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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.

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  • $\begingroup$ Maybe useful the post Are “if” and “iff” interchangeable in definitions? $\endgroup$ – Mauro ALLEGRANZA Jan 26 at 19:24
  • $\begingroup$ 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. $\endgroup$ – Mauro ALLEGRANZA Jan 26 at 19:27
  • $\begingroup$ @MauroALLEGRANZA what do you mean by described through? $\endgroup$ – user638203 Jan 26 at 20:38
  • $\begingroup$ 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. $\endgroup$ – Mauro ALLEGRANZA Jan 27 at 8:47
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Let's consider two different words: axiom and definition.

An axiom is something that is accepted to be true without proof. For example, one axiom of Euclidean geometry is "Two parallel lines never meet". This is never proved in Euclidean geometry; it's a fact that's accepted. Any type of math has a set of axioms. This is what a type of math is, a set of axioms. Then, as mathematicians, we ask ourselves, "What can we prove given the set of axioms that we have?"

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 $x\in\mathbb{R}$"

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