How can an axiom ever be dependent

Question

An axiom is defined, officially, as 'a statement or proposition which is regarded as being established, accepted, or self-evidently true.'

Yet an Independent axiom is one where it is not derived from other axioms within an axiomatic system, meaning a dependent one is. But doesn't this directly conflict with the definition of an axiom in the first place? If an axiom is dependent, surely it's not axiom?

Answer 1

There is nothing wrong per se in a theory with dependent axioms - they are just redundant.

Actually, you can think of axioms as special inference rules that allow you to infer the content of the axiom in one step.

And if you think of them that way, you will realize that mathematicians are adding axioms constantly, in the form of well-known theorems that they invoke without proof!

Note that this use of axioms does not conflict with the definition you gave, as they are established results.

Answer 2

Your "official" definition of "axiom" is rather outmoded and doesn't fit well with how the word has been used in contemporary mathematics for the last 100 years or so.

Today, axioms are simply the statements that we have decided to reason from for a particular purpose, no matter whether they are "self-evident" or not. (Or, in some other situations, axioms are just there to define what a model is, where it is the models we're really interested in -- such as the group axioms).

Once we have selected a set of axioms one of them is said to be independent of the rest of the set if one cannot derive it from the rest of the axioms. Conversely it is dependent if it can be derived from the other axioms.

But just because it is possible to derive such-and-such axioms from the other ones doesn't mean there is anything wrong with considering to be an axiom anyway. Being dependent means that it doesn't matter whether we do, in the sense that the statements we can derive are the same in the two situations. But there may be reasons to include the axiom anyway -- for example, that it is simpler to describe the full set of axioms that it would be to explicitly exclude all of the not-strictly-necessary ones.

In many situations, such as theories with infinitely many axioms (both PA and ZFC are examples) it is usually not even possible to know for sure which of the axioms are independent of the rest, and which aren't. So insisting on only wanting independent axioms in one's theory is counterproductive.