The visualization is of the power set, which is uncountable.

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.

TREE(3) is unimaginably small, compared to ω

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.

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.

It&#x27;s a very interesting idea; if you want to learn more about it, look up &quot;ultrafinitism&quot;.

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.

&gt; The power set lattice (P(N)) of all sets of natural numbers, not to scale, some sets omitted...

What a great visualization!Now can your favorite LLM make me a similar one for the Real #s?

The lattice of sets of natural numbers is rich (2021)