Strong second-order set theories do not have least transitive models
Shepherdson and Cohen independently showed that (if there is any transitive model of ) there is a least transitive model of . That is, there is a transitive set so that and if is any transitive model of then . We can ask the same question for theories extending . For some fixed set theory , does have a least transitive model?
I will look at this question where is a second-order set theory. Two major second-order set theories of interest are Gödel–Bernays set theory and Kelley–Morse set theory . The weaker of the two is , which is conservative over while is much stronger.
As an immediate corollary of the Shepherdson–Cohen result we get that there is a least transitive model of . The case for is more difficult and indeed, has a negative answer. I will show that there is no least transitive model of . Along the way we will build Gödel’s constructible universe above sets and into the proper classes, unroll models of second-order set theory into first-order models, and dip our toes into Barwise theory and the admissible cover. Time permitting I will mention some results and open questions about + Elementary Transfinite Recursion, which is intermediate between and in strength.