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The Proper Forcing Axiom kills CH in a particularly specific way: it implies that $2^{\aleph_0}=\aleph_2$. However, its impact on the continuum function above $\aleph_0$ is much less clear. It is known, for example, that it implies the Singular Cardinal Hypothesis, SCH.

In this paper (see page 2), Aspero writes that Magidor showed the following:

Any model of ZFC+PFA has a forcing extension satisfying ZFC+PFA+"$2^\kappa=\kappa^+$ for all uncountable $\kappa$".

However, he does not give a reference; the bibliography mentions a set of (unpublished) lectures by Magidor. Googling around, I haven't found anything more specific.

I was wondering whether a proof of this exists in the literature, or - failing that - if someone could sketch an outline of it.

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The proper forcing axiom is known to be indestructible by ${<}\aleph_2$-directed closed forcing, and since the forcing of the GCH for uncountable cardinals admits this degree of closure (iteratively add a Cohen set to the successor cardinals that are still there), it follows that one can simply force the GCH above $\aleph_1$ while preserving the PFA.

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  • $\begingroup$ Nice! Is it obvious that it's indestructible by such forcing? $\endgroup$ – Noah Schweber Dec 15 '16 at 3:09
  • $\begingroup$ No, it's not obvious, but it isn't that hard. Unfortunately, I couldn't find an easy reference for it online. David Aspero once showed this to me years ago... $\endgroup$ – Joel David Hamkins Dec 15 '16 at 3:12
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In Koenig and Yoshinobu's paper, they prove (Theorem 6.1) that $\sf PFA$ is preserved under $\omega_2$-closed forcings, and this should give the wanted result, as described by Joel.

Bernhard König and Yasuo Yoshinobu, Fragments of Martin’s maximum in generic extensions, MLQ Math. Log. Q. 50 (2004), no. 3, 297--302. MR 2050172

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