how math works. You extrapolate from a bounded set that is countable to the collection of all those bounded sets - and then claim because every one set is bounded, the collection must be bounded as well - and therefore countable.

Let me make a similar argument just to show how silly your argument is. Here we go: 1. Every set of integers with only one element is finite. Call it a trivial set, for example: {1}

2. Every finite union of such trivial sets is finite

3. Every finite union of such a finite union of trivial sets is finite

4. and so on - after many iterations, your finite set will look pretty big to the human eye - almost as if it was the set of integers.

5. Therefore, the set of integers must be finite.

Sounds pretty silly, right? Thats what you've done for a different property of a different set.

That is nonsense. This is why arguing with illiterates about math is so annoying. They think they can write non-stringent arguments and then claim they have "somewhat proof". In math, there is no such thing as "somewhat proof". You are either right or you are wrong. The reals are the reals by definition. You can not make them countable.

What you can make countable is sets of numbers that you work with. Sure. You wont ever create a computer that has access to all the reals, PRECISELY because it operates from a position of countable-ness (even quantum computers). But thats not what I care about when I talk math.

The real numbers are not accessible to in a "reality" way. They lie fully in the realm of arcane mathematics. Like many other things do. You would never claim that the hyper-exponentiation operator had anything to do with reality, and it doesn't. Neither do the reals. You're just confused about them because they appear so prominently in middle school mathematics.

Accept it and move on.

Mathematics is the realm of absolute truth. Everything we have proven is de facto true. You can make an argument that we are doing the "wrong kind of mathematics". And that may actually be true. But it doesn't change anything about our discipline. If you want to get philosophical and create a new Mathematics discipline based on different arbitrary rules (axioms), you are free to do so. But you will have to prove a lot of very mundane things and go through centuries of early day fuckups and its very likely that you will soon seek to merge with the already established mathematics.

Call it math2.0. I'll play with it. If its interesting, why not. But don't expect your new math to be more interesting than the one we've got.