In his ASL Godel lecture (Las Vegas, Nevada, 2002), Harvey Friedman asked the following question:
Are there fundamental principles of a general philosophical nature which can be used to give consistency proofs of set theory, including the so called large cardinal axioms?
Recently, he is able to isolate the following two fundamental philosophical principles and use them to prove the interpretability of various common sense thinking and set theory (with large cardinals).
(i) Plenitude Principle (PP): Anything that can happen will.
(ii) Indiscernibility Principle (IP): Any two horizons are indiscernible to observers on the basis of their extent.
(See his Concept Calculus article, e.g. link text)
My main questions are: How confident are we that PP and IP are “true” ? More specifically, is it possible to “prove” or justify PP and IP rigorously? If yes, how? If not, why not?
In my view, the ultimate justification of PP and IP would be to construct a (meta) system S based on PP and IP, and then prove its consistency and completeness. In the light of Godel’s incompleteness theorem, I’m not sure that this can be done. But perhaps S is not recursively axiomatizable, and so Godel’s incompleteness theorem would not apply to S.
My secondary question is: are there any logician (beside Friedman) who are working on this kind of research?
Update: July 2011
Here is a rephrasing of the question by Timothy Chow that makes it closer to mathematical logic:
Is there some precise mathematical statement, that has the flavor of IP or PP, which proves the consistency of all (or most) set-theoretic axioms that are generally accepted today (e.g., large cardinal axioms)?
Update: The question has now been open. It is now time for people who can relate to the problem to answer it.