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 PFA imply there is an inner model with a supercompact?
Does MM^++ imply that there are no divergent models of AD?
Is AD^{L(R)}+Theta^{L(R)}>\omega_3 consistent?
Can you have 4 consecutive measurable cardinals (under ZF)?
Can you force 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 HOD of a model of AD_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.
Grigor
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