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At least as I understand the motivation behind rigorous definitions of the foundations of mathematics (the only contender with which I'm familiar being ZFC and extensions), the idea of an axiomatic system is that its axioms are, necessarily, unprovable and "arbitrary." However, they have still been chosen to accord with and extend our intuition.

What I am wondering, is what kind of interest has been had in trying to explore an axiomatic system that doesn't necessarily correspond with intuitive abstract concepts like e.g. "sets" (ZF) and "numbers" (PA)? I know that there are areas like model theory which explore the general structure of consistency and decidability of general axiom systems, but it seems like that is very much distant from the "depths" of any particular axiomatization. Such theorems like the Schwarz inequality, as I understand, can only be described from the depths of ZFC after e.g. defining $ℂ$ to be the splitting field for $ℝ$ of $x^2+1$, beforehand having defined $ℝ$ in terms of Cauchy sequences or Dedekind cuts, beforehand having defined $ℚ$ as the field of fractions of $ℤ$ which is a ring completion of some weird set called $ℕ$, which arises as a niche object of the theory. Very deep exploration of our current logic for its own sake, but generally can fruitful theory like this be obtained from random but formal axiomatizations?

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  • $\begingroup$ Maybe something like this? $\endgroup$
    – Neil W
    Commented Mar 21, 2014 at 23:59

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The question you pose is an interesting one: what can a random set of axioms derive? To be fair, we will often have contradictory systems (e.g. if we take a postulate and its negation, or a postulate and the negation of a theorem it implies in the system, etc.). Still, the question is intriguing,

I can't pretend to know the ins and outs of a theory of this stuff would go, but I do know a little bit about differing axiomatizations of our set theory. My sense is that one of the reasons that ZFC is so robust derives from your comment that it maps onto the world around us. We have a good intuition for how certain axioms work, and we have poor intuition for how others do; this is not to say that either system is better, just that we know how to work with one as opposed to the other. Few systems will be beyond the realm of our experience, as this would make discovering theorems therein difficult (but not impossible).

Hope this sort of answered your question, or some aspect thereof!

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