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Oct 9, 2014 at 15:24 comment added Mohammad Golshani 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.
Oct 9, 2014 at 10:52 comment added Joel David Hamkins 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?
Oct 9, 2014 at 4:09 vote accept Thomas Benjamin
Oct 9, 2014 at 4:07 answer added Mohammad Golshani timeline score: 3
Oct 9, 2014 at 3:52 comment added Mohammad Golshani 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
Oct 9, 2014 at 2:47 history asked Thomas Benjamin CC BY-SA 3.0