A mathematical system of logic seems to prove statements, or theorems, by combining other statements according to the inference rules defined as part of the logical system. But in order to get started, it is desirable to have statements that don't need to be proved. These are called axioms, and form part of the definition of the system of logic. (It is possible for a system of logic to have no axioms, but most do.)
For example, if we want to devise a system of logic to prove statements about arithmetic, we might define rules of inference like "if a is an integer, then a+1 is also an integer" and "if a and b are both integers, and a = b, then a+1 = b+1". But this does not get us very far unless we also define some axioms as starting points, e.g.:
- 0 is an integer.
- 0 = 0
By combining these axioms with the rule of inference, we can prove theorems like
- 0+1 = 0+1
- 0+1+1 = 0+1+1
- 0+1+1 is an integer
and so forth.