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.

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

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