[Scarblac]

Which axioms you take as true is a free choice. They aren't true or false by themselves.

What's irrefutably proven is that if you take this particular set of axioms, then these conclusions hold.

But you are free to choose other axioms, that will lead to other conclusions.

Some statements people use as axioms are equivalent (you can include one, and then derive the other and vice versa). Some are contradictory: you can include the axiom of choice or the axiom of determinacy, but not both as that will lead to a contradiction and thus an unsound system.

In a sense it's a matter of taste, mathematicians choose a set of axioms that leads to interesting things to think about.

[mathgradthrow]

you cannot prove the consistency of a system of proof within that system, ie at all.

[eru]

There are systems that we can proof the consistency of just fine. See eg https://en.wikipedia.org/wiki/Presburger_arithmetic

Presburger Arithmetic cannot prove its own consistency inside of itself, but that doesn't mean we can't prove its consistency 'at all'.

[mathgradthrow]

What on earth do you mean by prove, If not within a system of proof?

You can prove the consistency of PBA, But you cannot prove that you cannot prove the inconsistency of PBA, because The inability to do both is the definition of consistency in the larger system.

[Scarblac]

But it's not really relevant for anything to be able to prove that you cannot prove the inconsistency.

[Scarblac]

But proving inconsistency can be done - show that a contradiction follows from the axioms.

[nyc111]

"What's irrefutably proven is that if you take this particular set of axioms, then these conclusions hold."

This is what I tried to say in my comment. It's the author who talks about the truth of the axioms. I'm objecting to his claim that we end up with "something we can know for sure". No. Your truth depends on your assumptions.

[Scarblac]

We do end up with something we know for sure: the whole proposition "if we take these axioms as true, then these statements hold."