FOM: ZFC+MC vs ZFC+Con(ZFC+MC)
Joe Shipman
shipman at savera.com
Mon May 15 15:04:59 EDT 2000
Maddy, paraphrasing Steel:
>>Suppose, in the simplest case, we're to choose between ZFC + MC and
ZFC + Con(ZFC + MC). John suggests that choosing the second is like
adopting the physical theory that the world behaves as if there are
atoms or that the world came into existence five minutes ago with all
the fossils, etc., that lead us to believe it's older than that, etc.<<
I find this analogy very questionable. It cannot even be justified by
the principle "consistency equals existence". For one thing, the
consequences of the two axiom systems are different, as the first
disproves V=L while the second, if consistent, doesn't. Furthermore,
assuming MC is an ontological presumption far more serious than assuming
that the world is 20,000,000,000 years old rather than 0.00001 years
old.
Maddy:
>> In fact, the mathematical case strikes me as fairly simple. We have
two proposed theories, as above. We want our theories to be consistent,
and measurable cardinals are in some (remote) danger of being
inconsistent, but both candidate theories take the step of assuming that
they are consistent with ZFC. After that, what's to worry? What seems
to motivate a preference for the weaker theory is a concern that
'measurable cardinals might not exist, really!', in other
words, an ontological worry; related worries could be phrased in terms
of realism or anti-realism, in semantic terms (e.g., about meaning) or
in other metaphysical terms (e.g., about truth). My naturalist takes
such all worries to be irrelevant to methodology, to the question of
which axioms to adopt.<<
This would be hard to disagree with if the consequences if the two
proposed theories were the same, but of course they're not.
ZFC+Con(ZFC+MC) is indeed the weaker theory in the sense that it has
nothing new to say about infinite sets and can't settle questions about
V=L and determinacy that ZFC+MC can; on the other hand, it is the
stronger theory as far as pi^0_1 consequences are concerned (it proves
the consistency of the other as well as all the pi^0_1 consequences of
the other).
My preference for ZFC+Con(ZFC+MC) is not primarily ontological, it
really is methodological: I don't worry that ZFC+Con(ZFC+MC) will be
inconsistent with any other axioms mathematicians might want to adopt
someday, but I do worry that ZFC+MC will be inconsistent with other
axioms mathematicians might want to adopt someday.
This is not to deny that there are good naturalistic grounds to prefer
ZFC+MC, but the case is not easily settled on methodological grounds
alone.
-- Joe Shipman
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