[mstoehr]

Although it does lead to a mathematically uninteresting paradox: if you let A be the axioms of set theory and you add an axiom P which states that A proves x and not x, (i.e. set theory is inconsistent) then A' = A and P is still consistent.

[d0mine]

For clearity: A' = (A and P)

[cchooper]

Ah, that makes a lot more sense now!