February 11, 2018

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Let λ = |P(κ)|. Suppose cf (λ) ≤ κ. 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}.

A cardinal κ is said to be inaccessible whenever both κ is regular and ∀λ < κ |P(λ)| < κ. An inaccessible cardinal is sometimes said to be strongly inaccessible, and the term weakly inaccesible is given to a regular cardinal κ such that ∀λ < κ λ+ < κ. Under the GCH these two notions are equivalent. Axiom of Inaccessibles ∃κ κ > ω and κ is an inaccessible cardinal This axiom is a stronger version of the Axiom of Infinity, but the mathematical community is not quite ready to replace the Axiom of Infinity with it just yet.

In particular, ω is regular. Lemma. 1. For each limit ordinal κ, cf (κ) is a regular cardinal. 2. For each limit ordinal κ, κ+ is a regular cardinal. 3. Each infinite singular cardinal contains a cofinal subset of regular cardinals. Exercise 22. Prove this lemma. Theorem 27. (K¨onig’s Theorem) For each infinite cardinal κ, |cf (κ) κ| > κ. Proof. We show that there is no surjection g : κ → δ κ, where δ = cf (κ). Let f : δ → κ witness that cf (κ) = δ. Define h : δ → κ such that each h(α) ∈ / {g(β)(α) : β < f (α)}.

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An Introduction to Set Theory by W. Weiss

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