# A natural strengthening of Kelley-Morse set theory

## Joel David Hamkins

### The City University of New York

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.

Professor Hamkins (Ph.D. 1994 UC Berkeley) conducts research in mathematical and philosophical logic, particularly set theory, with a focus on the mathematics and philosophy of the infinite. He has been particularly interested in the interaction of forcing and large cardinals, two central themes of contemporary set-theoretic research. He has worked in the theory of infinitary computability, introducing (with A. Lewis and J. Kidder) the theory of infinite time Turing machines, as well as in the theory of infinitary utilitarianism and, more recently, infinite chess. His work on the automorphism tower problem lies at the intersection of group theory and set theory. Recently, he has been preoccupied with various mathematical and philosophical issues surrounding the set-theoretic multiverse, engaging with the emerging debate on pluralism in the philosophy of set theory, as well as the mathematical questions to which they lead, such as in his work on the modal logic of forcing and set-theoretic geology.