A Course on Set Theory by Ernest Schimmerling

By Ernest Schimmerling

Set concept is the math of infinity and a part of the center curriculum for arithmetic majors. This ebook blends idea and connections with different components of arithmetic in order that readers can comprehend where of set conception in the wider context. starting with the theoretical basics, the writer proceeds to demonstrate functions to topology, research and combinatorics, in addition to to natural set thought. strategies comparable 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 routines designed for homework assignments, stories and checks. it really 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 realizing of easy set idea.

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11, since (1)γ holds, (1)δ holds too. Also, since (1)γ holds, if y ∈ Vγ , then y ⊆ Vγ , so y ∈ Vδ . This shows that Vγ ⊆ Vδ . This conclusion together with (2)γ implies that (2)δ holds. Limit case δ is a limit ordinal. In this case, Vδ = {Vγ | γ < δ}. 12 and the assumption that (1)γ for γ < δ. Also, (2)δ follows from (2)γ for γ < δ. Here is an interesting fact that plays no role in the rest of the book. It turns out that the Foundation Axiom is equivalent to the statement ∀x ∃α (α is an ordinal and x ∈ Vα ).

Also, ω is an ordinal. We have also seen transitive sets that are not ordinals. For example, for every n ∈ ω, if n > 2, then Vn is a transitive set that is not an ordinal. It is important to have a reasonably good picture of where we are headed before plunging into technical facts about ordinals. 2 Ordinal numbers 31 addition in the next section. Here is the picture you should have in mind. Starting from the empty set, we use the operation α → α ∪ {α} at successor stages and take unions at limit stages to generate all the ordinals beginning with the natural numbers 0 = ∅, 1 = {0}, 2 = {0, 1}, 3 = {0, 1, 2}, etc.

Then α ⊆ γ since γ is transitive. Hence β = α ∪ {α} ⊆ γ. 19, β ≤ γ. 21, it is natural to write α + 1 = α ∪ {α} for ordinals α. We call α + 1 a successor ordinal. Non-zero ordinals that are not successor ordinals are called limit ordinals. 22 Let A be a set of ordinals and β = A. Then β is an ordinal and, for every α ∈ A, α ≤ β. Moreover, if γ is an ordinal and α ≤ γ for every α ∈ A, then β ≤ γ. 12. Recall that supremum is another way to say least upper bound. 22 says that if A is a set of ordinals, then sup(A) = A.

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