Can one have a theory that includes its own consistency as an axiom?

Question

Consider the theory with the following axioms:

• The axioms of ZFC

• The "axiom of consistency": "This theory, including this axiom and all of the theory's other axioms, is consistent." Phrased differently (and equivalently):

  • This theory does not prove $\text{False}$
  • This theory states its own consistency as an axiom.

Can one have such a theory, one that includes its own consistency as an axiom? Can this theory be consistent?

Answer 1

What is this "axiom of consistency"?

Is it the statement $\operatorname{Con}(T)$? Because $T+\operatorname{Con}(T)$ is not the same as $T$. Or do you mean that $T=T+\operatorname{Con}(T)$? Which is just to say that $T$ proves its own consistency.

And if $T$ proves its own consistency, then it must violate one of the conditions of Gödel's theorem:

1. So either $T$ is not recursively enumerable,
2. or it is not consistent,
3. or it is not strong enough to interpret arithmetic.

If by adding one axiom to $\sf ZFC$ you managed to violate any of these, it has to be the second condition. So either $\sf ZFC$ was inconsistent in the first place, or that $\sf ZFC$ proves the negation of your new axiom, and you've added a new contradiction to the system.

Answer 2

Yes, one can have a theory that includes its own consistency as an axiom.

That theory is always (by Gödel's Second Incompleteness Theorem) inconsistent.