I have thought about this, but it isn't entirely clear to me. For example, what is the bit encoding of the real numbers? Is it really accurate to say we have an encoding of them by just saying we have the set R?

> But we can't enumerate those sets.

Well, that's kind of the point of my question. Is this not an issue?

> We can represent these sets of cardinality infinities of infinities in our brain and on paper, but representing these sets with all the available bits in the universe is not possible. This seems like a paradox, because our brains and paper are part of the universe.

Eh, I can write a recipe that calls for multiple universes-worth of flour and 600 trillion years in the oven, and I can write that recipe in a minute on an index card that weighs a couple grams. I can write small programs that loop infinitely or (attempt to) use infinite memory. I can write a song that repeats ad infinitum that nobody can ever sing to completion in the entire lifetime of the universe. That's not much different than the words "the Integers" requiring 2 words but describing something that could not possibly be "fully materialized" in any finite universe. Who cares?

What those all have in common is that despite being very big or long in some sense, they have low Kolmogorov complexity. It's very easy to describe "recipes" for infinite things that would exceed the bounds of the universe if they were ever fully realized; you don't even need higher cardinalities for that.

https://en.wikipedia.org/wiki/Kolmogorov_complexity

What higher cardinalities do tell us is that there must be some real numbers that have infinite Kolmogorov complexity, and in fact 100% of the real numbers have this property. (Of course all the numbers you have/will ever have heard of are in the remaining 0%.) There are only countably many (the smallest infinity) descriptions in any language (programming or natural), so most reals must not be "describable" at all. We will never be able to meaningfully use any of those numbers, because we couldn't possibly "compute" them -- we can't run the infinitely long program. It's not clear to me that we'd ever be able to learn anything at all about such numbers, or indeed that they "exist" in any meaningful sense.

Mathematics is a human activity, and humans are part of the universe.

> Math is bigger than our universe.

What does that mean?

> If our universe were completely different, math would still be the same.

That is more than debatable.

> Eh, I can write a recipe that calls for multiple universes-worth of flour and 600 trillion years in the oven, and I can write that recipe in a minute on an index card that weighs a couple grams. I can write small programs that loop infinitely or (attempt to) use infinite memory. I can write a song that repeats ad infinitum that nobody can ever sing to completion in the entire lifetime of the universe. That's not much different than the words "the Integers" requiring 2 words but describing something that could not possibly be "fully materialized" in any finite universe. Who cares?

I think you're skipping over the point. No one is going to argue with you about "multiple universes-worth of flour" existing because to our current knowledge, the universe is finite and thus there is a finite maximum to how much flour can exist. My question revolves around the philosophical question of the existence of mathematical objects and whether they can or cannot be actually realized with the given information in the universe. This is not a question out of left field. The acceptance of the real number continuum is not even settled.

Thanks for the reference to Kolmogorov complexity.

What do you mean by that?

The "paradox" arises because you're using the word "represents" in two different meanings.

For example, the number TREE(3) is extremely big (to put it mildly.) If you haven't heard of it you can search Youtube, there's a few Numberphile videos on it that are wonderful.

Anyway, TREE(3) is extremely big but finite. I can explain exactly what TREE(3) is. You can understand what it is. I can therefore "represent" it, and you can have an exact picture of your head of how to write a computer program that calculates it.

It is, of course, much too big to "represent" it by writing down all its digits in base 10. There's not enough bits in the Universe.

But these are two different senses of the word "represent".

In a similar way, I can "represent" sets of infinites very easily, in the sense that I can explain to you what they are and how they behave. I mean, every schoolchild knows about a set that's infinite - just the regular old counting numbers. The set of counting numbers if infinite - there is no way to actually write down all of these numbers, within our universe.

But I can still explain to you what that set is, and how it behaves. Everyone understands that the behavior of it is that you can keep pulling out more numbers from it. That's all that "representing" it is in this context.