FOM: What "we" can answer
Joe Shipman
shipman at savera.com
Wed Feb 17 14:08:22 EST 1999
Kanovei:
>S=ZFC is a quite adequate formalization of
>*our present methods* in mathematics.
>Could you please present any one question
>which ZFC can't answer *but we can*.
Tait:
that I had in mind was the question `` Consis(ZFC)?''. If by ``we'' you
allow me to refer to those of us who believe in the existence of an
inaccessible cardinal, then we can answer this question and so also
prove
that ZFC cannot answer it.
I like Kanovei's emphasis on "methods". By what "method", Bill, do you
arrive at your belief in the existence of an inaccessible cardinal? I
was surprised that you used such a strong assumption, because it is not
something you will persuade most other mathematicians with, and thus
requires a restrictive interpretation of "we". I think that Consis(ZFC)
can be established by the method (not formalizable in ZFC) of reflecting
on one's acceptance of the ZFC axioms as "true" -- this type of process
has been discussed regularly on FOM and it is not clear how far it gets
you, but I don't think it ever gets you any further than an inaccessible
does. One of Harvey's statements that's equivalent to a consistency
statement about higher cardinals like Mahlos and subtles has a different
status from Consis(ZFC) because it is not socially possible to force the
great majority of mathematicians to assent to it, since the generally
accepted "methods" for this don't exist. On the other hand, I am
certain the percentage of mathematicians who could be persuaded to
assent to the arithmetical statement Consis(ZFC) is in the high 90's.
Harvey, if you are listening, do you believe that your "concrete
independent statements" that are equivalent to the 1-consistency of
subtle cardinals are true? Why? If you are very sure they are true,
how would you go about persuading most of the rest of the mathematical
community of this? Or is your certitude noncommunicable? Which are you
more subjectively certain of, Goldbach's conjecture or the 1-consistency
of subtle cardinals?
-- JS
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