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As I was trying to find a list of mathematical axioms used in modern branches of mathematics, I wondered if there's any meaning to the question of "how many mathematical axioms are there ?", and then I wondered if the list, eventually, could grow infinitly.

I suppose there are multiple ways you could cut through such a question, a few would be:

  • If by axiom we take a proposition which is true (or considered so) without proof, then if A is an axiom and P is true proposition (true, proven and non-related to A), (A and P) could be considered an axiom (since true, and you can't at least prove the A part); such mechanism could grow the list of axioms, and since the list of Ps is infinite, then the list of axioms is infinit (could be a mistake in this reasoning);
  • if by axiom we take an irreducible atom of logical truth (some proposition which cannot be made into parts in any understandable way);
  • if by axiom we take any proposition we stumble upon which is not decidable, which we choose to consider true;
  • etc. (?)

Obviously, I am an having issue with the definition of an axiom, outside of its practical use in foundational mathematical constructs, the notion is very challenging (to me at least). However, even if we go by the most down-to-earth understanding of an axiom, that is "whatever we start from to deduce theorems by a thinking process which avoids contradiction", can we make anything of the question "how many axioms are there ?" or "how many axioms could we find ?" (different questions, I am sure, but in some ways related, I guess).

  • "there's any meaning to the question of "how many mathematical axioms are there ?" " NO. There are many mathematical theories with their own axioms, but we have also many alternative axiomatizations of the same math theory: e.g. set theory and Euclid's geometry. – Mauro ALLEGRANZA Apr 29 '20 at 8:19
  • I understand that the pratical exercice of mathematics is more in the spirit of 'accepting' an axiom and working with it, hence, the situation where anyone would persue an over-proliferation of axioms never arises. Let me ask a similar and different question: is there an infinite number of (definitly-)umprovable/none-provable propositions (in math or in more generic logic) ? – Gloserio Apr 29 '20 at 8:59
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    There are no "irreducible atoms of logical truth" in predicate logic. The standard axiomatization of ZFC has axiom schemas that already encode infinitely many axioms. And mathematical theories have infinitely many equivalent reformulations with non-overlapping sets of axioms of any number one wishes. Godel's theorem provides an inexhaustible generator of "definitely undecidable" propositions. Just take the Godel sentence of a theory and add it as a new axiom, then repeat. – Conifold Apr 29 '20 at 9:04
  • If I may, how does ZFC encore infinitely many axioms ? Otherwise, Godel's generative mechanism for undecidable propositions does in fact suggest you can have "axioms" at will. – Gloserio Apr 29 '20 at 10:00
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    plato.stanford.edu/entries/logic-infinitary is highly relevant to your question, @Gloserio. – Kristian Berry Apr 29 '20 at 16:27

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