Consistency

Consistency is an internal property of theories that we use, but it is also a feature that we can define on theories.

In relation to forcing.

Speed-up theorems
Sometimes a theorem will have a very long though finite proof; a speed-up theorem shows that this theorem has a much shorter proof in a slightly stronger axiomatic system, one which uses a consistency assumption.

Also, the technique used to show the existence of insanely long proofs is very similar to the technique for showing the undecidability of certain axiomatic systems. Recall that a system is decidable when the provability of every proposition in the system is decidable by an algorithm (so either it comes out with ``there is no proof or this is the proof for every proposition).