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It is known that axioms are the building blocks of mathematics. Differents sets of axioms different "games". What I don't understand is how do we know that we pick axioms that are consistent? . Does this mean that some day we may find a theorem that is true and false at the same time? Then what is the point trying to prove theorems if we can't know that our starting points are consistent? The word "proof" also doesn't make sense in that case.

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    "Does this mean that some day we may find a theorem that is true and false at the same time?" Yes, and it did happen historically. The original infinitesimal calculus assumptions were inconsistent, the system originally proposed by Frege was also, due to Russell's paradox. But so what? The purpose of axioms is to streamline what we do, if one system of them does not work out it is modified or replaced by another. "If a contradiction were now actually found in arithmetic -- that would only prove that an arithmetic with such a contradiction in it could render very good service", Wittgenstein – Conifold Apr 28 at 20:19
  • Actually, the run-up to Godel's Incompleteness Theorem is the Completeness Theorem, which proves that the set of axioms he chose cannot result in a contradiction in first-order logic. The Incompleteness Theorem, despite their opposite names does not contradict this, so bringing it up here seems misleading. – hide_in_plain_sight Apr 28 at 21:59
  • I would say you are confusing terminology. If you understood the concepts given before axioms you would see there are only TWO truth values possible: true or false. The term contradiction necessarily MEANS both statements cannot be true simultaneously.So if one of the axioms turns out objectively TRUE the others can't legitimately contradict it. The other axiom which you call contradictory is JUST CALLED FALSE. There is no need to go into if it is a contradiction of the true axiom or not. You should focus on TRUE STATEMENTS because that will continue to lead to OTHER true STATEMENTS. – Logikal Apr 29 at 1:46
  • "how do we know that we pick axioms that don't contradict each other?" Finding a model that satisfies them. – Mauro ALLEGRANZA Apr 29 at 6:31
  • "Can we pick whatever laws of logic (thought) we want?" More or less... some logical laws are so basic that we hardly can avoid them. – Mauro ALLEGRANZA Apr 29 at 6:32
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This is certainly an issue: whereas the consistency of weak axiom systems (like no axioms at all, just the inference rules of first-order logic) can be demonstrated in a very strong sense (basically, if we can exhibit a model and verify that those axioms are satisfied in it in an "appropriately constructive" way), this isn't satisfactory at all since per Godel's theorem once we hit really interesting axiom systems there's a level of uncertainty that can't really be argued away in any satisfying sense.

  • OK, "satisfying" is subjective. Some people, including many (most?) excellent mathematicians, take a Platonist stance and claim that there's some particular axiom system A which is "sufficiently rich" to serve as a foundation of mathematics but also has the property that its consistency is somehow "innate knowledge." However, while something like this is usually adopted and pretty satisfying on a pragmatic level, to my mind it's utterly unsatisfying once we actually think about foundations in a serious way.

There is however a response which addresses all but the most extreme skepticisms: namely, reframe mathematical activity as the manipulation of certain strings of finite symbols according to certain rules. Inconsistency of an axiom system amounts to the corresponding* symbol string having the property that we can get any other string from it via those rules. The facts we claim in this context are positive instances of deductions: if we prove p from X, the later discovery of an inconsistency of X doesn't retroactively make that discovery false, it just makes it less interesting. So there's no issue posed by Godel here.

Note however that this formalist response itself runs into some issues: namely, why are some instances of this "symbol game" incredibly interesting and useful while others aren't? More poignantly, this response doesn't address your sub-question:

"what is the point trying to prove theorems"?

From a formalist standpoint, the most common answers to that are the aesthetic one ("what's the point of creating art?") and the pragmatic one ("even if we're not sure why, it seems to be useful for manipulating and understanding the physical reality we live in").


*There's a slight subtlety here around foundational theories like (first-order) PA or ZFC, which have infinitely many axioms. However, such theories are "finitely describable" in a certain technical sense, so we can take "corresponding finite string" to refer to the description of the theory, not the theory as a whole. Alternatively, we can restrict attention to finitely axiomatized theories. This is all a bit technical, but ultimately easily handleable so I'm not going to go into more detail about it here.

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Zermelo faced this problem (SEP, "Zermelo's Axiomatization of Set Theory," sec. 1):

I have not yet even been able to prove rigorously that my axioms are “consistent”, though this is certainly very essential; instead I have had to confine myself to pointing out now and then that the “antinomies” discovered so far vanish one and all if the principles here proposed are adopted as a basis. But I hope to have done at least some useful spadework hereby for subsequent investigations in such deeper problems. (1908b: 262)

So that's why we do what we do: to try to make progress. If we were omniscient, we wouldn't need to do that; but we're not omniscient; so rather than let the possibility of being proven wrong someday stop us in our tracks, we plow ahead in hope that we might be proven right.

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