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## Upcoming talks and events:

# 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.