Generalized side conditions

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\(_\), sketching the consistency proof of one instance of PID\(_\). 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.