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What if our axioms are false? What happens then?

Bumble
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asdfasfasdgf
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  • Truth of the axioms is not a deductive system's concern, it only matters whether the theorems follow from them. Only when it is used as a model of something *then* one should care if the axioms hold. If not, it is not applicable as a model of that particular situation, as they say, garbage in garbage out. – Conifold May 26 '20 at 23:03
  • @Conifold Interesting. When is it not used as a model of something? – asdfasfasdgf May 26 '20 at 23:13
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    Whenever it is a formal game of symbols, large cardinals are not used as models of anything so far, for example. The point is to generate formal systems with rich properties in hope that something they apply to might be encountered eventually. Then after checking simple facts (axioms) more complex ones (theorems) can be inferred and used. And if not, it is still good for mental gymnastics and recreation. – Conifold May 26 '20 at 23:18
  • It happens that there is no assurance that the consequences of the axioms (the theorems) are right. – Mauro ALLEGRANZA May 27 '20 at 06:27
  • You are confusing mathematics with deductive reasoning. You may think that all math is PART OF A DEDUCTIVE REASONING SYSTEM but doesn't that imply some deductive reasoning is NOT MATH? Surely it does because the category of deductive reasoning is a larger set. Recall Math IS A PART OF DEDUCTIVE REASONING not the other way around. For example Human beings are PART OF THE ANIMAL KINGDOM but all animals are NOT human beings. You need to understand distinctions between mathematics & other subjects. There is no equivalent that is why it stands alone as a subject. Deductive logic uses no axioms. – Logikal May 27 '20 at 14:39
  • @Logikal Deductive reasoning is defined to be forward inference from axioms. Deductive reasoning is a part of logic, which is apart of math. – asdfasfasdgf May 27 '20 at 17:54
  • You are absolutely wrong. You do know historically Mathematical logic was invented in the 19 century right? Logic has been apart of only Philosophy up until the standardization of Mathematical logic. Deductive reasoning is not just math. As I stated philosophy as well as other fields have used deductive reasoning without advanced mathematics. Other animals as well have the capability of using deductive reasoning. Who are the people telling you otherwise? Do you understand context when people communicate to you as in school teachers? What a teacher may say may only apply to that classroom. – Logikal May 27 '20 at 18:02
  • @Logikal Thanks for the comment; all of it makes sense.What other animals can use deductive reasoning? – asdfasfasdgf May 27 '20 at 20:26
  • @asdfasfasdgf science has shown pigs, octopus, dogs, cats, birds, lions, hyenas, etc. Deductive reasoning does not have to be learned at school. – Logikal May 27 '20 at 20:31

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"Wrong" is not the correct term.  We'd simply rather axioms not be "Inconsistent".

This happened to be the case with the axioms of Cantorian Set Theory.  It was found to derive several contradictions, or paradoxes, such as the infamous Russell's Paradox.

As a result, the axioms of Cantorian set theory were formalised and refined in the hopes of eliminating such paradoxes; resulting in the axioms of Zermelo-Fraenkel Set Theory.

So that is what happens when the axioms of a theory are inconsistent.

Graham Kemp
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  • Very interesting. Thanks for the example. – asdfasfasdgf May 27 '20 at 01:11
  • @asdfasfasdgf: This answer is incorrect. Any foundational system generates theorems, each of which can have a truth value only under interpretation (which you have to decide for yourself). For example, an arithmetical sentence can be assigned a truth value based on the structure of natural numbers embedded in the real world as binary strings in some computing device. But there is no known real-world structure that obeys the axioms of ZFC, so sentences over ZFC may be completely meaningless. Furthermore, it may be that ZFC is consistent but proves a false arithmetical sentence. – user21820 Jul 31 '20 at 09:03