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As stated in Woodin, Davis, and Rodriguez - The HOD dichotomy, a longstanding open problem in set theory is to construct a canonical inner model for supercompactness. In general there are various ways of spelling out what a “canonical inner model of large cardinal axiom A” means. The linked paper, as well as other work of Woodin, suggests the following weak thesis:

Thesis 1: If there are canonical inner models of supercompactness, at least one of them should be a weak extender model (as defined in 1).

As it is said in the last paragraph on page 16 in the linked paper, supported by Example 27, the exhibition of a weak extender model of supercompactness is not sufficient to solve the inner model problem for supercompacts. Of course, $V$ is trivially such a model. This is why Thesis 1 is a weak thesis.

Now, due to difficulties involved in spelling out in a general context what a canonical inner model is, the consensus view seems to be that a “canonical inner model of large cardinal axiom A” should be a proper-class iterate of a mouse. A mouse is a small iterable model of a sufficient fragment of ZFC + A, which is canonical because of the Comparison Lemma. In particular, it makes sense to speak of “the smallest mouse for ZFC + A”. For example $M_1^\sharp$ is the smallest mouse above a Woodin cardinal, and $M_1$ is the canonical inner model of one Woodin cardinal obtained by iterating the top measure of $M_1^\sharp$ out of the ordinals. So based these general considerations, we have:

Thesis 2: All canonical inner models of sufficiently large cardinals are iterates of mice.

The low-level versions of these theses are in harmony. For example, assuming the existence of a measurable and a sharp for measurability, a weak extender model for measurability is also an iterate of a mouse. But they seem to come into conflict around supercompactness. It is shown in 1 that if $\delta$ is extendible and $M$ is a weak extender model for the supercompactness of $\delta$, then there is no $j : M \to M$ with critical point $\geq \delta$. On the other hand, iterates of mice (through all ordinals) are, by their construction, self-embeddable with arbitrarily high critical point.

Question: How do we resolve the “paradox” and clarify the inner model problem?

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    $\begingroup$ There is something tricky about the example with a measurable. In order to get a weak extender model for measurability, you're iterating the measure itself, until it gets high enough. Only then, you start to iterate the sharp out of the ordinals. This might work up to a strong cardinal, but it is unclear to me how you're going to get higher, even to a measurable above a strong. $\endgroup$
    – Yair Hayut
    Commented Oct 28, 2020 at 8:13

2 Answers 2

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  1. Inner model theorists use the word "canonical" to explain the problem in intuitive terms, it is indeed a vague problem, though it is as precise as anything in the region of superstrong cardinals.

  2. Inner model theorists understand the importance of the problem, the way it sits within our understanding of models of set theory, which is why they always aim to explain what the problem is about instead of saying what solving it would mean.

  3. One could very easily make the inner model problem to be any of the test questions Gabe had, and indeed answering those questions is the main motivation.

  4. While reading the comment above, I didn't feel offended, I did find it amusing. I am a bit puzzled why forcing people are not registering that the inner model problem is as much a problem of inner model theory as forcing.

  5. Continuing 4, how about "do large cardinals imply that $\omega_1$ carries a precipitous ideal"?, how about is there a poset that kills all precipitous ideals? this is an open problem from FMS.

  6. Is $\Sigma^2_2$ absoluteness conditioned to generic diamond true? Is the $\Omega$ conjecture true?

  7. Does $\sf PFA$ imply there is an inner model with a supercompact?

  8. Does $\sf MM^{++}$ imply that there are no divergent models of $\sf AD$?

  9. Is $\mathsf{AD}^{L(\Bbb R)}+\Theta^{L(\Bbb R)}>\omega_3$ consistent?

  10. Can you have 4 consecutive measurable cardinals (under ZF)?

  11. Can you force $\sf GCH+\neg\square_{\omega_3}+\neg\square(\omega_3)$ from a large cardinal weaker than a Woodin cardinal that is a limit of Woodin cardinals?

  12. The only known consistency proof of one of Hamkins' maximlaity principles is forcing over $\rm HOD$ of a model of $\sf AD_\Bbb R+\Theta$ is reg.

  13. I do share the view that IMT is different, the reason being that it feels more like doing physics than doing math. Within the subject there are two groups (not necessarily disjoint). There are those who think about the models themselves and those who think about how to build these models. Doing the second is much more like doing physics than doing math. The first is as mathematical as it can be. Most talks one hears these days are on constructing the models rather than studying them.

  14. While I understand that the set theoretic community respects the area, I do feel that people often are being unfair to it (I do not mean anyone here). The fact that inner model theorists put the time to explain their area seems to be not fully appreciated, while often one sits through talks that give incredibly technical constructions and the only motivation for it is that "it is interesting to have such a thing". It would have been nice if all mathematicians tried to explain their area, the main problems and the importance. By this I don't mean history of the problems, but why one actually wants to solve it. Then many areas also would look quite vulnerable. I do agree with the point that "canonical inner model" is bad terminology, but if you have the courage to explain why you want to do something you inevitably run into the issue of sounding imprecise and vague.

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    $\begingroup$ I hope you don't mind, Grigor, I've made the $\rm\LaTeX$ code into actual $\rm\LaTeX$. $\endgroup$
    – Asaf Karagila ♦
    Commented Jul 2, 2021 at 15:42
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The issue is Thesis 2. The notion of a canonical inner model is vague, but I don't think that when inner model theorists use the term they mean to suggest that the model is obtained by iterating a sharp. In common usage, the canonical inner models include every proper class Mitchell-Steel premouse whose countable $\Sigma_1$-elementary substructures are $(\omega_1+1)$-iterable mice. (Yes, you can iterate the measurables into weird configurations or take stationary tower ultrapowers in forcing extensions, but you can't mess with the internal theory, which is all one really cares about.) It is known that these models can be close to $V$ – actually equal to $V$ – in the presence of many Woodin cardinals.

The Inner Model Problem ("Build a canonical inner model with a supercompact cardinal") is vague. If one knew the general definition of the term "canonical inner model," one would have answered half the question. The vagueness of the Inner Model Problem is not an issue because in actuality the problem has more than one component:

  1. A pattern that is known to hold quite high into the large cardinal hierarchy with many applications.
  2. The vague problem of extending this pattern to larger cardinals.
  3. A host of precise yes-or-no test questions that likely can only be answered positively by solving (2) and to which a negative answer would strongly suggest that (2) cannot be solved, or at least that our current conception of a canonical inner model must be drastically altered.

These test questions include the HOD Conjecture, the Ultimate $L$ Conjecture, Con(Supercompact + $(\Sigma^2_1)^{\text{Hom}_\infty}$ wellorder of the reals), Con(Supercompact + Ultrapower Axiom), $\text{AD}^++V = L(P(\mathbb R))$ implies GCH in $\text{HOD}$, and more!

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    $\begingroup$ I think one could add that our understanding of canonicity gets revised as we climb up. This is perhaps tautological but deserves discussing precisely because our understanding can radically change our goals. For instance, before reaching Woodin cardinals, common wisdom was that canonical inner models had a $\Sigma^1_3$ well-ordering of their reals. The issue was that we really did not understand iterability well enough at that point, and the missing ingredient (iteration trees) had not manifested itself yet. $\endgroup$
    – Andrés E. Caicedo
    Commented Nov 2, 2020 at 22:14
  • $\begingroup$ Three questions. (1) Are you saying that it is plausible that all large cardinals can exist in Mitchell-Steel premice whose $\Sigma_1$ hull is iterable? (2) Do you not think that the rigidity result about weak extender models already says something in the direction of “our current conception must be drastically altered”? (3) Is it fair to say reword the Inner Model Problem as, “Develop analogues to what we’ve already done for larger cardinals”? $\endgroup$
    – Monroe Eskew
    Commented Nov 3, 2020 at 7:51
  • $\begingroup$ Is being a Mitchell-Steel proper class premouse with all countable $\Sigma_1$-elementary substructures iterable internally expressible? (Or do we want to consider countable subsets and iteration strategies that are not members?) If so, this would seem to fall prey to the anti-canonicity observations linked above. $\endgroup$
    – Monroe Eskew
    Commented Nov 3, 2020 at 9:58
  • $\begingroup$ (1) No, I meant just that when people say canonical inner model, they include all Mitchell-Steel models. (2) The WEM result does say something new about inner models at the level of supercompact cardinals, but first of all, that was about 15 years ago, and second it did not rule out using the same basic methodology (comparison of backgrounded extender models via some kind of iteration trees) in the construction. (3) Yes, although then you’d have to say what has already been done. (4) No it is not internally expressible, although I still don’t think it will meet all your canonicity criteria. $\endgroup$
    – Gabe Goldberg
    Commented Nov 3, 2020 at 13:53
  • $\begingroup$ Thanks for your responses. On (4), can you give an example of such usage? Regarding (3), my concern is that it becomes not really a mathematical problem per se, but just a kind of label for an area of research, whose goals are basically to keep doing the kind of thing they’re doing, and whose particular concrete objectives shift as needed. (Is that offensive? I think it characterizes most math research actually.) $\endgroup$
    – Monroe Eskew
    Commented Nov 3, 2020 at 14:46

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