@Herald,
Epistemology deals with what we mean by the word "knowledge" but recent philosophers have turned towards the
contextual use of words like "knowledge" rather than making authoritative absolutist claims about what knowledge
is. The pragmatists would say that our "knowledge of the big bang" is based on a
consistent research paradigm (Kuhn) using "big bang" as a central hypothesis, which
yields useful and interesting data. In short "scientific knowledge" is about
scientific functionality, science being a human activity, not a set of absolutist claims, since all paradigms tend to be transient and limited in the long run. Indeed the "big bang" paradigm is already under attack from various quarters but dissenters must account for any data revealed by
bigbangism if they are to establish an alternative.
Obviously, not all aspiring philosophers or scientists have that view of "knowledge", but irrespective of that, consensual functionality is a
de facto limit to
ad hoc speculation. In your case you don't seem to appreciate the essential factors of functionality and consensus which, like a rolling snowball, govern epistemological progress.
EDIT
Note that extrapolation from Godel's Incompleteness Theorem makes your request for
proof about any axioms, vacuous.
Proof proceeds from axioms whose truth value is already assumed. That is which
functionality is a preferable criterion for axioms.