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    But we haven't got infinite time, so a few brief theories is definitely the way to go. Another word for axiom is assumption, and you know what happens when we ass/u/me. Commented Jan 6 at 12:44
  • "What is of interest is the theory generated by their deductive closure; which may have multiple distinct axiomatizations." Is this a claim inherent in this that the same deductive conclusions can be reached through different, but similar ontologies? Commented Jan 6 at 23:02
  • @JD I'm not sure, to be honest my answer is very mainstream model theory / proof theory, not a philosophical analysis. It's an interesting idea though. But all I'm saying is that there can be different sets of axioms that are logically equivalent. Commented Jan 9 at 4:34
  • @JD There's so much to learn / research / think about, and so little time, but one thing your comment reminds me of is IEML by Pierre Levy - it is a formal ontology language based on a certain collection of ontological primitives. But long ago when I encountered it, I had the sneaking suspicion that the choice of ontological primitives was arbitrary. What matters is that you can build the same universe of concepts, even starting from a different chosen "basis". Concepts emerge by contrast from other concepts. I think this is similar to what you're saying. Commented Jan 9 at 4:43
  • @JD But on the other hand... we might say the ontology is better given by the signature - this is where we state "what things exist". So we might have 2 equivalent sets of axioms but over the same signature. Or we might have 2 equivalent theories in totally different languages. And I think that's called "bi-interpretable". So yes, I would say... we can have interesting insights about different "ontologies" expressing the same "ultimate thing". Like, when I have the time I want to study NGB set theory, since you get a "class" of "all sets". Seems... ontologically intriguing. Commented Jan 9 at 5:04