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) --> A_M is satisfied?
 I asking also for an explanation of the paradox in the link
 
cs.nyu.edu/pipermail/fom/2007-October/012035.html
 
of the case S=ZFC+ omega-model?