Strong large cardinals as reflection principles
Large cardinal principles up to totally indescribable cardinals can be easily justified in terms of reflection. However, for measurable cardinals and above the situation is less clear. In this talk we will consider a form of reflection for classes of structures and will show that large cardinal principles, from the level of supercompact cardinals and up to Vopěnka's Principle, are indeed equivalent to natural forms of reflection.
Yurii Khomskii (Joint work with Jörg Brendle)
Polarized partitions on the second level of the projective hierarchy.
Polarized partition properties for sets of real numbers are variants of the classical Ramsey property which have come under attention recently, in the work of DiPrisco, Todorcevic, Zapletal and Shelah among others. Just like other regularity properties, they are true of all Borel and analytic sets, but the question whether this can be extended to the next level, i.e., the Δ12 and Σ12 sets, is independent of ZFC. We investigate the logical strength of the statements "the polarized partition property holds for all Δ12/Σ12 sets" and compare it to other well-known regularity properties on the second level of the projective hierarchy.
Peter Koepke (Joint work with Ioanna Dimitriou)
It is relatively consistent that the infinite cardinals are alternately regular and singular
In the classical Feferman-Levy model, the successor of ℵ0, namely ℵ1, is singular, since it is obtained by collapsing predecessors of some (larger) singular cardinal in the ground model to ℵ0. One can use any regular cardinal κ instead of ℵ0 and get a singular κ+. By a finite support product of class many Feferman-Levy collapses and moving to a symmetric submodel we obtain the situation described in the title.
Complexity of Ramsey-null sets
We show that the set of codes for Ramsey positive analytic sets is Σ12-complete. This is a one projective-step higher analogue of the Hurewicz theorem saying that the set of codes for uncountable analytic sets is Σ11-complete. This shows a close resemblance between the Sacks forcing and the Mathias forcing. In particular, we get that the σ-ideal of Ramsey null sets is not ZFC-correct. This answers a question posed by Ikegami, Pawlikowski and Zapletal.
Abstract: By a strong logic I mean a logic which is as strong as second order logic, at least in some inner models of set theory. I relate definability in set theory and definability in strong logics to each other by looking at various invariants related to logics, such as decision problem, Lowenheim number, Hanf number and the Delta-extension. I compare such invariants to large cardinals. I discuss sort logic, the ultimate strong logic. Finally, I discuss stronger versions of the invariants.
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