Two weeks ago a conference was held on precisely the topic
of your question, the Workshop on Set Theory and the
Philosophy of
Mathematics
at the University of Pennsylvania in Philadelphia. The
conference description was:
Hilbert, in his celebrated address to the Second International
Congress of Mathematicians held at Paris in 1900, expressed the
view that all mathematical problems are solvable by the application
of pure reason. At that time, he could not have anticipated the fate
that awaited the first two problems on his list of twenty-three,
namely, Cantor's Continuum Hypothesis and the problem of the consistency
of an axiom system adequate to develop real analysis. The Gödel
Incompleteness Theorems and the Gödel-Cohen demonstration of the
independence of CH from ZFC make clear that continued confidence in
the unrestricted scope of pure reason in application to mathematics
cannot be founded on trust in its power to squeeze the utmost from
settled axiomatic theories which are constitutive of their respective
domains. The goal of our Workshop is to consider the extent to which it
may be possible to frame new axioms for set theory that
both settle the Continuum Hypothesis and satisfy reasonable
standards of justification. The recent success of set
theorists in establishing deep connections between large
cardinal hypotheses and hypotheses of definable determinacy
suggests that it is possible to find rational justification
for new axioms that far outstrip the evident truths about
the cumulative hierarchy of sets, first codified by Zermelo
and later supplemented and refined by others, in their
power to settle questions about real analysis. The Workshop
will focus on both the exploration of promising
mathematical developments and on philosophical analysis of
the nature of rational justification in the context of set
theory.
Speakers included Hugh Woodin, Justin Moore, John Burgess,
Aki Kanamori, Tony Martin, Juliette Kennedy, Harvey
Friedman, Andreas Blass, Peter Koellner, John Steel, James
Cummings, Kai Hauser and myself. Bob Solovay also attended.
Several speakers have made their slides available on the
conference page, and I believe that they are organizing a
conference proceedings volume.
Without going into any details, let me say merely that in
my own talk (slides
here) I argued
against the position that there should be a unique theory
as in your question, by outlining the case for a multiverse
view in set theory, the view that we have multiple distinct
concepts of set, each giving rise to its own set-theoretic
universe. Thus, the concept of set has shattered into
myriad distinct set concepts, much as the ancient concepts
of geometry shattered with the discovery of non-euclidean
geometry and the rise of a modern geometrical perspective.
On the multiverse view, the CH question is a settled
question---we understand in a very deep way that the CH and
$\neg$CH are both dense in the multiverse, in the sense
that we can easily obtain either one in a forcing extension
of any given universe, while also controlling other
set-theoretic phenomenon. I also gave an argument for why
the traditionally proposed template for settling CH---where
one finds a new natural axiom that implies CH or that
implies $\neg$CH---is impossible.
Meanwhile, other speakers gave arguments closer to the
position that you seem to favor in your question. In
particular, Woodin described his vision for the Ultimate L,
and you can see his slides.
The following is due to Woodin:
Theorem. Assume $ZF$+ there exists a Reinhardt cardinal + there exists a proper class of supercompact cardinals is consistent.
Then there exists a genric extension of the universe which satisfies $ZF$ + the axiom of choice + there exists a proper class of supercompact cardinals, and such that in it Woodin's $HOD$ conjecture fails.
Best Answer
Harvey Friedman has recently produced some results in this area. See for example Friedman, Invariant Maximal Cliques and Incompleteness, 2011. There is also a draft of a text book titled Boolean Relation Theory and Incompleteness also by Harvey Friedman which is apparently also in this area.
In the paper it seems that Friedman has produced a graph theoretic theorem which is somewhat natural and requires a certain large cardinal axiom to prove.
Unfortunately I'm not particularly familiar with either of these works, so I'm not able to give a better explanation (although maybe another poster will be able to put a good explanation in their answer).