Let con(ZFC) be a sentence in ZFC asserting that ZFC has an omega-model M. Let $A\_{M}$ be an wff over M. Let S be the theory ZFC+con(ZFC). Is the reflection for S: $Bew\_{S}(A\_{M}) \implies A\_{M}$ is satisfied?
 I asking also for an explanation of the paradox in the link


<http://cs.nyu.edu/pipermail/fom/2007-October/012035.html>


of the case when ZFC is replaced on S=ZFC+(ZFC has omega-model)? 



Reflection principles
I suppose the "paradox" you're asking about is the passage marked with >> at the link you gave, but with "$\omega$-model" in place of "model" and with "has an $\omega$-model" in place of "is consistent". But then there is no longer any justification for the statement (on lines 9 & 10) that there's a proof in ZFC of the negation of con(ZFC) (which now becomes the negation of "ZFC has an $\omega$-model"). What you have is rather that this negation holds in all $\omega$-models of ZFC, but that doesn't immediately translate into a syntactic fact about existence of a proof, which you could then translate into English.


I conjecture that, if you write down carefully just what the "paradox" is supposed to be, it will disappear.


