"unprovability" is not a property that it is necessary to prove to pick something as an axiom. There is generally a project to reduce axioms to the simplest an

"unprovability" is not a property that it is necessary to prove to pick something as an axiom. There is generally a project to reduce axioms to the simplest and weakest forms required to make a proof. This is does result in axioms that are unprovable but does not mean the "unprovable" is a necessary property of axioms.