munchler6 hours ago
What a beautiful illustration. It makes intuitive the very abstract concepts discussed in the text. It’s fun to zoom in and browse around the structure.
michael0church6 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.
voidmain6 hours ago
The visualization is of the power set, which is uncountable.
michael0church6 hours ago
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.)
aeneasmackenzie2 hours ago
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.
__MatrixMan__2 hours ago
"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.
zaebal6 hours ago
TREE(3) is unimaginably small, compared to ω
tromp4 hours ago
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
zygentoma5 hours ago
Well, any natural number is unimaginably small, compared to ω …
nphardon2 hours ago
If the universe contains a finite amount of information, would that disprove the existence of an infinite set? I.e. if the representation of a number contained more information than the amount of information available in the entire universe.
benmandrewop2 hours ago
It's a very interesting idea; if you want to learn more about it, look up "ultrafinitism".
scythmic_waves3 hours ago
> The power set lattice (P(N)) of all sets of natural numbers, not to scale, some sets omitted...
flobosg7 hours ago
(2021)
genxy4 hours ago
math is timeless
flobosg2 hours ago
Blog entries, alas, are not.
gregw26 hours ago
What a great visualization!
Now can your favorite LLM make me a similar one for the Real #s?
stavros6 hours ago
Why can't yours?
MarkusQ3 hours ago
Nope.
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