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

[Ali Enayat]

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:

http://www.cs.nyu.edu/pipermail/fom/2013-March/017169.html

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).

http://www.cs.nyu.edu/pipermail/fom/2013-April/017209.html

Best regards,

Ali Enayat





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End of FOM Digest, Vol 149, Issue 18
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