Wiesław Kubiś - Generic Mathematical Structures

May 24, 2026

A mathematical object can be called “generic” if it appears, up to isomorphism, with probability one as the result of a natural stochastic process. Instead of [using] probability, one may [give] its topological counterpart, using the Baire category theorem. Yet another option is using a natural infinite game for two players, declaring an object U ”generic” if one of the players has a suitable winning strategy leading to the isomorphic copy of U....

... The story of generic mathematical structures goes back to Cantor, who was the first to identify the set of rational numbers as the generic countable linearly ordered set. About half a century later, Fraïssé developed an abstract theory of universal homogeneous structures (nowadays called ”Fraïssé limits”) which until recent years was viewed as a part of model theory. As it happens, Fraïssé limits are particular cases of generic mathematical objects which can be found in several branches of mathematics, starting from model theory, algebra, functional analysis, and geometric topology. We will try to explain why pure and enriched category theory is the suitable language and framework for studying these objects.

So who's afraid of stochastic processes now?

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My comment:

@computablesecrets This sounds interesting! I don't think there are any good talks on ordinal logics on Youtube. But isn't first order PA a non-finite axiomatic system? I think the usual completeness proofs like Gödel's use an axiom schema for induction indexed by natural numbers. Or do you mean something else by non-finite axiom system?

10:27 But this is the thing: BB(6) may be well-defined, but it is far from well-known. So using the expression BB(6) to denote some definite number is not actually meaningful. The same applies to the other terms of the series. So how is this different to the Berry paradox? The trouble is, as you have pointed out, that as the lengths of strings grow, the number of sequences grows exponentially, so if you try to intensionally define a function that picks out any of them, then you can't do better than just listing their digits. So whatever the set of intensional functions we construct (call them finite ordinal notations) to specify an injection into Nat, once the rate of growth of the terms is sufficiently high the notation ceases to produce meaningful terms in the sense that we don't know how to translate the terms from one notation to another, except in a vanishingly small proportion of cases where we can construct endomorphisms on some subset of Nat. See that talk by Wiesław Kubiś I mentioned on your Random Graphs video for some ideas about these sorts of constructions on rationals etc. There is a whole field of intensional type theory that is about these sort of constructions. Steve Awodey has done some recent talks about it. In case you're still reading, Jade Tan Holmes (Up and Atom channel on YT) is making a video about Goodstein's Theorem and is looking for fact checkers, in case you're interested. Goodstein's theorem was one of the first written down that couldn't be proved in first-order PA. It's all the same, once you have the right isomorphism class.

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See Dana Scott's 2014 paper Stochastic λ-calculi: An extended abstract which changed the world, but nobody seems to have noticed yet:

We would also like to say that computability notions can also be inherited from P(N). This is true, but we have to take some care, because a particular space T might have several homeomorphic embeddings into P(N) by different choices of bases giving us different notions of computability. Usually, in the author’s experience, there is a preferred embedding, however, and then the notions of computability are well behaved and have desirable properties as expected.

From 2020: