FOM: Conservative extensions

Adrian Mathias amathias at
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
constructible reals. 

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
from alpha 
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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