Consider the the Powerset Size Axiom, that is, the following:


(PSA) ($\forall$x,y) |x|$\lt$|y|$\Rightarrow$$2^{|x|}$$\lt$$2^{|y|}$.


Does there exist a class $\mathscr M$ of models of ZFC such that the following holds:


ZFC+PSA$\vdash$"The Whitehead problem is answered in the Affirmative (i.e. Every Whitehead group is free)"


ZFC+$\lnot$PSA$\vdash$"The Whitehead Problem is answered in the Negative (i.e. there are nonfree Whitehead groups)"


If so, can a model of ZFC+$\lnot$PSA be a generic extension of a model of ZFC+PSA?



A Question Regarding the Powerset Size Axiom
In this answer, I will concentrate on groups of size $\aleph\_1.$ By Shelah, we have the following:


1. Diamond at $\omega\_1$ implies all $W-$groups of size $\aleph\_1$ are free,
2. $MA+2^{\aleph\_0}=\aleph\_2$ implies there exists a $W-$group of size $\aleph\_1$ which is not free.


Note that if for example we force with forcings which are $\aleph\_3-$closed, we can preserve either diamond at $\omega\_1$ or $MA+2^{\aleph\_0}=\aleph\_2$.


So we can easily find a proper class consisting of generic extension of the ground model, which satisfy diamond at $\omega\_1+PSA$. Similarly we can find a proper class consisting of generic extension of the ground model, which satisfy $MA+2^{\aleph\_0}=\aleph\_2+ \lnot PSA$.


For your last question, just start with $GCH$ and add $\aleph\_2-$many Cohen reals.


