MOPA

A model $(M, < ,\ldots)$ is said to be $\kappa$-like if $|M| = \kappa$ but for all $a \in M$, $|\{x \in M \mid x < a\}| < \kappa$. Based on the paper, the theory of $\kappa$-like models of arithmetic by R. Kaye, we will identify some axiom schemes true in such models of $I\Delta_0$ and investigate their interesting properties. ​

Peano’s original investigations into the foundations of arithmetic led to the observation that the axioms of PA are categorical for the Natural Numbers when treated in 2nd order logic. In this talk we will give a proof of Wilkie’s theorem which states that PA is in a sense the least such theory with this property. Arguably this justifies its significance in foundational studies. The proof uses the method of (L, n)-models, which I have previously talked about and I’ll discuss it’s use here as well.

The axiom of disjunctive correctness (DC) asserts that the truth predicate commutes with disjunctions of arbitrary size in the sense of PA. In this talk we follow an unpublished paper of Enayat and Pakhomov discussing the strength of the theory CT^{-}(PA)+DC, and in particular showing that it implies Con(PA).

In this talk, I will give some key definitions related to automorphisms of countable models of Presburger arithmetic, and summarize some results (such as quantifier elimination and DP rank) for an expansion by a specific modest automorphism. At my second talk, I will prove, or sketch proofs of, those results.

In this talk, I will give some key definitions related to automorphisms of countable models of Presburger arithmetic, and summarize some results (such as quantifier elimination and DP rank) for an expansion by a specific modest automorphism. At my second talk in two weeks, I will prove, or sketch proofs of, those results.

In 1989, Stuart Smith proved that for every full truth predicate T on a nonstandard model M of PA, there is a nonstandard formula that, according to T, defines an undefinable class of M. Recently, Bartosz Wcisło improved this by showing that there is also a nonstandard formula that defines an inductive partial truth predicate. The proof uses the technique of disjunctions with stopping conditions. I will discuss the proof, and, if time permits, I will also talk about the end extension problem for full truth predicates.

I will talk about a recent paper by Jim Schmerl which provides an alternative proof that every resplendent model of PA has a full truth class. The proof, surprisingly, boils down to studying kernels of digraphs. A kernel of a digraph is a set K such that for any a, b in K, (a, b) is not an edge, and for any a not in K, there is some b in K such that (a, b) is an edge. Schmerl shows that one can view the result that resplendent models have full truth classes as a particular instance of the result that resplendent digraphs which have local finite height have kernels.

We will continue discussing Shelah's technique of $(\mathcal L, n)$ models and their applications to independence results in PA. This includes a new proof of Paris-Harrington and an example of a concrete true but unprovable $\Pi^0_1$ sentence. Time permitting we'll also discuss the connection of these ideas to Wilkie's theorem on Why PA? and Kripke's notion of fulfillment.

We'll discuss Shelah's technology of $(\mathcal L, N)$ models and its applications to independence results in PA. This includes an alternative proof of the Paris-Harrington theorem and a sharpening to a $\Pi^0_1$ true but unprovable statement of some mathematical interest. Time permitting, we'll also connect these ideas to Kripke's notion of Fulfillment and Wilkie's theorem on Why PA.