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NOTES ON SET THEORY

We will now axiomatize set theory. From this point on, we cannot use any previous knowledge of mathematics, aside from providing intuition or motivation. Everyday words such as “finite” or “number” no longer have any technical meaning for us, except as specified by our axioms and definitions. These axioms and definitions will become our entire world, and we must reconstruct mathematics within it. The only serious gap in our formalization will be the use of terms such as “and”, “or”, “implies”, “there exists”, etc. from logic. We’ll deal with this gap in the second part of the course, but for now we will take logic for granted.