[FOM] Eliminating AC and GCH for statements of analysis
Ali Enayat
ali.enayat at gmail.com
Sun Apr 7 14:00:26 EDT 2013
In his third posting on this topic, Paul Levy has asked (April 6th) whether
ZFC + GCH is conservative over ZFC for statements of analysis (second order
arithmetic).
The answer to the above question is in the positive. Indeed, something
stronger is true, namely:
Theorem. ZFC+ GCH is conservative over ZF+ DC for statements of analysis,
where DC is the usual axiom of dependent choice of countable length.
The above theorem follows the fact that for every countable model M of ZF+
DC, there is a model N of ZFC with exactly the same natural numbers and
real numbers as M in which GCH holds.
N is obtained in two steps: use forcing over M to arrange CH (and in
particular: the well-orderability of the reals) without adding any new
reals to obtain the model M[G] of ZF. This can be done by adding a Cohen
subset G of omega_1 of M using countable conditions. Then N can be chosen
as L[G], as computed within M[G], where L[G] is the universe constructible
from G.
It appears that the above result has not been explicitly noted; however,
the 1969-paper of Platek in JSL that I mentioned in my earlier posting
contains a result with a stronger hypothesis and a stronger conclusion,
namely:
Theorem (Platek). If ZFC + GCH proves a statement S of the form "for all X,
A(X)", where X ranges over *subsets of reals*, and A(X) is a statement in
the language of second order number theory augmented with a (third-order)
predicate symbol X, then S is also provable in the theory obtained by
augmenting ZF with DC(aleph_1) [Dependent Choice of length aleph_1].
Best regards,
Ali Enayat
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