FOM: Conservative extensions
amathias at rasputin.uniandes.edu.co
Tue Oct 6 09:21:13 EDT 1998
On Mon, 5 Oct 1998, Joseph Shoenfield wrote:
> 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. 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 don't think I agree; though there is ambiguity.
The class of sets constructible from R as a predicate, (L[R] in
the notation of Jech's book) equals L, so it contains just the
The class of sets constructible from R as a set contains all the
reals and (e.g. if there is a supercompact) is a model of AD + DC.
In Jech's notation that is L(R).
If alpha is a subset of omega, the class of sets constructible
is the same whether we say as set or as predicate, and models AC and
contains the "real", alpha. So L(alpha) = L[alpha].
If by "the class constructible from a real" is meant the
union over all reals alpha of the classes L(alpha), that might well
not be a model of ZF, since though it contains each real it might not
contain the set of reals, though it will be a model of the
well-ordering principle, that every set has a well-ordering.
A. R. D. Mathias
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