[abtinf]

Why did the axioms bit have you worried? OP is pretty much exactly right, except for the part where we accept it by our "logic". Axioms are simply true, end of story; logic does not apply to axioms themselves, only how they can be used in relation to other axioms.

[shkkmo]

Axioms are simply something that you accept as true to build a model. There is no 'simply true, end of story' or 'proven by logic' to axioms.

Many axioms maybe picked because they seem 'obviously true', (or more likely, because they are useful) but that doesn't make their truth simple or make them the result of logic. (For an example take a look at the existence of infinity).

Additionally, the 'axioms' he lists are all what I would generally consider tautologies. (Although you might argue that the first one is actually an axiom of bivalent logic systems).