Can one have a theory that includes its own consistency as an axiom? 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?
 A: Let me address a point which is implicit in the question but hasn't been squarely addressed by the other answers.  Your description of the "axiom of consistency" is not precise: it is not at all obvious how to encode this axiom in the language of set theory.  In particular, there is no direct way to encode the self-reference involved in saying "this axiom".
Here is one way you might try to make it precise.  Let us say that a sentence $\varphi$ in the language of set theory can encode your "axiom of consistency" if $$ZFC\vdash \varphi\Leftrightarrow\text{"ZFC$+\varphi$ is consistent"}.$$
Now unfortunately, this criterion is rather weak.  For instance, if $\varphi$ is any sentence that ZFC proves to be false, then it satisfies this criterion, with both sides of the implication simply being false.
Nevertheless, there still is a reasonably "natural" candidate for such a $\varphi$: you can construct a sentence $\varphi$ that can naturally be interpreted as saying that ZFC$+\varphi$ is consistent, by essentially the same trick as is used in Gödel's incompleteness theorem.  More generally, such self-referential statements can be constructed by the diagonal lemma.
So there is a reasonable candidate for the theory you are proposing.  Unfortunately, this theory is inconsistent by Gödel, as discussed in Asaf's answer.  In fact, Asaf's answer shows that the sentences $\varphi$ satisfying the criterion above are exactly the sentences which ZFC disproves!
A: 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.
A: 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: 


*

*So either $T$ is not recursively enumerable, 

*or it is not consistent, 

*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. 
