On any storage system you will eventually run out of memory, so there will always be a maximum integer that you can write down with what you have, and we can only write down a vanishingly small subset of the natural numbers.

Of course, this isn’t what mean when they talk about “writing down” a number, but I’ve long thought that assuming the existence of natural numbers that you can’t write down is poetically apt.

In some sense, the set of natural numbers is “too big” to be about everyday finite numbers alone and it seems fitting to acknowledge that our usual axioms can’t exclude weirdo large finite numbers that are simply too big to ever reach.