Everyone knows that there is a least transitive model of ZFC. Is the same true for second-order set theories? The main result of this talk is that the answer is no for Kelley-Morse set theory. Another notion of minimality we will consider is being the least model with a fixed first-order part. We will see that no countable model of ZFC has a least KM-realization. Along the way, we will look at the analogous questions for Gödel-Bernays set theory.

Slides

It’s well-known that there is a least transitive model of ZFC. $L_\alpha$, where $\alpha$ is the least ordinal which is the $\mathrm{Ord}$ of a model of set theory is contained in every transitive model of ZFC. With a little bit of effort, one can extend this to see that there is a least transitive model of GBC; its first-order part is $L_\alpha$ and its second-order part is the definable classes. Can we extend this result further to get that there is a least transitive model of KM? The purpose of this talk is to answer this question in the negative.

A model $M$ of ZFC is rather classless if every class of $M$ all of whose bounded initial segments are in $M$ is definable in $M$. In this talk, we will construct rather classless end extensions for every countable model of set theory. As an application of this construction, we will see that there are models of ZFC with precisely one extension to a model of GBC and that there are models of set theory which admit no extension to a model of GBC. If time permits, we will look at some related constructions with models of KM + the axiom schema of class choice.

Kelley-Morse (${\rm KM}$) set theory is one of the standard axiomatic foundations for set theory with classes as well as sets. Its defining feature is a strong class existence principle which states that any collection defined by a second-order assertion is a class. Choice schemes are choice/collection axioms for classes. The full choice scheme states for every second-order assertion $\varphi$ that if for every set $x$, there is a class $X$ such that $\varphi(x,X)$, then there is a single class $Z$ collecting all the witnesses, so that $\varphi(x,Z_x)$ holds, where $Z_x$ is the slice on coordinate $x$ of $Z$. The full choice scheme can be weakened in a number of ways. For instance, the set-sized choice scheme allows only set many choices to be made and the $\Sigma^1_n$-choice scheme restricts the complexity of $\varphi$. Study of the choice schemes dates back to the work of Marek and Mostowski from the 1970s. They have recently found application in nonstandard set theory with infinitesimals and analysis of properties of class forcing extensions. We show that even the weakest fragment of the choice scheme, where $\omega$-many choices must be made for a first-order assertion, may fail in a model of ${\rm KM}$ and that it is possible for the set-sized choice scheme to hold, while the full choice scheme fails for a first-order assertion. We argue that the theory ${\rm KM^+}$ consisting of ${\rm KM}$ together with the full choice scheme is more robust than ${\rm KM}$ because it is able to prove the Łoś Theorem for second-order ultrapowers and the absorption of first-order quantifiers in second-order assertions. We show that both these properties can fail in a model of ${\rm KM}$: the second-order ultrapower of a ${\rm KM}$-model may not even be a model of ${\rm KM}$ and a second-order assertion of complexity $\Sigma^1_n$ with a set quantifier in front may fail to have complexity $\Sigma^1_n$. This is joint work with Joel David Hamkins and Thomas Johnstone.