TITLE: What is exactly the difference between a definition and an axiom?
QUESTION [61 upvotes]: I am wondering what the difference between a definition and an axiom.
Isn't an axiom something what we define to be true?
For example, one of the axioms of Peano Arithmetic states that $\forall n:0\neq S(n)$, or in English, that zero isn't the successor of any natural number. Why can't we define 0 to be the natural number such that it isn't the successor of any natural number?
In general, why is an axiom not a definition? 
Are there cases where we can formulate definitions as axioms or vice versa?

REPLY [2 votes]: None of the answers satisfied me. I had to look at Enderton "A mathematical Introduction to Logic" Section 2.7, the section entitled "Defining functions", especially theorem 27A, to be satisfied.   The point is that the question deserves a formal answer in the form of a  theorem and a proof, not only examples. Examples are useful to convey the basic idea, but most people already have the basic idea, which is that, as Enderton says, "unlike axioms, we do not expect definitions to provide substantive information".   Enderton covers the general case where functions are defined. For simplicity, he considers a function $f$ of arity $1$, but it is the same for arity $n$.  We want to add a function symbol $f$ to the language, introducing it by a definition of the form
$$\forall v_1\; \forall v_2[f(v1) = v2 \leftrightarrow \phi] \tag{$\delta$}\label{def}$$
where $\phi$ is a property that does not contain the symbol $f$ and in which $v_1$ and $v_2$ (and only $v_1$ and $v_2$) occur free (i.e. without being bounded by a quantifier). Because formula \eqref{def} looks like an axiom, the question arises what makes it different from an axiom. The first ingredient of the answer is that equation \eqref{def} adds a new symbol $f$, so it creates a different language, whereas an axiom does not add a new symbol. Instead, it relates existing symbols. But there is more to say. Because equation \eqref{def} is only a definition of the new symbol $f$, the new theory (i.e., the original theory + the definition) should not contain additional "substantive information". But this is something that deserves to be proved. Formally, what needs to be proved is that, if any sentence $\sigma$ in which the new symbol $f$ does not occur is true when we add \eqref{def} to the original theory, then it is also true in the original theory without \eqref{def}.  In practice, when a function is defined using equation \eqref{def}, one simply shows that indeed $f$ is well defined by this equation, that is, one shows
$$\forall v_1 \, \exists!v_2\; \phi. \tag{$\epsilon$} \label{welldefined}$$
One does not show that no additional information is provided. It is not necessary because theorem 27A in Enderton's book says that the condition \eqref{welldefined} implies that the definition provides no additional substantive information.  The proof takes about half a page.