# What does Martin's Maximum imply for $P(\mathbb{R})$?

Prompted by this question: of course Godel's constructibility axiom implies that $P(S)$ is minimal for any set $S$ and so handily answers the question of the size of the power set of the continuum in $L$. Martin's Maximum is the only other principle I know of that has implications for the size of the continuum (caveat: most of my knowledge comes from Kanamori's The Higher Infinite, a couple of other set theory books and the occasional dip into Wikipedia), and it's not clear to me what its implications are for power set operations on higher cardinals, in particular $2^{\mathbb{R}} (=2^{\aleph_2})$ - my impression is that it implies that $2^{\aleph_1} = \aleph_2$ but I'm not 100% sure on that front either. Can anyone point to good information on the implications of MM for other cardinalities than just $\mathfrak{c}$?

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Martin's Maximum is known to be consistent relative to a supercompact cardinal $\kappa$ by a revised countable support forcing iteration that results in $\mathfrak{c}=\aleph_2=\kappa$. This forcing does not affect the values of the continuum function $\gamma\mapsto 2^\gamma$ for values of $\gamma$ at $\kappa$ or above. But it is known by the Laver preparation that if $\kappa$ is supercompact, then there is a forcing extension in which the supercompactness of $\kappa$ is preserved by all ${\lt}\kappa$-directed closed forcing. Such forcing can modify the values of the continuum function at $\kappa$ or above to be in accordance with any patter that conforms with Easton's theorem.

So by combining the two facts, if it is consistent with ZFC that there is a supercompact cardinal, then it is consistent with ZFC that MM holds, and the continuum function has any desired pattern at $\aleph_2$ and above, in acordance with Easton's theorem. In particular, MM does not determine the value of $2^\mathbb{R}$.

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Perfect! That was basically my presumption, but I wasn't 100% sure on that front. What about my presumption that it implies $2^{\aleph_1}=\aleph_2$ as well - is that correct, or is $2^{\aleph_1}$ similarly unprescribed? –  Steven Stadnicki Feb 9 '11 at 2:07
Yes, it is correct. MM implies PFA, which implies $MA+\neg CH$, which already implies $2^{\omega_1}=2^\omega$, by the almost disjoint coding method (in Jech), and also PFA implies $2^\omega=\omega_2$, so you've got $2^{\omega_1}=\omega_2$. –  JDH Feb 9 '11 at 2:18
Hmm, and just to check my brief confusion - the forcing that collapses our supercompact $\kappa$ to $\aleph_2$ has to destroy its supercompactness, doesn't it? (since obviously $\aleph_2$ isn't even inaccessible) Doesn't this of necessity imply some cardinal collapse above $\kappa$? Certainly it seems like e.g., the $\aleph_3$ of our original $V$ can't be the $\aleph_3$ of the extension... –  Steven Stadnicki Feb 9 '11 at 2:33
Everything below $\kappa$ is collapsed to $\aleph_1$, but not $\kappa$ itself, which is why $\kappa=aleph_2$ in the extension. So $\aleph_3=\kappa^+$ and $\aleph_4=\kappa^{++}$ and so on. –  JDH Feb 9 '11 at 2:41
Your answer is perfectly opaque to me :) –  Mariano Suárez-Alvarez Feb 9 '11 at 4:58

About the size of $2^{\mathfrak c}$: MM is preserved by $\omega_2$-directed closed forcing, so we can change the size of $2^{\aleph_2}$ by forcing directly over a model of MM and preserving MM. This is a result of Paul Larson. (To contrast with Joel's answer: He shows how we can ensure models of MM with different sizes for $2^{\aleph_2}$ by manipulating the exponential function above a supercompact and then forcing MM. Paul's argument allows us to manipulate the relevant cardinal regardless of how the model of MM is originally obtained.)

As for implications beyond the continuum: MM implies PFA, and PFA implies SCH, the singular cardinal hypothesis (a very nice recent result of Matteo Viale), so it has a direct influence on (singular) cardinals larger than $\aleph_2$. For example, this implies that $2^\kappa=\kappa^+$ holds in models of MM for a proper class of cardinals $\kappa$.

(The same is true in ZFC above a strongly compact.)

PFA also implies the failure of $\square_\kappa$ at all $\kappa>\omega$. This was first shown by Todorcevic.

(Again, this also holds above strongly compact cardinals. This is all indirect evidence that the strength of PFA ought to be in the neighborhood of strong compactness, and that the method we know for establishing its consistency is essentially the only possible method. There are recent results of Matteo Viale and Christoph Weiss strengthening this connection, showing that the standard forcing argument requires supercompactness. On the other hand, there are also recent results of Itay Neeman exhibiting a different method of forcing PFA, although also from a supercompact and using properness.)

For a result that follows from MM and not from PFA: Magidor showed that MM implies that the principle "very weak square" fails at singulars of cofinality $\omega$. This shows that the singular cardinal combinatorics in models of MM is very interesting (and not yet well understood.)

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Andres, I was hoping you'd chime in too - thank you! This is a really nice complement to JDH's answer. –  Steven Stadnicki Feb 9 '11 at 7:28
Thanks, Andres, this is the right way to do it. –  JDH Feb 9 '11 at 11:27