[FOM] Platonism and Formalism
steel at math.berkeley.edu
Sun Sep 14 17:30:14 EDT 2003
There has been some discussion as to the likelihood that
#) ZF + "there exists a nontrivial elementary embedding from some V(kappa)
>> into V(kappa), where kappa is an inaccessible cardinal"
is consistent. My personal, not very strong, opinion is that (#)
is consistent. Woodin's relative consistency results using the
consistency of # as a hypothesis constitute some evidence.
It is evidence of the same sort as the many relative consistency
results using the existence of supercompacts as a hypothesis provide
for con(supercompacts)--there just isn't as much in this case.
(The hypothesis need not be necessary; in fact, the use of supercompacts
in getting con(GCH fails at a singular strong limit) in a way
provides even stronger evidence of con(supercompacts), now that we know
this consequence only requires many measurables of high Mitchell
The strongest evidence of consistency for large card. hypos. comes from
the inner model program. In a general sense, it is like the evidence
above, in that you are doing something nontrivial with the hypothesis,
developing a theory based on it. The theory of a canonical inner model
of P is particularly systematic, thorough, and detailed, and hence
provides especially good evidence of con(P). At the moment,
inner model theory can't reach supercompacts, much less #.
Finally, it is interesting to ask what the behavioral evidence that
X beieves con(P) would be, for P a large card. hypothesis.
Announcing that you believe it isn't much--you could be lying, or mistaken
about your own beliefs. Perhaps developing ZFC +P, or supporting others
who do that, is the most important evidence that you believe it
consistent. Ironically, that is exactly what one might try to do in
order to show ZFC + P is inconsistent. Behaviorally, it seems the
most important divide is between those who think ZFC +P is
interesting, and those who don't.
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