A natural strengthening of Kelley-Morse set theory, CUNY Logic Workshop, May 2014

This will be a talk for the CUNY Logic Workshop on May 2, 2014.

Abstract. 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 folklore fact that KM implies Con(ZFC). This is joint work with Victoria Gitman and Thomas Johnstone.

4 thoughts on “A natural strengthening of Kelley-Morse set theory, CUNY Logic Workshop, May 2014”

1. Will there be a ‘powerpoint’ presentation to go along with the talk? If so, will you make it available on a pdf file for general viewing?

• I don’t plan to use slides, but the paper itself will be ready soon, since we are currently working on it, and I’ll certainly make it available here.

2. By the way, what is that “certain class collection principle”, exactly? Also, can KM+ be interpreted in the positive theory $\GPK_\infty^{+}$ ?

• The class collection principle is the following:

If for all sets i there is a class A with phi(i,A), then there is a class B, consisting of pairs, such that for all i phi(i,B_i), where B_i = { x | (i,x) in B } is the i-th section of B.

This principle, which can also be viewed as a choice principle, is not provable in KM, and so KM+ is strictly stronger than KM. The formula phi here is allowed to be any assertion in the (second-order) language of KM.

I don’t know what GPK is, so I can’t answer the last part of our question.