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

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.

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 "". 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.

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.

  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|