# Blog Archives

# Topic Archive: choice scheme

# Choice schemes for Kelley-Morse set theory

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.