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

We prove a theorem on iterated forcing that can be used for preservation of ${\mathrm{\aleph }}_2$ and ${\mathrm{\aleph }}_1$ in iterations with supports of size ${\mathrm{\aleph }}_1$ 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 ${\mathrm{\aleph }}_1$. 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 ${\mathrm{\aleph }}_1$ 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 ${\mathrm{\aleph }}_1$. 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 $\le \kappa$.