FOM: corrected conservation result

Stephen G Simpson simpson at
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.

-- Steve

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