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Good Math by Mark C. Chu-Carroll

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The Axioms of ZFC Set Theory

In this section, we’re going to walk through the process of creating a sound set theory, a version of set theory that preserves the intuition and simplicity of naive set theory but won’t fall into the trap of inconsistency.

Keep in mind as we go that the point of what we’re doing is to produce something that is a foundation. As a foundation, it cannot depend on anything except for first-order predicate logic and the axioms themselves. Until we can construct them using the axioms, there are no numbers, no points, no functions! We can’t assume that anything exists until we show how it can be constructed using these axioms.

What do we need to create a sound version of set theory? In naive set theory, we started ...

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