User blog comment:P進大好きbot/New Issue on Traditional Analyses/@comment-34422464-20190828003038/@comment-39541634-20190828125702

That's actually a pretty good way to look at it.

The tricky part is that these building blocks must be manufactured somehow. Set theory manufactures these blocks, but set theory is also a language with a finite description and a finite set of symbols.

So the question remains, how does a finite language give rise to an infinity-sized building block?

The answer, sort-of, is that the building block we're working with is not exactly the real thing. It's like we're working with toy blocks that have ω's on them, rather than actual infinite sets. But this isn't accurate either, because these "toy blocks" are so realistic that the difference between them and the real thing breaks down. You certainly can't distinguish them by a crude measure such as size. It's not that the blocks are "finite" while the real thng is "infinite". The difference is far more subtle than that. It's like infinity is a perfectly ordered crystal, while our block is a slightly imperfect version of the same crystal. Same size, but it has some hazy parts which which represent the limit of what our finite concepts can "fetch" from the infinite world.