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Unformatted text preview: Lecture 1: Introduction and Axioms January 19, 2009 1 Introduction Remark. Generally speaking, there are two sorts of mathematical theories: those, like number theory, that seek to describe and explore some particular structure; and those, like group theory, that seek to generalize structures found elsewhere. In this respect, we conceive of set theory as being more like number theory: it seeks to describe the cumulative hierarchy of sets. The cumulative hierarchy of sets begins with just the empty set; then we use the power set operator to build further levels. V = ∅ V n +1 = P ( V n ) . Each of these stages is finite. We can then construct a limit stage as follows: V ω = [ n ∈ N V n . V ω is known as the collection of hereditarily finite sets : any set in V ω is finite, and so are its elements, and the elements of its elements, and the elements of elements of elements. . . and so on. We also want to be able to deal with infinite sets—in fact, set theory was introduced by Cantor in the 1880s partly to deal with infinity in the context of real analysis. So we can introduce V ω +1 = P ( V ω ) and so on, leading to the notion of transfinite ordinals . Much more will be said about this in time. So set theory is about saying as much as possible about this cumulative hi erarchy of sets; characterizing both the “height” (ordinals) and “width” (power set operator). 2 ZermeloFrankel Axioms Remark. This axiomatization of set theory was developed first by Zermelo in 1908 and extended by Frankel and Skolem in 1922/3. The ZermeloFrankel axioms with the Axiom of Choice are often abbreviated “ZFC”. This is a first order theory with = and ∈ the only nonlogical symbols. The axioms are as follows. 1 1. Extensionality. ∀ x. ∀ y. ( ∀ z.z ∈ x ⇔ z ∈ y ) ⇒ x = y. Two sets are equal if they have the same elements. 2. Pairing. ∀ x. ∀ y. ∃ z. ∀ w. ( w ∈ z ⇔ ( w = x ∨ w = y )) . We can form a set with two given sets as elements (i.e. an unordered pair). Note that we can now form “ordered pairs” by identifying the ordered pair ( x, y ) with the set {{ x } , { x, y }} ; note that the set { x } exists by the axiom of Pairing (pair x with itself). 3. Union. ∀ x. ∃ y. ∀ z.z ∈ y ⇔ ( ∃ w.w ∈ x ∧ z ∈ w ) . For every set x there exists a set y (which we will abbreviate S x ) whose elements are exactly the elements of the elements of x . 4. Power set. ∀ x. ∃ y. ∀ z.z ∈ y ⇔ z ⊆ x. For every set x there exists a set y (abbreviated P ( x )) whose elements are precisely the subsets of x . Note that we use z ⊆ x as an abbreviation for ∀ w.w ∈ z ⇒ w ∈ x . 5. Comprehension. If ϕ is a firstorder formula which does not contain y free, then ∀ x. ∃ y. ∀ z.z ∈ y ⇔ ( z ∈ x ∧ ϕ ( z )) ....
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 Fall '10
 RussellMiller
 Set Theory, Number Theory

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