arXiv subjects
Publications and source records attributed to Boban Veličković.
We give a game-theoretic characterization of when a model of an infinitary propositional formula can be added by a proper, semiproper, and stationary-set-preserving poset. In the latter case, we also give a general sufficient condition for the existence of such a poset. We use this condition to give a somewhat different proof of the theorem of Asperó and Schindler, which states that $\mathsf{MM}^{++}$ implies Woodin's axiom $(*)$.
By a virtual model, we mean a model of set theory which is elementary in its transitive closure. Virtual models are first used by Neeman \cite{neeman2014forcing} to iterate forcing. That paper is concerned with proper forcing. The method was then adjusted by Veličković to the case of semi-proper forcing and this was drafted in \cite{velickovic2021iteration}. We here straighten the details and further elaborate on Veličković's method. The first section collects facts about virtual model, the second section describes the iteration, and the third one illustrates the method in the case of getting saturation of $\mathsf{NS}_{ω_1}$ (loosely relying on \cite{schindler2016nsomega1}).
A theorem of Galvin asserts that if the unordered pairs of reals are partitioned into finitely many Borel classes then there is a perfect set P such that all pairs from P lie in the same class. The generalization to n-tuples for n >= 3 is false. Let us identify the reals with 2^omega ordered by the lexicographical ordering and define for distinct x,y in 2^omega, D(x,y) to be the least n such that x(n) not= y(n). Let the type of an increasing n-tuple {x_0, ... x_{n-1}}_< be the ordering <^* on {0, ...,n-2} defined by i<^*j iff D(x_i,x_{i+1})< D(x_j,x_{j+1}). Galvin proved that for any Borel coloring of triples of reals there is a perfect set P such that the color of any triple from P depends only on its type. Blass proved an analogous result is true for any n. As a corollary it follows that if the unordered n-tuples of reals are colored into finitely many Borel classes there is a perfect set P such that the n-tuples from P meet at most (n-1)! classes. We consider extensions of this result to partitions of infinite increasing sequences of reals. We show, that for any Borel or even analytic partition of all increasing sequences of reals there is a perfect set P such that all strongly increasing sequences from P lie in the same class.