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As we know, an axiom cannot be proven. It is an assumption that certain properties exist, and then we can learn more interesting properties (theorems) from anything that obeys these axioms.

So although you cannot prove an axiom, is there a way of proving whether an axiom is even allowed, whether it is sort of compatible with other axioms? Is this something that should be done before studying properties that are derived from having this axiom?

An example that comes to mind is the Axiom of Completeness, which states that every subset that is bounded above has a least upper bound. (Depending on the other axioms of the system, this may actually be a property that can be derived, but we will assume that this is one of the axioms). Is it necessary to show that such an axiom is compatible with the other axioms of the system and does not lead to contradictions, or is one allowed to just assume that the axiom holds? It doesn't feel right to take it as an axiom without any evidence that it fits in.

It feels pointless to study a system where an axiom might make the system actually impossible to exist. Then again, I remember hearing from someone that all axiomatic systems eventually lead to contradictions somewhere.

Can someone give me more information on all of this?

David Callanan
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    A theory is inconsistent if its axioms lead to contradictions. This is generally considered to be a bad thing, and to be avoided. In many situations, we can prove the axioms are not inconsistent. In some important cases, e.g., the usual axioms for set theory, we don't have a consistency proof, but we haven't found any contradictions yet, so we forge ahead. – Gerry Myerson Mar 13 '22 at 11:53
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    This being said, it is worth noting that logically speaking, inconsistent theories pose no problem. They just don't have any models, and can prove any statement (and its negation, of course). So they are just very boring. – Captain Lama Mar 13 '22 at 12:10
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    It may be the case that the axioms can prove that if they are consistent, then adding the new axiom won't change that. That's probably the best you can do. – eyeballfrog Mar 13 '22 at 13:07
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    There are certainly horror stories about people doing PhDs or whatever, proving all kinds of interesting things about objects that turned out not to exist. This kind of thing does happen sometimes. – Patrick Stevens Mar 13 '22 at 13:11
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    @Patrick Stevens: interesting things about objects that turned out not to exist --- An example by the highly respected mathematician Dini is described in this answer. – Dave L. Renfro Mar 13 '22 at 15:40

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Suppose that a system of axioms is consistent. An axiom is compatible with this system if after adding it, the resulting system is still consistent. Yes, you should check whether adding this additional axiom preserves consistency. The technical name is known as equi-consistency. We say that the larger axiom system is equi-consistent with the smaller axiom system if the existence of a model of the smaller system implies the existence of a model for the larger system.

For the Axiom of Completeness, if the base axiom system is that of an ordered field, then we start with a model of the rational numbers and then construct a model of the real numbers using either Dedekind cuts or Cauchy sequences.

If you add an axiom to a base system and the resulting larger system is inconsistent, it means that the negation of the axiom is provable from the base system.