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Upcoming talks and events:
Definability versus Algebraicity
In a recent paper, Hamkins and Leahy introduce the concept of algebraicity in the set theoretic context. Thus, a set is algebraic in a model of set theory if it belongs to a finite set definable in the model. Clearly, algebraicity is a weak form of definability, and it can be varied in similar ways as definability, for example by allowing parameters. While the authors showed that the class of hereditarily ordinal algebraic sets is equal to the class of hereditarily ordinal definable sets, many fundamental questions on the relationship between algebraicity and definability were open: in particular, the question whether these concepts can be different in a model of set theory. I will show how to produce models of set theory in which there are algebraic sets that are not ordinal definable, and construct a model in which there is a set which is internally algebraic (i.e., which belongs to a definable set the model believes to be finite), but not externally.
Maximal Automorphisms
Given a model M, a maximal automorphism is one which fixes as few points in M as possible. We begin by outlining what the correct definition of “as few points as possible” should be and then proceed to study the notion. An interesting question arises when one considers the existence of maximal automorphisms of countable recursively saturated models. In particular an interesting dichotomy arises when one asks whether for a given theory T all countable recursively saturated models of T have a maximal automorphism. Our primary goal is to determine which classes of theories T lie on the positive side of this dichotomy. We give several examples of such classes. Attacking this problem requires a detailed understanding of recursive saturation, which we will also review in this talk.
A natural strengthening of Kelley-Morse set theory
I shall introduce a natural strengthening of Kelley-Morse set theory KM to the theory we denote KM+, by including a certain class collection principle, which holds in all the natural models usually provided for KM, but which is not actually provable, we show, in KM alone. The absence of the class collection principle in KM reveals what can be seen as a fundamental weakness of this classical theory, showing it to be less robust than might have been supposed. For example, KM proves neither the Łoś theorem nor the Gaifman lemma for (internal) ultrapowers of the universe, and furthermore KM is not necessarily preserved, we show, by such ultrapowers. Nevertheless, these weaknesses are corrected by strengthening it to the theory KM+. The talk will include a general elementary introduction to the various second-order set theories, such as Gödel-Bernays set theory and Kelley-Morse set theory, including a proof of the fact that KM implies Con(ZFC). This is joint work with Victoria Gitman and Thomas Johnstone.
The Humean Thesis on Belief
I am going to make precise, and assess, the following thesis on (all-or-nothing) belief and degrees of belief: It is rational to believe a proposition just in case it is rational to have a stably high degree of belief in it.I will start with some historical remarks, which are going to motivate calling this postulate the “Humean thesis on belief”. Once the thesis has been formulated in formal terms, it is possible to derive conclusions from it. Three of its consequences I will highlight in particular: doxastic logic; an instance of what is sometimes called the Lockean thesis on belief; and a simple qualitative decision theory.
Submodel Lattices of Nerode Semirings
$p$-adic model theory uniformly in $p$, and applications to algebra and number theory
In 1949, Julia Robinson proved that the field of rational numbers is undecidable. Later, in 1965, motivated by a conjecture of Artin on zeroes of p-adic forms, Ax and Kochen proved that the field of p-adic numbers is decidable. This enabled development of model theory for $p$-adic and related fields. These theories have turned out to have a surprising feature, namely, that most of the properties that hold do not depend on the prime $p$, and are in turn controlled by other universal theories. Later, Ax developed a model theory of finite fields for almost all $p$, and this theory beautifully relates to the theory of $p$-adics for almost all $p$. The uniform logical behaviour was then showed by Pas-Denef-Loeser to govern many properties of $p$-adic integrals uniformly in $p$, which enabled a theory of motivic integration, and it is is believed that this is a feature of many of the naturally occurring concepts and structures in number theory.
Recently, in continuation of this line of developments, new results have been obtained in the following topics:
1. On counting conjugacy classes and representations (and other counting problems) in algebraic groups over local fields (this is joint work with Mark Berman, Uri Onn, and Pirita Paajanen) where one can translate asymptotic questions in group theory formulated in terms of a generating Poincare series to questions on p-adic and motivic integrals approachable by model theory.
2. A model theory has been developed for the adeles of a number field (this is joint work with Angus Macintyre). The ring of adeles of the rational numbers is a locally compact ring made of all the $p$-adic fields for all $p$ and the real field and enables using results on the local fields to derive results for the global rational field. It is intimately related to questions on various kinds of zeta functions in arithmetic and geometry.
Going from the local ($p$-adic and real) fields back to the rationals has long been a fundamental local-global transition both for the logic and the algebra. The above results give new tools and results on this. I will give a survey of some of the results and challenging open problems.