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In his paper "Some Second Order Set Theory", Joel Hamkins asked whether there is a model of set theory $V$ that is elementary equivalent to $V[G]$, Whenever $G$ is $V$-generic for the collapse of a cardinal $\delta$ to $\omega$? It seems that the existence of such a model will have large cardinal strength. Assume such model exists, then ;

1.There is no switches(a statement $\phi$ in language of set theory such that both $\phi$ and $\neg \phi$ can be forced by set forcing-here by collapse forcing)for collapse forcing over this model.

2. $\diamondsuit\phi \longleftrightarrow \phi \longleftrightarrow\square \phi$(By collapse forcing).

3. $V \models$ MP(Hamkins' maximality principle)

Assume such a model exists, my questions are :

Question 1. What is valid principle for collapse forcing? Is it modal logic of provability?

Question 2. What are other straightforward consequences?

Question 3. In general, is this model strange? Is it ugly or no, it's nice? will we have nice consequences or not(not necessarily related to modal logic of forcing)?

Question 4. Can this question be asked about other forcing notions, I mean are they interesting?

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    $\begingroup$ I realize that this is quite a technical question which a non-specialist like me cannot answer anyway. But still it would be nice if I could at least understand what is asked. Could you please be just slightly less cryptic? $\endgroup$ – მამუკა ჯიბლაძე Feb 15 '15 at 20:23
  • $\begingroup$ @ მამუკა ჯიბლაძე which part?what do you mean by cryptic? $\endgroup$ – Rahman. M Feb 15 '15 at 20:59
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    $\begingroup$ It is difficult - for me at least - to figure out exactly what you are asking. For example, your question 1 makes no sense to me - what does "valid principle for collapse forcing" mean? $\endgroup$ – Noah Schweber Feb 15 '15 at 21:31
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    $\begingroup$ I find the question to be clear enough, if one has some familiarity with the modal logic of forcing. I'll write up an answer soon. $\endgroup$ – Joel David Hamkins Feb 16 '15 at 1:40
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    $\begingroup$ I posted an answer, trying to include some general background that might also help to make the question more understandable. $\endgroup$ – Joel David Hamkins Feb 16 '15 at 2:28
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In a joint work with Mitchell, which is under preparation, we succeeded to give a full answer to Hamkins-Löwe question.


Edit:

The paper is ready. See

On a Question of Hamkins and Löwe on the modal logic of collapse forcing.

The proof uses Radin forcing with interleaved collapses. Also it is shown (by Mitchell) that some large cardinals are needed to obtain a consistent answer.

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$\newcommand\possible{\diamondsuit}\newcommand\necessary{\square}$The forcing modalities defined for a given definable class $\Gamma$ of forcing notions are

  • $\varphi$ is $\Gamma$-possible, written $\possible\varphi$, if $\varphi$ holds in some forcing extension by forcing in $\Gamma$.
  • $\varphi$ is $\Gamma$-necessary, written $\necessary\varphi$, if $\varphi$ holds in all forcing extensions by a forcing notion in $\Gamma$.

The main question is, for a given class $\Gamma$ of forcing notions, what are the modal validities of $\Gamma$ forcing? For example, which modal assertions are correct for c.c.c.-forcing? for collapse forcing? For proper forcing? And so on. To give an easy example that may illustrate the point: when $\Gamma$ is closed under finite iterations, then $\necessary\varphi\to \necessary\necessary\varphi$ is valid for $\Gamma$-forcing.

I introduced these forcing modalities in A simple maximality principle, because the maximality principle itself is easily expressed in this modal language as the assertion that every possibly necessary statement is true. Benedikt Löwe and I proved in The modal logic of forcing that the ZFC-provably valid principles of forcing, for the case where $\Gamma$ has all forcing notions, is precisely the modal logic S4.2. That work made use of the concepts of buttons and switches:

  • $\varphi$ is a switch, if $\varphi$ is necessarily possible and also $\neg\varphi$ is necessarily possible. In other words, you can turn $\varphi$ on and off by further $\Gamma$ forcing. $$\necessary\possible\varphi\qquad\qquad\text{ and }\qquad\qquad\necessary\possible\neg\varphi$$
  • $\varphi$ is a button, if $\varphi$ is necessary possibly necessary. In other words, you can force $\varphi$ in such a way that it remains true after any further forcing. $$\necessary\possible\necessary\varphi$$

Our more recent work is Structural connections between a forcing class and its modal logic, joint with George Leibman, expands on the general connection between the structure of the forcing notions in a definable class $\Gamma$ of forcing notions and the modal logic of forcing to which it gives rise. In order to do this, we introduce a number of other types of modular control statements, including ratchets, weak buttons and others.

For the case of collapse forcing, we establish in the paper the following (thm 24), and this answers your question 1:

Theorem. (Hamkins, Leibman, Löwe) If ZFC is consistent, then the ZFC provably valid modal principles of collapse forcing are precisely the assertions of the modal logic S4.3.

This doesn't mean, however, that every model of ZFC has S4.3 as its validities. For an upper bound on most of the modal logics, we had proved in the paper that if a class $\Gamma$ of forcing notions necessarily has a family of independent switches, then the modal logic of $\Gamma$-forcing is contained in the modal logic S5. And what Benedikt and I had noticed earlier was that for almost all the natural definable classes of forcing $\Gamma$, we were able to identify families of independent switches. For example, with c.c.c. forcing, we can always make the continuum equal to $\aleph_\alpha$ for very specific $\alpha$, either even or odd (or to have some particular binary bit pattern), and so for any class $\Gamma$ containing the forcing to do this, there will be independent switches. In particular, there will always be forcing extensions $V\subset V[G]$ having different theories. And we had similar switches for most of the standard classes of forcing.

But in the case where $\Gamma$ consists of all collapse-to-$\omega$ forcing, we couldn't seem to identify any natural switches. Thus, we wondered whether there necessarily were switches. An extreme negative case of this would be a forcing extension $V\subset V[G]$ where $V\equiv V[G]$ are elementary equivalent. And so we had asked whether this was possible.

At the Będlewo set theory meeting in 2007, I mentioned the problem to Paul Larson, who proposed an approach by which we should add a club to a suitable large cardinal and then collapse cardinals between members of the club. At Będlewo, we mentioned the idea to Bill Mitchell, who later carried out such an idea with his proposed solution, On a question of Hamkins and Löwe, but afterwards backed off of the proposal. (Meanwhile, see Mohammad Golshani's answer for later news on this approach.)

So to summarize the answers to the four questions:

Question 1 is answered in full by the theorem stated above. The provable validities of collapse forcing are precisely S4.3, and this is different from provability logic.

For question 2, if you mean consequences of the situation where necessarily $V\equiv V[G]$ in every collapse extension, then the modal logic trivializes, since if $\possible\varphi\iff\necessary\varphi\iff\varphi$, then all the modal logic in effect evaporates.

For question 3, I'm not sure about the nature of the model apart from the lack of switches, which I find fairly strange. It is a very different situation than occurs with most other classes of forcing, since with other kinds of forcing we can generally affect the theory of the resulting model.

For question 4, we point out in the "Structural connections" paper that all the other natural classes of forcing do not have this situation. Indeed, it was our inability to provide switches in the class of collapse forcing that led to the question. For example, it is a good exercise in forcing to show that there are switches for c.c.c. forcing, or for countably closed forcing, etc. Give it a try!

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  • $\begingroup$ Dear Prof. Hamkins, I went through Mitchell's argument and presented a proof of his claim by essentially the same ideas as him, but with some substantial changes in his forcing argument. It is submitted now (with Mitchell's permission), and if the proof be accepted, then we will have a weak answer to your question. $\endgroup$ – Mohammad Golshani Feb 16 '15 at 3:58
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    $\begingroup$ May I ask you what is the modal logic of collapse forcing in such a model? It seems that it should be very wide, as in such a model we have $\phi \equiv \Diamond \phi \equiv \Box \phi.$ $\endgroup$ – Mohammad Golshani Feb 16 '15 at 4:00
  • $\begingroup$ Thanks a lot, excellent explanation! Just one thing remains unclear for me (well, four things in fact) - do you thus leave all four OP questions open? :) $\endgroup$ – მამუკა ჯიბლაძე Feb 16 '15 at 7:16
  • $\begingroup$ Does this mean that modal principles of collapse forcing valid in a particular model of ZFC form a (quasinormal, hence normal) extension of S4.3? These have a fairly simple structure. Anyway, there is only one consistent modal logic including the schema $\phi\leftrightarrow\Box\phi$, namely the logic Triv axiomatized by this schema over K. $\endgroup$ – Emil Jeřábek Feb 16 '15 at 10:19
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    $\begingroup$ @მამუკაჯიბლაძე I have edited to make the answers explicit. Basically all four questions are answered. $\endgroup$ – Joel David Hamkins Feb 16 '15 at 14:04

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