Math 558 – Homework #4

Due December 10, 2009

1. Define ordinal exponentiation by transfinite recursion as follows.

\begin{aligned}α^{0} & = & 1, \\ α^{β+1} & = & α^{β} · α, \\ α^{δ} & = & sup_{β<δ} α^{β} \end{aligned}

for all limit ordinals δ. In this exercise we shall obtain a more explicit representation of α^{β}.

Let α = type((A, R)) and β = type((B, S)). Let a_{0} be the R-least element of A. Let C be the set of functions f : B → A such that f(b) = a_{0} for all but finitely many b ∈ B. Let T be the ordering of C by last difference (see Exercise 4.4.4 in the lecture notes). Show that α^{β} = type((C, T)).

2. Let X be a set of sets. By a chain within X we mean a set C ⊆ X such that for all U, V ∈ C either U ⊆ V or V ⊆ U. A chain within X is said to be maximal if it is not properly included in any other chain within X.

Use the Axiom of Choice plus transfinite recursion to prove that there exists a maximal chain within X.

Note: This is a version of Zorn’s Lemma.

3. Let λ be an infinite cardinal. The cofinality of λ, abbreviated cf(λ), is defined to be the least κ such that λ can be written as the sum of κ-many cardinals each less than λ. Note that cf(λ) ≤ λ.

(a) Show that cf(λ) is an infinite regular cardinal.

(b) Show that for any infinite regular cardinal κ there exist arbitrarily large strong limit cardinals λ such that cf(λ) = κ.

(c) Show that λ is regular if and only if cf(λ) = λ.

(d) For any infinite cardinal κ, show that cf(μ^{κ}) > κ for all μ ≥ 2. Two special cases of this are cf(2^{κ}) > κ and λ^{cf(λ)} > λ.

(e) Assuming the GCH, prove that for all infinite cardinals κ and λ we have

λ^{κ} = \{ \begin{matrix}λ & if κ < cf(λ), \\ λ^{+} & if cf(λ) ≤ κ ≤ λ, \\ κ^{+} & if κ ≥ λ. \end{matrix}