Consider the sentence $\mathtt{PSP}_\mathfrak{c}$: "Every subset of $\mathbb{R}$ having the cardinality of the continuum contains a Cantor set".
A priori this sentence is weaker than the usual $\mathtt{PSP}$, since $\mathtt{PSP}_\mathfrak{c}$ requires the set not only to be uncountable, but to be of the size of the continuum.
My questions are:
- Given a model $M$ of $\mathtt{ZF}+\mathtt{DC}+\mathtt{PSP}_\mathfrak{c}$, can we find a forcing notion $\mathbb{P}$ in $M$ such that for any $G$ $\mathbb{P}$-generic over $M$, the extension $M[G]$ satisfies $\mathtt{ZF}+\mathtt{DC}+\mathtt{PSP}$?
- More in general, can we prove that $\mathtt{ZF}+\mathtt{DC}+\mathtt{PSP}_\mathfrak{c}$ is equiconsistent with $\mathtt{ZF}+\mathtt{DC}+\mathtt{PSP}$?
- Is it consistent $\mathtt{ZF}+\mathtt{DC}+\mathtt{PSP}_\mathfrak{c}+\neg\mathtt{PSP}$, modulo the consistency of some large cardinal?
Ideas?
Thanks!
Best Answer
Here are the answers you're looking for:
Look no further than John Truss' paper:
There he shows that by just adding many Cohen reals to $L$ we can have a model of $\sf ZF+DC$ in which every set of reals is well-orderable or contains a perfect set. Moreover, we can choose an arbitrarily high Hartogs numbers for the reals, so there can be many well-orderable cardinals intermediate to the continuum.
This provides us with a model of $\sf PSP_{\frak c}$ that requires no large cardinals. However, since $\sf PSP$ implies that $\omega_1$ is inaccessible to reals, that means we cannot extend further to a model of full $\sf DC+PSP$ without an inaccessible cardinal present.