When I refer to the "experience axiom" as "the only" axiom, I am only being "rigorous enough".
I suspect that the experience axiom must be but a placeholder for however many axioms it takes to formally axiomatize our personal experiences.
Like how accepting logic as a thing that "works" axiomatically actually cashes out into however many axioms it takes to formally axiomatize logic.
But with logic, we are freed from having to do that by defining logic empirically from our experiences instead of axiomatically in the abstract. And I'm pretty sure that's what brains actually do anyway to get their sense of logic, so we might as well do it in our rigorous philosophical foundations too.
With experiences... we don't have that luxury.
But this gets... tricky. I am fumbling with slippery nuances as I try to think about it now, so I do not have a fully coherent view just yet about what exactly "axiomatizing experiences" even means.
However, I have realized just now that it might be intrinsically a subjective problem, or at best a border that non-relativistic philosophy cannot cross.
I can only formally axiomatize my experiences. At best, you might then empirically observe that my formalisms correctly model your experiences in so far as you are able to introspect them.
This direction of thinking intrigues me, because I find "profound epiphanies" to be exquisitely rewarding both emotionally and practically, and this has all the signs of a profound epiphany to discover.
But I am not confident that going deeper in this case will have much practical value. The experience axiom seems to be an "axiom bottleneck" of sorts, the point where necessarily personal and subjective philosophy stabs through to objective philosophy.
Even if a relativistic axiomatization of experiences can be found that empirically seems to apply to all minds, it can only be derived from the starting point of our personal experiences. Maybe that is okay? But it is inherently circular in its derivation.
Maybe that itself is an insight. Maybe there can be no true axioms, only at best loops of tentative axioms not being ruled out by empirical observation. Never a starting point, just a fuzzy starting region that we can only ever refine to an ever tighter approximation.
But does that not itself state a formalizable rule, and thus an axiom? It does! But how did I get here? With logic empirically defined by my raw experiences. Oops, we're back to the experience axiom, however under-defined and un-rigorous it may be.
Returning back to the slippery nuance I was experiencing:
There are moments when it is so clear to me that the experience axiom really is just one axiom in the most rigorous sense possible, because everything, everything, rests on the inescapable act of us using raw experience itself as our foundation (mind the relevant language hole).
Then there are other moments where it seems obviously unclear and opaque, because what, precisely, does it even mean to ascribe truth value (whether in boolean or quantum or whatever formal logic) to all "experience" in one single axiom? Do we not need a rigorous definition of "experience"?
What remains clear is that anything and everything, including the notion of truth, logic, the idea of a true reality that might not match experience, and any further axiomatizations of experience, can only ever be derived by a mind from its raw experience.
The point: The experience axiom might arguably not be rigorous, but there is a fundamental problem with axiomatizing experience further that I am not sure is worth solving, or is even solvable at all.