By K.T. Leung
Read Online or Download Elementary set theory PDF
Similar pure mathematics books
Fractals, Scaling and Growth Far From Equilibrium
This publication describes the growth that has been made towards the advance of a finished realizing of the formation of advanced, disorderly styles lower than faraway from equilibrium stipulations. It describes the applying of fractal geometry and scaling ideas to the quantitative description and knowing of constitution shaped lower than nonequilibrium stipulations.
Introduction to the Theory of Sets
Set conception permeates a lot of latest mathematical suggestion. this article for undergraduates deals a ordinary advent, constructing the topic via observations of the actual international. Its revolutionary improvement leads from finite units to cardinal numbers, endless cardinals, and ordinals. workouts look during the textual content, with solutions on the finish.
This can be the complaints of the AMS detailed consultation on nonstandard versions of mathematics and set thought held on the Joint arithmetic conferences in Baltimore (MD). the amount opens with an essay from Haim Gaifman that probes the concept that of nonstandardness in arithmetic and offers a desirable mixture of ancient and philosophical insights into the character of nonstandard mathematical constructions.
Extra info for Elementary set theory
Example text
Then λ = |P(κ)| = |κ 2| = |(κ×κ) 2| = |κ (κ 2)| = |κ λ| ≥ |cf (λ) λ| > λ. Cantor’s Theorem guarantees that for each ordinal α there is a set, P(α), which has cardinality greater than α. However, it does not imply, for example, that ω + = |P(ω)|. This statement is called the Continuum Hypothesis, and is equivalent to the third question in the introduction. 64 CHAPTER 7. CARDINALITY The aleph function ℵ : ON → ON is defined as follows: ℵ(0) = ω ℵ(α) = sup {ℵ(β)+ : β ∈ α}. We write ℵα for ℵ(α). We also sometimes write ωα for ℵ(α).
This leads to the contradiction ON ∈ ON. Theorem 11. (Trichotomy of Ordinals) (∀α ∈ ON)(∀β ∈ ON)(α ∈ β ∨ β ∈ α ∨ α = β). Proof. The reader may check that a proof of this theorem can be obtained by replacing “N” with “ON” in the proof of Theorem 7. Because of this theorem, when α and β are ordinals, we often write α < β for α ∈ β. Since N ⊆ ON, it is natural to wonder whether N = ON. In fact, we know that “N = ON” can be neither proved nor disproved from the axioms that we have stated (provided, of course, that those axioms are actually consistent).
Theorem 22. (∀X)(∃ <) [ X, < is a well ordered set]. Proof. We begin by using Theorem 5 to obtain a choice function f : P(X) \ {∅} → X such that for each nonempty A ⊆ X we have f (A) ∈ A. By recursion on ON we define g : ON → X ∪ {X} as: g(β) = f (X \ {g(α) : α < β}), if X \ {g(α) : α < β} = ∅; X, otherwise. 1) Now replace each x ∈ X ∩ran(g) by the unique ordinal β such that g(β) = x. The Axiom of Replacement gives the resulting set S ⊆ ON, where S = {β ∈ ON : g(β) ∈ X}. By Theorem 10 there is a δ ∈ ON \ S.