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I am trying to understand the hiearchy of foundational concepts in mathematics/logic.

Specifically, is it true that objects that are left undefined or primitive or atomic are in some sense "below" axioms? As an example, in Euclid's geometry a point is primitive (undefined, at least mathematically precisely) whereas the first postulate (axiom) talks about how to define a line segment from two points.

Halmos' Naive Set Theory (p2) seems to get the "natural" picture above backwards since it makes a point of using an axiom regarding the equality of sets as more fundamental (implying) membership in a set which it previously referred to as a primitive concept in axiomatic set theory.

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    I don't know if the following might be helpful: http://math.stackexchange.com/questions/140681/where-to-begin-with-foundations-of-mathematics2012-05-17
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    In some sense, yes, membership of sets is more primitive than equality of sets.2012-05-17
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    Some philosophers would hold that the fact that any language has meaning would be prior to the statement of any axiom ... !2012-05-17

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