What if our axioms are false? What happens then?

  • 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
  • 1
    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 6:27
  • 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

If an axiom in a deductive system is contingently false (i.e. could be true, but isn't), you can come to some false conclucions. For example, if you axiomatically assume all mammals are viviparous, you can come to the conclusion that momotremes are viviparous, which they are not.

If an axiom in a deductive system is logically false (i.e. cannot be true for logical reasons), you can come to any conclusion you want. This effect has been known in the latin ancient world as "ex falso (sequitur) quodlibet (EFQ)". The reason is the way the deductive argument form of modus ponens works. From the axiom "A and not-A" you can virtually conclude anything!

In an axiomatic system that deals with abstract objects -- think of mathematical systems -- it might not from the beginning be obvious whether the single axioms are true or false. It may then, in the process of deducing, by contradiction turn out that they are inconsistent, i.e. that not all of them can be true together. Then you are back to one of the two cases above.


"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.

  • Very interesting. Thanks for the example. – asdfasfasdgf May 27 '20 at 1:11
  • @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 9:03

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