Algebra Seminar talk
2016-09-09
Diego Alejandro Mejía Guzmán (Shizuoka University)
New reals in intermediate stages of FS iterations
Abstract:
It is well known that if $P$ is a finite support (FS) iteration
of length $\delta$ of Suslin ccc posets, it is possible to define $P|X$
(the iteration restricted to $X$) for all subsets $X$ of $\delta$. Also, for
such an iteration, it is known that, for
$\alpha < \delta$, any real in
$V^{P|(\alpha+1)} \setminus V^{P|\alpha}$ is not in $V^{ P|(\delta \setminus \{\alpha\}) }$,
which implies that $P$ forces that the groupwise density number $\mathfrak g$
is equal to $\aleph_1$.
I will talk about a generalization of these results to a wider class of FS iterations and even to the more general context of unbounded reals (instead of just new reals) and present some applications. This was a technicality developed by the speaker in the context of template iterations to force a value of $\mathfrak g$ (not necessarily $\aleph_1$) in the paper ``Template iterations with non-definable ccc forcing notions", see science direct or also the arxiv version.