# What is the difference between an axiomatization and a definition?

It is sometimes said that some things are not ever defined but instead axiomatized. What does this mean exactly and what is the difference?

For example in set theory the symbol $$\in$$ is not ever actually defined but instead "axiomatized" in set theory.

• Maybe useful the post Are “if” and “iff” interchangeable in definitions? – Mauro ALLEGRANZA Jan 26 at 19:24
• In every context, we cannot define everything. Thus, the basic concept of set theory, the relation $\in$, is not explicitly defined but only "described" through" the axioms of the theory. – Mauro ALLEGRANZA Jan 26 at 19:27
• @MauroALLEGRANZA what do you mean by described through? – user638203 Jan 26 at 20:38
• The axioms "describe" the proeprties of the "belongs to" relation, in the sense that we call sets every colelction of objects with a binary relation between them that satisfies the axioms of the theory. – Mauro ALLEGRANZA Jan 27 at 8:47

Given a set of axioms, we can create definitions. A definition is simply a meaning stated to a word, and it is logically equivalent to "if and only if". For example, we can say, "A value $$\gamma$$ is the supremum of a function $$f$$ that maps real numbers onto the real numbers means $$f(x)\leq\gamma$$ for all $$x\in\mathbb{R}$$". Logically, this is equivalent to saying "A value $$\gamma$$ is the supremum of a function $$f$$ that maps real numbers onto the real numbers if and only if $$f(x)\leq\gamma$$ for all $$x\in\mathbb{R}$$"