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In Naive Set Theory, Halmos states the following:

The principal concept of set theory, the one that in completely axiomatic studies is the principal primitive (undefined) concept, is that of belonging.

I'm surprised. Is $\in$ really not defined in axiomatic studies?

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    No; it is a primitive notion. You describe its properties, you don't say what it is. That's the nature of axiomatic theory: you have a bunch of undefined notions, and some axioms about how they behave and what their properties are. – Arturo Magidin Jun 10 '19 at 18:26
  • It is not defined.This gives great flexibility. For example Russell's Paradox is that it is illogical to assert the existence of a widget $W$ that digs every widget that does not dig itself and does not dig any widget that digs itself. (Replace "widget" with "set" and "dig(s)" with "has as a member"). It doesn't matter what widgets are or what "dig(s)" means. – DanielWainfleet Jun 11 '19 at 04:13
  • See The Frege-Hilbert Controversy for a discussion between founding fathers of modern axiomatic method. – Mauro ALLEGRANZA Jun 11 '19 at 19:08

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