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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?

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    $\begingroup$ For your last question, the answer is yes. If V satisfies GCH, then V satisfies PSA and we can easily force $\lnot$PSA just by adding $\aleph_2-$many Cohen reals $\endgroup$ – Mohammad Golshani Oct 9 '14 at 3:52
  • $\begingroup$ I don't understand the question. Does it make sense? You ask if there is a class of models $\mathscr{M}$ such that....and then you do not mention $\mathscr{M}$, but rather state an arithmetic condition on provability. Have you asked the question you meant to ask? $\endgroup$ – Joel David Hamkins Oct 9 '14 at 10:52
  • $\begingroup$ What I understood from the problem is that it asks if we can have a proper class of models of PSA+every W-group is free, and a proper class of models of $\neg PSA+$there are non-free W-groups. $\endgroup$ – Mohammad Golshani Oct 9 '14 at 15:24
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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.

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  • $\begingroup$ Is there a combinatorial principle $\mathrm X$ such that ZFC+$\mathrm X$+PSA$\vdash$"Every Whitehead group is free" and ZFC+$\mathrm X$+$\lnot$PSA$\vdash$"there exist nonfree Whitehead groups" since Mekler and Shelah show in their paper "Diamond and $\lambda$-systems that it is consistent with $2^{\aleph_0}$=$2^{\aleph_1}$ that every Whitehead group is free? This might address Prof. Hamkins' concern that my question does not make sense. $\endgroup$ – Thomas Benjamin Oct 22 '14 at 7:01
  • $\begingroup$ Maybe the paper A Combinatorial Principle Equivalent to the Existence of Non-free Whitehead Groups by Eklof-Shelah be relevant. $\endgroup$ – Mohammad Golshani Oct 23 '14 at 6:01

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