A Course on Set Theory by Ernest Schimmerling

By Ernest Schimmerling

Set concept is the math of infinity and a part of the middle curriculum for arithmetic majors. This e-book blends thought and connections with different components of arithmetic in order that readers can comprehend where of set concept in the wider context. starting with the theoretical basics, the writer proceeds to demonstrate purposes to topology, research and combinatorics, in addition to to natural set conception. innovations equivalent to Boolean algebras, bushes, video games, dense linear orderings, beliefs, filters and membership and desk bound units also are constructed. Pitched in particular at undergraduate scholars, the process is neither esoteric nor encyclopedic. the writer, an skilled teacher, contains motivating examples and over a hundred workouts designed for homework assignments, stories and assessments. it truly is applicable for undergraduates as a path textbook or for self-study. Graduate scholars and researchers also will locate it helpful as a refresher or to solidify their knowing of simple set idea.

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Then π(x) ∈ π(y) by the definition of π. On the other hand, π(x) ∈ π(x) by the Foundation Axiom. Hence π(x) = π(y). The proof is similar if y ≺ x. Every injection is a bijection with its range. It remains to see that π is order preserving. We already noted that if x ≺ y, then π(x) ∈ π(y). For the converse, suppose that π(x) ∈ π(y). Then there exists x ≺ y such that π(x ) = π(x). Since π is an injection, x = x. Hence x ≺ y as required. 2 Order π[A] is an ordinal. Proof If u ∈ v and v ∈ π[A], then there are x, y ∈ A such that u = π(x) and v = π(y), so u ∈ π[A].

8. 1. 3. 9 A is a transitive set iff for all x and y, if x ∈ y and y ∈ A, then x ∈ A. 2 Ordinal numbers 29 Equivalently, A is a transitive set iff for every y ∈ A, y ⊆ A. 15. First, here are three basic facts about how certain operations preserve transitivity. 10 transitive set. Let A be a transitive set. Then A ∪ {A} is also a Proof Let y ∈ A∪{A} and x ∈ y. We must show that x ∈ A∪{A}. In fact, we will show that x ∈ A. Either y ∈ A or y ∈ {A}. If y ∈ A, then x ∈ A since A is transitive. If y ∈ {A}, then y = A, so x ∈ A.

Then there is a unique function f with domain θ such that, for every γ < θ, ⎧ ⎪ if γ = 0 ⎨α f (γ) = f (β) + 1 if γ = β + 1 ⎪ ⎩ sup ({f (β) | β < γ}) if γ is a limit ordinal, namely, the function given by f (γ) = α + γ. Sketch of proof Use induction on γ < θ to see that that ordinal addition satisfies the three conditions we specified for f . Namely, • α + 0 = α, • α + (β + 1) = (α + β) + 1 and • if γ is a limit ordinal, then α + γ = sup ({α + β | β < γ}). 24. 5. Here is an entertaining false argument.

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