FOM: question about the axiom of choice
Stephen G Simpson
simpson at math.psu.edu
Mon Mar 15 19:45:37 EST 1999
For any cardinal kappa, let kappa-AC be the axiom of choice for kappa,
i.e., the statement that for any family of kappa nonempty sets there
exists a choice function. Let c = 2^{aleph_0} = the cardinality of
the continuum. Let T be the following theory:
ZF + c-AC + there is no well ordering of the reals
Question: Is T consistent relative to ZF? Or maybe, ZF + an
inaccessible cardinal ....
One conjectures yes, but does anybody know for sure? How does the
proof go?
Background: This came up because Shapiro (page 107 of his book on
second-order logic) needs to know that T is consistent, but he obtains
this only by noting that T is included in ZF + AD_R. Here AD_R is the
axiom of determinacy for games where the players choose reals. AD_R
implies inner models with large cardinals and so has consistency
strength much greater than that of ZF. It is therefore desirable to
show that T itself has the same consistency strength as ZF.
Shapiro uses the consistency of T to obtain an independence result for
second-order logic. The result is: D2 does not prove WOP. Here D2 is
a system of axioms and rules of inference for pure second-order logic,
including full comprehension and the axiom of choice. WOP is the
statement `there exists a well ordering of the universe' in the
language of D2. Trivially D2 + not-WOP is interpretable in T.
(Note: Shapiro's system D2 is not really a second-order system. It is
actually a two-sorted, first-order system.)
Actually, now that I look more carefully, it appears that G"unter
Asser and Christine Gassner have shown that T is consistent relative
to ZF. See Shapiro, pages 130-131. Has anyone seen the Asser/Gassner
paper?
-- Steve
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