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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’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.)
TREE(3) is unimaginably small, compared to ω
Well, any natural number 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

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.
Joel Hamkins is my new favorite mathematician :-)

There is a beautiful application of the power set of the Naturals to denotational semantics. I assume Prof. Hamkins's book will cover that topic, but it is not mentioned in the linked article.

Dana Scott (and apparently Gordon Plotkin independently) came up with a clever way to create a model of the lambda calculus that employs the power set of the Naturals.

The problem is that in lambda calculus, the formal language permits every expression to appear in the left-hand slot of the "Function_Application" operator. I.e., every term is simultaneously permitted to be given as an argument to a function, and also to be used as a function.

So we have the conundrum of finding some set "S" where every element of S is a function (not a problem so far), BUT, those functions all take elements of S as inputs and produce elements of S as outputs. So we need a set S that is isomorphic to the set of functions "S -> S". Cardinality arguments show that this is not possible: the function space for any non-trivial set has greater cardinality than the set itself.

So, Scott and Plotkin devised a "computationally sensible" way to interpret an arbitrary set of integers as a function over sets of integers.

By standard encodings, interpret any integer "n" as an ordered pair "<M,u>". Now, again via standard encoding, interpret M as a finite set of integers M_set.

The "function" defined by a singleton set { n } applied to some other set Q is: {u} if M_set is a subset of Q, Empty_set otherwise.

Then for a set of more than one element, take the union of the outputs of each of the elements applied to Q as above.

One can define a topology over the power set of the Naturals, and the above functions turn out to be exactly the continuous functions relative to that topology. The continuous functions so defined have the same cardinality as the power set of the Naturals.

An interesting historical side-note: historically, mathematical structures were the starting point, and axiometized formal languages a la Frege, Russell & Whitehead, etc. were built later. In the case of lambda calculus, it was the other way around: the formal language came first, and it was a multi-decade riddle what actual mathematical structure (if any) this formal language actually described.

> The power set lattice (P(N)) of all sets of natural numbers, not to scale, some sets omitted...
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.
Not really. Math uses no physical observation, only axioms. Nothing can "prove" or "disprove" axioms. However, if observation supports the axiomatic theory, then we use the theory for physical prediction. If observation doesn't, then we don't use the theory. Does that count as "disproof"?

In practice, infinite sets never exist as enumerations of every element, but as ways to generate more elements along with descriptions for which elements to include. Infinite set theories allow for equivocating a finite description with the infinite enumeration. In contrast, programming languages usually make a distinction between data (always finite) and data generation (possibly infinite). I would think that counts as a "disproof" in a way.

This was the answer I was looking for. You're reminding me that existence of infinity is an axiom, like parallel lines dont intersect. We can remove that axiom and get a new mathematical framework, but by their nature, axioms are not something to prove or disprove.
Exactly. For example, John Mayberry wrote "The Foundations of Mathematics in the Theory of Sets" (2000). Half of the book consists of philosophical arguments for his "Euclidean set theory" contrasted against the big bad "Cantorian set theory". He takes inspiration from Euclid's common notion 5 "the whole is greater than the part". On page 277, formula 8.3.1, his Axiom of Euclidean Finiteness goes like this: any injective endofunction is also surjective, ∀f∀Y((f:Y→Y ∧ 1to1(f)) ⇒ onto(f)).

I've come to believe that many related incompatible theories have interpretations between each other. For example, hyperbolic geometry has a Euclidean-like Poincare disk model, and Euclidean space exists locally in a hyperbolic space. Boolean logic contains intuitionistic logic (just add the law of excluded middle), but intuitionistic logic contains Boolean logic through the double negation translation. Similar might happen for finite set theories, infinite set theories, and neutral set theories. The fun includes finding the right translation so that we can all enjoy our different tastes in axioms.

There exist subsets of the natural numbers which are infinite, but which are not finitely definable in first-order arithmetic.
For example? And does "first-order arithmetic" mean ZFC?
I got this example from an LLM:

  1. Fix a formal system S. In the LLM example, it uses first-order arithmetic, but I don't see why we wouldn't be able to use ZFC.
  2. Let D be the set of subsets of the natural numbers N which are definable by a finite formula in S.
  3. There are countably many finite formulas, so |D| <= |N|.
  4. Cantor's theorem says that the size of the power set of N is greater than |N|.
  5. Therefore there must be subsets of N which are not definable by a finite formula in S.
If you disagree with this, I would be interested to know.
Not happy to respond to LLM talk, but you seem interested anyway. Some sleight of hand happens between "fixing a formal system" and using Cantor's theorem for the metamathematical analysis, as if we use classical set theory anyway. Note that you cannot construct any particular example of a non-definable set, which should cast doubt of existence. I'll disagree by pointing to anti-classical set theories. The axiom of infinity proves independence from ZFC, so I can freely replace the axiom of infinity with its negation, then the natural numbers no longer form a set. Some constructive analysis systems include an axiom that every real-valued function is continuous (as discontinuous functions are undecidable).

https://en.wikipedia.org/wiki/Axiom_of_infinity#Independence

https://en.wikipedia.org/wiki/Constructive_analysis#Anti-cla...

You’d have to define what you mean by “existence” here. Clearly, there are infinite sets we can represent with a finite sequence of symbols. We can also imagine and reason about alternative universes with an infinite amount of information. You’d have to argue about how doing so would somehow be an incorrect thing to do.
It's a very interesting idea; if you want to learn more about it, look up "ultrafinitism".
(2021)
math is timeless
Blog entries, alas, are not.
What a great visualization!

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

Why can't yours?
It was a joke/tease question.

I have yet to be able to create diagrams or visualizations I am happy with with LLMs. I can get a diagram, but tweaking it via prompts is extremely painful.

I do think it's a interesting moderately straightforward test case for an LLM that is beyond today's frontier.

Nope.