This is joint work with Alberto Marcone and Antonio Montalb'an. Finite Ramsey Theorem states that fixed \(n,m,k \in \mathbb{N}\), there exists \(N \in \mathbb{N}\) such that for each coloring of \([N]^n\) with \(k\) colors, there is a homogeneous subset \(H\) of \(N\) of cardinality at least \(m\). Starting with the celebrated Paris-Harrington theorem, many Ramsey-like results obtained by replacing cardinality with different largeness notions have been studied. I will introduce the largeness notion defined by Ketonen and Solovay based on fundamental sequences of ordinals. Then I will describe an alternative and more flexible largeness notion using blocks and barriers. Finally, I will talk about how the latter can be used to study a more general Ramsey-like result.

Research in infinitary combinatorics has shown that the specific cardinals \(\aleph_0\), \(\aleph_1\), \(\aleph_2\), etc\(.\) exhibit distinct properties. One way of studying these distinctions is to examine to what extent these cardinals can be turned into one another by forcing. Bukovsky and Namba independently showed that \(\aleph_2\) can be turned into an ordinal of cofinality \(\aleph_0\) without collapsing \(\aleph_1\), and this forcing and its variants for other cardinals are now known as Namba forcing. In this talk we will show that it is consistent from an inaccessible cardinal that classical Namba forcing has the weak \(\omega_1\)-approximation property, answering a question of Cox and Krueger. The exact statement we obtain is similar to Hamkins’ Key Lemma and has implications for weakly guessing models. Time permitting, we will discuss implications for the study of successors of singular cardinals like \(\aleph_{\omega+1}\).

I will address the classification of different fragments of the Galvin-Prikry theorem, an infinite dimensional generalization of Ramsey’s theorem, in terms of their uniform computational content (Weihrauch degree). This work can be seen as a continuation of [1], which focused on the Weihrauch classification of functions related to the Nash-Williams theorem, i.e., the restriction of the Galvin-Prikry theorem to open sets. We have shown that functions related to the Galvin-Prikry theorem for Borel sets of rank \(n\) are strictly between the \((n+1)\)-th and \(n\)-th iterate of the hyperjump operator \(\mathsf{HJ}\), which corresponds to \(\Pi^1_1\)-\(\mathsf{CA}_0\) in the Weihrauch lattice. To establish this classification we obtain the following computability theoretic result (along the lines of [2] and [3]): a Turing jump ideal containing homogeneous sets for all \(\Delta^0_{n+1}(X)\) sets must also contain \(\mathsf{HJ}^n(X)\). Similar results also hold for Borel sets of transfinite rank. This is joint work with Alberto Marcone.

Bibliography

1. Alberto Marcone, Manlio Valenti,The open and clopen Ramsey theorem in the Weihrauch lattice,The Journal of Symbolic Logic,vol. 86 (2021), no. 1, pp. 316–351.
2. Carl G. Jockush,Ramsey’s Theorem and recursion theory,The Journal of Symbolic Logic,vol. 37 (1972), no. 2, pp. 268–280.
3. Stephen G. Simpson,Sets which do not have subsets of every higher degree,The Journal of Symbolic Logic,vol. 43 (1978), no. 1, pp. 135–138.

In this talk, we explore the question whether some notion of logical pluralism can be motivated on behalf of the existence of non-classical algebra-valued models of ZF(C). We argue that we can derive two notions of pluralism: a goal-oriented pluralism and a proper pluralism. The salient idea is that there exist several non-classical set theories that employ different underlying logics and which serve equally well for one particular goal: providing a foundation for mathematics.We examine the resulting notion of pluralism and discuss several objections against our proposal such as meaning-variance and the classical meta-theory objection.

We prove monadic second order limit laws for ordinals stemming from segments of some prominent proof-theoretic ordinals like \(\omega^\omega,\varepsilon_0,\Gamma_0,\ldots\). The results are based on a combination of automata theoretic results, tree enumeration theory and Tauberian methods. We believe that our results will hold in very general contexts.

Some results have been obtained jointly with Alan R. Woods (who unfortunately passed away in 2011).