2013 seminar talk: Generalized side conditions

Talk held by Giorgio Venturi (Scuola Normale Superiore di Pisa, Italy) at the KGRC seminar on 2013-11-21.

Abstract

In this talk I would like present the method of generalized side conditions, first proposed by Neeman in 2011: a method that allows to give uniform consistency proofs for the existence of objects of size $\aleph_2$. Generally speaking a poset that uses models as side conditions is a notion of forcing whose elements are pairs, consisting of a working part which is some partial information about the object we wish to add and a finite $\in$-chains of elementary substructures of $H(\theta)$ (for some regular cardinal $\theta$) whose main function is to preserve cardinals. I will present in details the pure generalized side conditions poset and I will briefly present the poset that allows to force a club in $\omega_2$, the poset for forcing a Thin Tall Boolean algebra and the one for forcing an $\omega_2$ Souslin tree. In the end I will present a generalization of the combinatorial principle P-Ideal Dichotomty (PID) to ideals of uncountable sets, that I called PID$_{\aleph_1}$, sketching the consistency proof of one instance of PID$_{\aleph_1}$. If I will have time I will also discuss the possibility to generalize this method and its link with the problem of generalizing the Forcing Axioms.

References

[1] Itay Neeman: "Forcing with sequences of models of two types". Preprint.

[2] Boban Veličković and Giorgio Venturi: "Proper forcing remastered". In Appalachian Set Theory (Cummings, Schimmerling, eds.), LMS lecture notes series, 406, 331–361, 2012.