
> 
> If a $\Pi\_1$ sentence is independent from PA, then it is true.  
> 
> CON(ZFC) is a $\Pi\_1$ sentence and independent from PA.  
> 
> Therefore, CON(ZFC).
> 
> 
> 


If this is a valid argument in ZFC, it would violate Gödel's incompleteness theorem. What's wrong with it?



What's wrong with this argument for CON(ZFC)?
Well, if CON(FZC) is false, then it is not independent of PA - as PA can look at the ZFC-proof of falsity and confirm it.


