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By Terence Tao

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Fundamentals 41 demonstrates this non-emptyness. 12) we will show that given any finite number of non-empty sets, say Ab ... , An, it is possible to choose one element x1, ... , Xn from each set A1, ... , An; this is known as "finite choice". 4. 8. Note that the empty set is not the same thing as the natural number 0. One is a set; the other is a number. 6. 2 was the only axio~ that set theory had, then set theory could be quite boring, as there might be just a single set in existence, the empty set.

While the operation of union has some similarities with addition, the two operations are not identical. For instance, {2} U {3} = {2, 3} and 2 + 3 = 5, whereas {2} + {3} is meaningless (addition pertains to numbers, not sets) and 2 U 3 is also meaningless (union pertains to sets, not numbers). This axiom allows us to define triplet sets, quadruplet sets, and so forth: if a, b, care three objects, we define {a, b, c} := {a}U{b}U {c}; if a, b, c, d are four objects, then we define {a, b, c, d} := {a} U {b}U{c}U{d}, and so forth.

We have already introduced one type of number system, the natural numbers. We will introduce the other number systems shortly, but for now we pause to introduce the concepts and notation of set theory, as they will be used increasingly heavily in later chapters. ) While set theory is not the main focus of this text, almost every other branch of mathematics relies on set theory as part of its foundation, so it is important to get at least some grounding in set theory before doing other advanced areas of mathematics.

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