Comment by michael0church
6 hours ago
It’s also genuinely surprising. We’re used to thinking of the countable as the small infinity, which it is, and yet a structure we feel like we can visualize contains so much complexity.
There is also, weirdly, a way in which massive finite numbers like TREE(3) “feel” larger than N, and large countable infinities “feel” larger than w_1, even though the opposite is clearly true.
The visualization is of the power set, which is uncountable.
Right. But because it’s the smallest structure of its type (speaking loosely) it feels like something we should have a grasp on, even though it contains more complexity than we could ever describe or compute with (since both of those are countable.)
All describable or recognizable complexity is part of the subcountable set of computable subsets of N. Higher infinities thus mostly contain fake elements about which nothing can be said, so they don’t feel any bigger.
"Fake elements," feels right to me. They're allegedly in there but we can't find any of them. It's funny that these elements comprise the majority of the "real" numbers.
TREE(3) is unimaginably small, compared to ω
TREE(3) is also unimaginably tiny compared to the normal form size of (λa.aaa(λbλcλdλe.ebbbcde)aaaa)(λfλx.f(fx)) [1].
[1] https://wiki.bbchallenge.org/wiki/Lambda_Calculus#Champions
Well, any natural number is unimaginably small, compared to ω …