Cofinality and inaccessibles

1. Introduction

This page is being constructed. It will cover issues to do with cofinality, and some applications of König's inequality on cardinals.

We assume the Axiom of Choice throughout and identify cardinals with initial ordinals. Arithmetic, such as addition, multiplication and exponentiation is always cardinal arithmetic unless stated otherwise.

2. Regular and singular cardinals

Definition.

A cardinal $κ$ is a successor cardinal if it is $ℵ_{α}$ for a successor ordinal $α$ . It is a limit cardinal if it is $ℵ_{λ}$ for a limit ordinal $λ$ .

Definition.

A cardinal $κ$ is said to be regular if it is not the sum of a collection $κ_{i}$ of strictly smaller cardinals, where $i$ runs over an index set $I$ with $card (I) < κ$ . A cardinal is singular if it is not regular.

It is easy to see from ${ℵ_{α}}^{2} = ℵ_{α}$ that if $κ_{i} ⩽ ℵ_{α}$ and $card (I) ⩽ ℵ_{α}$ then $\sum_{i \in I}^{} κ_{i} ⩽ ℵ_{α}$ . Thus we derive the important result that

Theorem.

Every succesor cardinal $ℵ_{α + 1}$ is regular.

3. Cofinality

Definition.

The cofinality $cf (α)$ of a cardinal or initial ordinal $α$ is the least ordinal $β$ such that there are sets $A_{i} \subseteq α$ for all $i < β$ with each $A_{i}$ having cardinality strictly less than $α$ and with $α = ⋃ \{A_{i}: i < β\}$ .

Thus a regular cardinal is one, $α$ , such that $cf (α) = α$ . A singular cardinal has $cf (α) < α$ .

The following is easy.

Proposition.

The cofinality of a cardinal $α$ is an initial ordinal (i.e. a cardinal).

The next theorem is more interesting.

Theorem.

If $κ ⩾ ℵ_{0}$ is a cardinal then $κ < κ^{cf κ}$ .

Proof.

Suppose cardinals $α_{i} < κ$ are given for $i < cf κ$ such that $κ = \sum_{i < cf κ}^{} α_{i}$ . Then by the König inequality

κ = \sum_{i < cf κ}^{} α_{i} < \prod_{i < cf κ}^{} α_{i} ⩽ κ^{cf κ}

as required.

Theorem.

If $κ ⩾ ℵ_{0}$ and $λ ⩾ 2$ are cardinals then $κ < cf (λ^{κ})$ .

Proof.

Let $μ = λ^{κ}$ . If $κ ⩾ cf μ$ then by the previous theorem

μ < μ^{cf μ} ⩽ μ^{κ} = (λ^{κ})^{κ} = λ^{κ} = μ

a contradiction.

In particular $cf (2^{ℵ_{0}}) > ℵ_{0}$ so for example $2^{ℵ_{0}} \neq ℵ_{ω}$ .

4. Inaccessible cardinals

We saw that every succesor cardinal is regular. Whether every limit cardinal is singular is more difficult. A regular limit cardinal $λ$ is said to be (weakly) inaccessible. It is strongly inaccessible if in addition $μ < λ$ implies $2^{μ} < λ$ for all cardinals $μ$ .

The non provability of the existence of strongly inaccessible cardinals is rather easier to show.

Theorem.

If $κ$ is a strongly inaccessible cardinal then $V_{κ}$ is a model of $ZFC$ .

ZFC cannot prove the existence of a strongly inaccessible cardinal.

Proof.

Otherwise ZFC would prove the existence of a model of ZFC which by the Soundness Theorem (formalised and proved inside ZFC) would imply that $ZFC ⊢ Con (ZFC)$ contradicting Gödel's second incompleteness theorem.

If ZFC is consistent so is ZFC plus the assertion that there is no strongly inaccessible cardinal.

Proof.

If $V$ is a model of ZFC then take the smallest strongly inaccessible cardinal $κ$ in $V$ and observe that $V_{κ}$ satisfies the required theory, so this theory is consistent by soundness.

It turns out that by a construction using Gödel's constructible universe $L$ , similar results apply to weakly inaccessible cardinals too.