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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!

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Here are the answers you're looking for:

  1. No.
  2. No.
  3. Yes, no large cardinals needed! (Which explains the previous two answers.)

Look no further than John Truss' paper:

Truss, John, Models of set theory containing many perfect sets, Ann. Math. Logic 7, 197-219 (1974). ZBL0302.02024.

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.

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