# Why is this theorem (about $L(P(\omega_1))^V$ and $L(P(\omega_1))^{V[G]}$) nice?

I was recently told that the following (due to M. Viale) is a nice theorem:

Suppose there are arbitrarily large supercompacts, and $\mathrm{MM}$ holds in $V$. Let $G$ be generic for a proper forcing and $V[G] \vDash \mathrm{MM}$. Then $L(P(\omega_1))^V$ is elementarily equivalent to $L(P(\omega_1))^{V[G]}$.

My question (borne of ignorance, not skepticism) is:

Why is this theorem nice, and how does it fit into the bigger picture?

Some slightly-more-specific questions that refine my main question are: Do the hypotheses of this theorem often come up in natural settings? What's the upshot of the conclusion? Is it that proper forcing which preserves $\mathrm{MM}$, leaves the theory of a small but not-that-small chunk of the universe unchanged?

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My view is that questions about highly technical results such as this needn't be community wiki, even when they are asking for intuition or soft information about them. – Joel David Hamkins Jul 11 '11 at 0:43
It happens that Viale just spoke on his theorem today at a conference here in Singapore (www2.ims.nus.edu.sg/Programs/011aiic), and I had a chance to talk to him about it. The theorem, a little different than stated in the question, asserts that if MM++ holds and there are a proper class of inaccessible limits of supercompact cardinals, then the theory of the $\omega_1$-Chang model $L([\text{Ord}]^{\lt\omega_1})$ cannot be changed by stationary-preserving forcing that preserves MM++. The axiom MM++ strengthens MM by requiring that names for stationary sets have stationary values. – Joel David Hamkins Jul 21 '11 at 10:40
Oops, I meant $L([\text{Ord}]^{\lt\omega_2})$ as Matteo says. – Joel David Hamkins Jul 22 '11 at 4:06

By your tags you've asked for a soft answer, and so let me try to provide one.

The theorem is indeed very nice and engages with and reinforces a number of philosophical views in set theory.

First, there is the idea that large cardinal axioms are leading us towards the final, true set-theory, and so set-theorists are keenly interested when the existence of large cardinals causes a fragment of set-theoretic truth to stabilize in the lower part of the universe. One could take this to mean that that stabilized fragment is a part of the final theory we seek. Although Gödel's hopes that large cardinals would settle CH were dashed by the Levy-Solovay theorem, nevertheless the phenomenon does exist. Increasingly large large cardinal assumptions, for example, imply that increasingly large portions of the projective hierarchy have extremely nice properties (Lebesgue measurable, property of Baire, determinacy, etc.). If there are infinitely many strong cardinals, then it is consistent that projective truth is invariant by forcing. When there is a proper class of Woodin cardinals, then the theory of $L(\mathbb{R})$ is invariant by set forcing. This paper by Neeman and Zapletal shows that if there is a weakly compact Woodin cardinal, then for any proper forcing extension $V[G]$, there is an elementary embedding $j:L(\mathbb{R})^V\to L(\mathbb{R})^{V[G]}$. The idea is that once we have sufficient large cardinals, then one cannot change the universe too much by this kind of forcing.

Viale's theorem essentially extends this from $L(\mathbb{R})$ to $L(P(\omega_1))$, which is impressive.

Second is the philosophical idea that much of the indeterminism of set-theory is due to the anomalous effects of forcing. That is, the extreme flexibility of forcing causes so much set-theoretic chaos---we can turn the continuum hypothesis on and off like a lightswitch---and so when we come to know that a statement or class of statements is invariant by forcing, or by a huge natural class of forcing such as proper forcing, then this is really significant. The theorem and the others I have mentioned are all instances where the chaotic nature of forcing is controlled or restricted by the existence of large cardinals.

Third, there is the view of Martin's Maximum MM as expressing a fundamentally important truth about the universe. Justin Moore has emphasized its attractive nature as a fundamental axiom generalizing the Baire category theorem. Viale's theorem essentially says that when there are sufficient large cardinals, then Martin's maximum completely smoothes out the chaotic effects of (proper) forcing, since under statements in $L(P(\omega_1))$, which is a huge fragment of the universe, cannot be affected by proper forcing preserving MM. This result therefore underscores the idea that MM and large cardinals cause a measure of stability in the set-theoretic universe. (Note that the theorem is definitely false without the MM preservation, since proper forcing can switch the CH lightswitch, and CH is expressible in $L(P(\omega_1))$.)

Meanwhile, I expect that other set-theorists can add important insights.

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In the Neeman/Zapletal result, I should have said "for any small proper forcing," meaning the size of the forcing notion is below the weakly compact Woodin cardinal. – Joel David Hamkins Jul 11 '11 at 13:58
Thanks, this is great, as always! – Amit Kumar Gupta Jul 16 '11 at 2:54

Let me give a brief answer as the author of the mentioned theorem. I shall first say that (as you might imagine) I was very pleased to see that my theorem raised your interest. Now, as mentioned by Joel in his latest post, your statement of the theorem is not correct. However there is a typo also in Joel's post, namely the conclusion of the theorem holds for the Chang model $L([Ord]^{<\omega_2})$ and not for the Chang model $L([Ord]^{<\omega_1})$ as posted by Joel. For this Chang model the result is due to Woodin and was already known, it appears for example in Chapter 3 of Larson's book on the stationary tower forcing. To complement the very good answer Joel has already given you, I invite you to read the many survey papers related to the philosophical position subsumed by this theorem, here is a sample list: Woodin's two short papers for the "notices of the AMS" where he exposes the basic ideas behind the generic absoluteness results for $L(\mathbb{R})$:

Several of Peter Koellner's papers available at his webpage:

http://www.people.fas.harvard.edu/~koellner/

Most of them contains a long introductory part which motivates and explains very carefully and plainly the ideas at the heart of the $\Omega$-logic approach to absoluteness result.

It has to be noted that the kind of solution to the continuum problem prospected by my theorem follows Woodin's view as exposed in the papers on AMS notices. Currently Woodin has pursued a different approach towards the solution of the continuum problem that leads him to prospect a view of the universe (Ultimate $L$) which is radically different from the one given by MM.

Finally in case you are interested there are in my webpage several slides of talks I gave on this and related subjects, as well as a proof of the theorem you have mentioned in your question: http://www2.dm.unito.it/paginepersonali/viale/index.html (unfortunately my university is currently changing the websites locations, so for some time you may have troubles to consult it....)

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Matteo, welcome to MathOverflow! – Joel David Hamkins Jul 22 '11 at 4:12
Thanks! Yes, I've seen Koellner's great paper as well as Woodin's articles and the slideshows on your site, I really appreciate seeing the clear explanations of how these results fit into the broader perspective and their philosophical consequences for questions of foundations. Cheers! – Amit Kumar Gupta Jul 25 '11 at 22:25