[FOM] Uses of Replacement
ali enayat
a_enayat at hotmail.com
Mon Feb 27 13:08:43 EST 2006
In his posting of Feb 23, 2006, Friedman presented a number of significant
examples of various results whose proofs heavily rely on Replacement.
Freidman concluded his posting with the following query:
>SUBSCRIBERS - do you know other examples where the uses of replacement
>mayor may not be >necessary?
This prompted me to discuss the role of Replacement in the known proofs of
the conservativity of ZF+GC (global choice) over ZFC.
It is known that every countable model M of ZFC can be expanded to a model
(M,f), such that:
(1) f is a global choice function, i.e., for all nonempty x, f(x) is a
member of x.
(2) (M,f) satisfies Replacement in the extended language {epsilon, f}.
This result was independently proved via a forcing argument by a number of
people including Cohen, Solovay, Jensen, Kripke, and Felgner (whose proof
was published in [Fund. Math., 1971]).
The conservativity of ZFGC over ZFC was also established by Gaifman without
using forcing in [Israel J. Math, 1975]. However, both the forcing proof and
Gaifman's proof rely on Replacement (in the forcing proof, Replacement is
invoked indirectly in the guise of the reflection theorem).
This raises the following apparently open foundational question regarding
Replacement:
Question: Is Z + GC [Zermelo Set theory plus global choice] conservative
over ZC [Zermelo set theory with local choice]?
Regards,
Ali Enayat
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