FOM: corrected conservation result
Stephen G Simpson
simpson at math.psu.edu
Mon Oct 19 15:03:31 EDT 1998
Joe Shipman writes:
> You stated that ZFC+GCH is conservative over ZF+real choice for pi^2_1
> sentences, and conservative over ZF for pi^1_3 sentences.
> Can you give me an example of a sigma^1_3 sentence provable in ZF+real
> choice but not ZF?
Actually what I should have said is: ZF + real choice (in fact ZFC +
GCH, in fact ZF + there exists a real r such that V=L[r]) is
conservative over ZF for Pi^1_4 sentences. This follows from the
Shoenfield absoluteness theorem.
But even ZF + Sigma^1_3 countable choice is not conservative over ZF
for Sigma^1_4 sentences. An example showing this is the following
sentence: there exists a real r such that for all n, if aleph_n^L is
countable then r counts it.
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