FOM: Conservative extensions
Stephen G Simpson
simpson at math.psu.edu
Mon Oct 5 11:51:16 EDT 1998
Joseph Shoenfield writes:
> Steve's proof that ZF plus real choice is conservative over ZF for
> second order arithmetic seems to be marred by a confusion between two
> models of ZF.
No, no! That's not what I proved! What I proved is that ZFC is
conservative over ZF + real choice, for 2nd order arithmetic.
By the way, more is true: ZFC + GCH is conservative over ZF + real
choice, for 2nd order arithmetic. This needs an additional argument:
collapse c to aleph_1 by forcing with countable conditions, without
adding new reals.
What you said I proved isn't even true! ZF + real choice is *not*
conservative over ZF, for 2nd order arithmetic! (To see this, look at
the Feferman-Levy model of ZF in which aleph_1 = aleph_omega^L. In
this model, Sigma^1_3 choice fails.)
> The first, the class of sets constructible from a real, does not
> necessarily satisfy choice; in fact, a popular axiom is that this
> model satisfies AD. The second, the class of sets constructible
> from R (the set of reals) does not necessarily contain all the
> reals.
I didn't use either of these models. What I used was the smallest
transitive model of ZF containing R and W. Here R is the set of all
reals, and W is any well ordering of R. This model by definition
contains W, R, and all elements of R. It is a model of ZFC.
-- Steve
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