Note that I am not talking perse, about the definition of axioms. I know what that is, their these 'self-evidently true' statements which are the building blocks of all reasoning, meaning there untraceable.

This makes sense, because you can't have an infinite regress where reason after reason, you justify yourself perpetually. However, given the nature of axioms, how do we know when we have one? How do we know that something is 'self-evident' enough (though some may argue it need not be) and that it cannot be proven by any other truths?

I think that a good place to start is answering such questions, is by looking at their use. Since you can't just say 'this is an axiom' but 'this is not an axiom', maybe we can look for signs that we have obtained one in their deployment. Particularly in the fact that their untracable: Is there a solid method to prove that an axiom is the ultimate starting point in our reasoning?

Just to make clear, am speaking of axioms as being independent of what we think. Axioms are discovered, not created.


closed as off-topic by Shailesh, Adam Hughes, астон вілла олоф мэллбэрг, user91500, Erick Wong Dec 5 '16 at 7:00

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  • $\begingroup$ It has happened before that an axiom thought to be 'self evident' was, in fact, contradictory. That's the case of the axiom schema of comprehension (unrestricted) which leads to the so-called naive set theory. Hence the axioms are good untill they are proved not to be. Also, an axiom seemingly 'self evident' might have very surprising consequences, e.g. the Banach-Tarski paradox. Hence, sometimes, many axiomatizations of a same theory exist. $\endgroup$ – Guest Dec 4 '16 at 23:41
  • $\begingroup$ Also, an axiom can be proved to be independent of other axioms (see here). An example is the axiom of choice (AC) which is independent of the ZF axiomatic set theory. This is the axiom that leads to the Banach-Tarski paradox, and if a proof can be done without AC then it is generally considered a better one. $\endgroup$ – Guest Dec 4 '16 at 23:52
  • $\begingroup$ Aren't all axioms independent? $\endgroup$ – Jim Jam Dec 4 '16 at 23:54
  • $\begingroup$ Yes, you absolutely can say "this is an axiom" and " this isn't an axiom," at your own personal discretion. Enjoy the results you get. $\endgroup$ – mrob Dec 4 '16 at 23:55
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    $\begingroup$ @JimJam Just to be 100% clear, no they are not discovered any more than the rules of football were. $\endgroup$ – mrob Dec 5 '16 at 0:25

Axioms are like picking the rules of a game. If you pick them well, the game is fun and interesting. If you don't, no one wants to play. I wouldn't say axioms in mathematics are self-evident. They're simply accepted as the rules of play.

So how do you know when you have one? If assuming its truth makes the game more interesting to other humans.


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