Is there a way to prove the consistency of some chosen axioms? In the two senses following:
- In each mathematical logic book, there is a special kind of deduction system, which include some logical axioms(as opposed to nonlogical axioms). How does the proposer (inventor?) of such a deduction system know that his/her (pure logic) system is consistent?
- Once a deduction system is founded, one would like to introduce some nonlogical axioms to build a theory. How can one be sure that his/her chosen nonlogical axioms is consistent? For example, how do mathematicians know that the axiom of choice is consistent with the axioms in ZF.
To push the question further, what rules should we follow so as not to fall into inconsistence when we try to choose some axioms for a deduction system and for a theory, and when we try to introduce new nonlogical axioms to a consistent theory?
Any help is appreciated! Thanks in advance!