[FOM] On the arithmetical conservativity of ZFC + GCH over ZF

Ali Enayat ali.enayat at gmail.com
Sat May 23 15:43:49 EDT 2015

This is a response to a query of Roman Murawski (May 22, 2015) who has asked:

"I was told about Kreisel's result stating that ZF + AC + GCH is a
conservative extension of ZF with respect to sentences about natural
numbers. Is it true? Where one can find it?"

I will give a three-pronged answer:

1. The result is indeed true.  By modern standards it is a routine
consequence of the fact that, provably in ZF, (a) AC and GCH hold in L
(Goedel's constructible universe); and (b) arithmetical statements are
absolute between V (the universe) and L.

2. This topic was discussed on FOM a couple of years ago. You will
find bibliometric information about Kreisel's observation in the
following FOM posting of mine:


3. A similar conservativity result is true even in the realm of
analysis (second order number theory), namely:  if a theorem of
analysis is provable in  ZFC+GCH, then it is already provable in ZF+DC
(where DC is the usual axiom of dependent choice).   For more detail
see my FOM posting below (this conservativity results seems not to
have been explicitly noted earlier).


Best regards,

Ali Enayat


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