# Proving that an extension A* of A is consistent iff there is a well formed formula that is not a theorem of A* .

Let A* be a formal system obtained from A by enlarging the set of axioms of A. How do I prove that A* is consistent if and only if there is a well-formed formula that is not a theorem of A*?

Also, A is consistent.

The forward direction is trivial but I'm having trouble proving the reverse.

• Because A is consistent iff there is a formula that is not a theorem of A. – Mauro ALLEGRANZA Feb 23 at 8:57
• Having said that you have first to state the def of consistency. – Mauro ALLEGRANZA Feb 23 at 9:03