axioms

All true statements of the first order axioms are true in the model of the second order axioms.

The axioms are assumed - in particular, the free will of experimenter B is assumed - and a conclusion is drawn.

Within reason, more axioms increase the number of things that definitely are the case but fewer axioms increase the number of things that might be the case.

An axiomatic system, in the context of Goedel's theorems, is a bunch of axioms with some inference rules.

It would only create other axioms and/or other non-dividable entities, at what point would you be satisfied?

In one meaning, the first-order logic is called complete because every logically valid statement is provable from the logical axioms.

Now there is a only one model of the second order Peano axioms (up to isomorphism).

All tautologies like that are better be provable with logical axioms alone, and Goedel's Completeness Theorem ensures that they are.

When someone ignores, prohibits or shame a field using name-calling and fallacious axioms, arguments/debates are irrelevant and you have to run.

Valid proofs of course typically also make use of non-logical axioms of the ambient theory.

You could have axiomatic systems that are built on axioms some of which are known to be false, and yet they're consistent.

In math, the model and axioms are sound (by definition) within the mathematician's world.

The other, remaining non-logical axioms say things like "a+0=a" and "every number except 0 has a predecessor".

But the problem isn't only that the conclusion isn't really "God exists" if you want that to match the god of any actually existing religion; it's also that the axioms are...

Typically, some of the axioms and all of the inference rules are called "logical" and are independent of the particular math structure we're trying to investigate (natural numbers, in this case).

The logical axioms and inference rules say things like "x=x" and "if you proved "for all x,...", you can also count as proved any version of...

Axioms are statements that are automatically taken as proved, while inference rules allow one to infer more proved statements from some already proved statements.