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Inner model theory aims to construct canonical inner models which captures as much of V as possible, which now is formulated more concretely as to build (fine structural) mice that contain many large cardinals. I understand that supercompact is the next step.

But, what is the motivation for achieving this goal? I have some ideas, but I'm wondering if there's a motivation I'm missing completely:

  1. It provides a tool for showing lower consistency bounds of axioms of interest by constructing inner models with certain large cardinals. Most notably, finding such a lower bound for PFA.
  2. It supplies us with more knowledge about V, especially if it turns out that, say, no mouse with a supercompact can be found.
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Grigor Sargsyan, towards the end of this talk, discusses the "motivational problem" in some detail:

One of the main open problems in set theory is the conjecture that "PFA is equiconsistent with a supercompact cardinal". As it is already known that one can force PFA from supercompact cardinals, the direction that is open is whether one can produce a model of supercompactness from a model of PFA. We know essentially one method of doing such things and that is via solving the inner model problem for large cardinals. The approach is via developing two things at the same time.

  • Develop tools for proving determinacy from hypothesis such as PFA.
  • Develop tools for proving equiconsistencies between determinacy hypothesis and large cardinals.
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    $\begingroup$ Thanks for the answer; however, this is more or less my point (1) in my question. I'm wondering if the Inner Model Problem, an open problem since the 60s, is really only to show an equiconsistency result. What happens if we manage to construct a mouse with a supercompact from PFA? Is the field of inner model theory then "finished"? What about constructing mice with, say, extendibles? There seems to be no motivation for constructing these mice in general - only the vague notion of "learning how V behaves". $\endgroup$
    – Dan Saattrup Nielsen
    Commented Mar 17, 2016 at 10:37

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