Property B: A Baumgartner-style Property that Applies to Preservation of ${\aleph }_1$ and ${\aleph }_2$ under Iterations with Supports of Size ${\aleph }_1$

Abstract

We prove a theorem on iterated forcing that can be used for preservation of ${\aleph }_2$ and ${\aleph }_1$ in iterations with supports of size ${\aleph }_1$ of forcings that have amalgamation properties similar to those present in the perfect set forcing. The work is modelled after Baumgartner's Axiom A and his proof that iterations with countable support of the same preserve ${\aleph }_1$. In honour of James E. Baumgartner, the property introduced here is called Property B$(\kappa )$. The known additional difficulties when forcing at cardinals higher than ${\aleph }_1$ make for a less general theorem and a more complex theorem on the iteration, which is not an iteration theorem in the classical sense. The results extend to other cardinals $\kappa$ such that ${\kappa }^{<\kappa }=\kappa$, in place of ${\aleph }_1$. We give examples of individual forcings that have Property B$(\kappa )$ and their products. In particular, we introduce a correct version of the generalised perfect set forcing, which we call Perfect Set Forcing with Respect to a Filter. We give its basic properties and show that for the right kind of filter $\mathcal{F}$ this kind of forcing is iterable with supports of size $\le \kappa$.

A second version after submission. Fixed the definition of the fusion sequence by removing the unnecessary and wrong equipment that connected C_zeta's and their cardinality with the support of p_zeta's. On the occasion, improved the presentation of the whole section 7 by putting more details in various arguments