> How does this work then? We can represent these sets of cardinality infinities of infinities in our brain and on paper, but representing these sets with all t

> How does this work then? 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. 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.