FOM: question about the axiom of choice

Stephen G Simpson simpson at
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

-- Steve

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