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)?