Measurable Cardinals and Ultrafilters

Measurable cardinals mark a crucial threshold in the large cardinal hierarchy, connecting set theory with measure theory and the theory of elementary embeddings.

Definition

Definition8.4Measurable cardinal

A cardinal $\kappa$ is measurable if there exists a $\kappa$ -complete non-principal ultrafilter $\mathcal{U}$ on $\kappa$ . Here:

Ultrafilter: For every $A \subseteq \kappa$ , either $A \in \mathcal{U}$ or $\kappa \setminus A \in \mathcal{U}$ .
Non-principal: $\{a\} \notin \mathcal{U}$ for any $a \in \kappa$ .
$\kappa$ -complete: If $\{A_\alpha\}_{\alpha < \gamma}$ with $\gamma < \kappa$ and each $A_\alpha \in \mathcal{U}$ , then $\bigcap_\alpha A_\alpha \in \mathcal{U}$ .

Theorem8.2Properties of measurable cardinals

Every measurable cardinal $\kappa$ is:

Strongly inaccessible (and hence a limit of inaccessibles).
Weakly compact (and hence Mahlo, and a limit of Mahlos).
The $\kappa$ -th inaccessible cardinal.

Thus measurable cardinals are far above inaccessible and Mahlo in the large cardinal hierarchy.

Ultrapower Embeddings

Definition8.5Ultrapower

Given a $\kappa$ -complete ultrafilter $\mathcal{U}$ on $\kappa$ , the ultrapower $\mathrm{Ult}(V, \mathcal{U})$ is defined as follows. Consider all functions $f: \kappa \to V$ and define the equivalence relation $f =_{\mathcal{U}} g$ iff $\{\alpha : f(\alpha) = g(\alpha)\} \in \mathcal{U}$ . The universe of the ultrapower is $V^\kappa / \mathcal{U}$ , with membership $[f] \in_{\mathcal{U}} [g]$ iff $\{\alpha : f(\alpha) \in g(\alpha)\} \in \mathcal{U}$ .

By Los's theorem: $\mathrm{Ult}(V, \mathcal{U}) \models \varphi([f_1], \ldots, [f_n])$ iff $\{\alpha : V \models \varphi(f_1(\alpha), \ldots, f_n(\alpha))\} \in \mathcal{U}$ .

Theorem8.3Scott's theorem

If there exists a measurable cardinal, then $V \neq L$ (the universe is not the constructible universe). More precisely, the ultrapower embedding $j: V \to M$ with critical point $\kappa$ cannot factor through $L$ .

The Embedding Characterization

RemarkMeasurability via embeddings

A cardinal $\kappa$ is measurable if and only if there exists an elementary embedding $j: V \to M$ (where $M$ is a transitive class and $j$ preserves all first-order properties) with critical point $\kappa$ : $j(\alpha) = \alpha$ for all $\alpha < \kappa$ and $j(\kappa) > \kappa$ .

This embedding characterization unifies the large cardinal hierarchy: stronger large cardinals correspond to embeddings with more closure properties on the target model $M$ .

ExampleLarge cardinals via embeddings

Measurable: $j: V \to M$ with critical point $\kappa$ .
Strong: $j: V \to M$ with $V_{j(\kappa)} \subseteq M$ .
Supercompact: $j: V \to M$ with ${}^{j(\kappa)}M \subseteq M$ (closure under $j(\kappa)$ -sequences).
Extendible: $j: V_{\kappa+\alpha} \to V_{j(\kappa+\alpha)}$ for every $\alpha$ .

Each level demands more of the target model $M$ , yielding strictly stronger large cardinal notions.