Is there a proof for no proof ?



what is logic without a proof system
If I understand your question you are asking can something be proved as unprovable? If so I'd suggest that Gödel's incompleteness theorems for a starting place.



> 
> "Gödel's incompleteness theorems state
>  that any effectively generated formal
>  theory in which all arithmetic truths
>  can be proved is inconsistent; hence,
>  any such consistent formal theory that
>  can prove some arithmetic truths can't
>  prove all arithmetic truths."
> 
> 
> 


http://en.wikipedia.org/wiki/Gödel's\_incompleteness\_theorems


