On the one hand, I've often understood that it would be nice if we could assemble a list of "all the axioms" and prove it's consistent - at least that's how it

On the one hand, I've often understood that it would be nice if we could assemble a list of "all the axioms" and prove it's consistent - at least that's how it was presented to me and roughly my interpretation of what eg. Russell and Whitehead were trying to do before being derailed by Godel proving it's impossible. On more recent reflection, it seems a little silly to want a proof in system X of the consistency of system X, even before we get to Godel, because if system X is inconsistent then it can definitely provide a proof that it is consistent by the principle of explosion.