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Over my life, I've encountered three different definitions for mathematical axioms:

  • Axioms are statements that must be accepted on faith. Unbelievers shall be punished by eternal damnation grade reduction or course failure.
  • Axioms are statements that, while not provable in a mathematical sense, are obviously true to any reasonable and smart person. You're a reasonable and smart person, right?
  • Axioms are worldbuilding statements that define what mathematics even is. The idea of an axiom being false is either undefined or represents an entirely different system of mathematics. For example, considering "What if the Axiom of Choice is actually false?" is equivalent to considering "What if Darth Vader isn't actually Luke's father?". That might make an interesting story but it wouldn't be Star Wars.

None of these statements are entirely satisfactory to me. The third one seems the most reasonable, but it implies that mathematics and reality may not be as linked as most people think.

Coming back from the definitions, I want to consider the possibility that an axiom (e.g. Euclid's axioms, or the Axiom of Choice, etc.) might actually be false. Are there mathematicians or schools of mathematics that seriously doubt the truth of one or more of the major mathematical axioms?

Robert Columbia
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    I assume you know about non Euclidean geometry, which negates the parallels axiom – lhf Feb 28 '23 at 12:22
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    For example, Non-Euclidean geometry, Also see Aczel's anti-foundation axiom. – David Lui Feb 28 '23 at 12:22
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    "doubtful axioms" means axioms not universally agreed on by the relevant scientific community. See Axiom of Choice, Large Cardinals. Axiom of Infinity (for Finitism). The level of disagreement varies with time. – Mauro ALLEGRANZA Feb 28 '23 at 12:30
  • The axiom of choice is most famous for being "doubtful" as you say it. Quite a good deal of research has gone into what happens when this axiom is not used. One alternative is "dream mathematics", which in some sense is much better behaved, in particular all subsets of $\mathbb{R}$ are measurable. It's called Solovay model on wikipedia – student91 Feb 28 '23 at 12:36
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    None of your three "definitions" for axioms is anything like the way mathematicians use the term. – Gerry Myerson Feb 28 '23 at 12:52
  • The word “doubtful” does not have a precise meaning, do you mean axioms whose consistency has been proven and yet is rejected by some mathematicians? Or do mean axioms whose consistency has not been proven? – Vivaan Daga Feb 28 '23 at 12:53
  • There are two senses in which an axiom can be false. In a strong sense, if an axiomatic system can prove a contradiction, one of its axioms must be false. In a weaker sense, an axiom can be considered false if it proves a statement considered to be false in the intended interpretation of the theory. We can add the axiom $\exists n(n \textrm{ encodes a proof of 0 = 1})$ to Peano arithmetic and it won't prove a contradiction, but that statement is known to be false in the intended model $\mathbb N$. – eyeballfrog Feb 28 '23 at 15:45
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    See constructive mathematics for an approach to mathematics which denies The Law of the Excluded Middle. https://plato.stanford.edu/entries/mathematics-constructive/ may get you started. – Gerry Myerson Feb 28 '23 at 23:39

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