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Hi I am new to proofs of consistency and independence with ZFC of some claims. I have read "The uses of set theory" by Judith Roitman, in that article it is mentioned that the Whitehead conjecture ($\mathsf{WC}$) is independent of ZFC. In fact, it is mentioned that $\mathsf{MA} + \neg \mathsf{CH}$ implies $\mathsf{WC}$ and Jensen's diamond principle implies $\neg\mathsf{WC}$. Does anyone have a bibliography on the proofs of these facts? Taking advantage of the opportunity, someone knows more results of independent or consistent results with ZFC, demonstrated in a similar way (for example, "every subset of $\mathbb{R}$ with cardinality $\aleph_1$ has measure zero" is independent of ZFC, to prove this fact Martin's axiom shows that statement and CH shows the negation).

Thanks

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    $\begingroup$ A very good reference is Whitehead's Problem is Undecidable. For more results in this direction, you may look at the book Almost Free Modules: Set-theoretic Methods. $\endgroup$
    – Mohammad Golshani
    Commented Feb 2, 2021 at 2:31
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    $\begingroup$ The book Consequences of Martin's Axiom contains a lot of consequences of MA $\endgroup$
    – Mohammad Golshani
    Commented Feb 2, 2021 at 2:33
  • $\begingroup$ Thanks for the information Professor Mohammad. $\endgroup$
    – Gabriel Medina
    Commented Feb 2, 2021 at 2:33
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    $\begingroup$ just Mohammad is sufficient! $\endgroup$
    – Mohammad Golshani
    Commented Feb 2, 2021 at 2:34
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    $\begingroup$ Nik Weaver's book "forcing for mathematicians" has two whole chapters on Whitehead's problem, together with all the background on forcing and other machinery that is needed. I haven't read those two chapters myself, but I used other parts of the books and they were very well written. $\endgroup$
    – Alessandro Codenotti
    Commented Feb 2, 2021 at 10:05

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