Set Theory Seminar: Ur Yaar (HUJI)

Wed, 29/12/202114:00-16:00
Title: Iterating the cofinality-\omega constructible model
Consider C* - the inner model constructed in an L-like fashion, but using first order logic augmented with the "cofinality \omega'' quantifier. C* has some canonicity properties similar to L, but a notable difference is that C* does not necessarily satisfy the axiom V=C*, that is - constructing the C* of C* may result in a smaller inner model. If this happens, one can iterate this construction, taking intersections at limit stages and ask at what stage it stabilizes or "breaks" (i.e. that the result is no longer a model of ZFC). This type of construction was considered with regards to HOD, where it was shown by McAaloon, Harrington, Jech and Zadrozny that "everything is possible" - on one hand there are models with iterated HODs of any ordinal length (and even of length Ord), and on the other hand it is possible that after \omega many stages the intersection either satisfies ZF without AC or even doesn't satisfy ZF at all.
In this talk we will discuss iterating the C* construction, and show that under ZFC alone we can only reach finitely many steps, while a sequence of length $\omega$ is equiconsistent with an inner model with a measurable cardinal.
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