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?
Best Answer
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.
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.
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.
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.
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.
Is $\Sigma^2_2$ absoluteness conditioned to generic diamond true? Is the $\Omega$ conjecture true?
Does $\sf PFA$ imply there is an inner model with a supercompact?
Does $\sf MM^{++}$ imply that there are no divergent models of $\sf AD$?
Is $\mathsf{AD}^{L(\Bbb R)}+\Theta^{L(\Bbb R)}>\omega_3$ consistent?
Can you have 4 consecutive measurable cardinals (under ZF)?
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?
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.
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.
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.