nLab
measurable cardinal

Contents

Definition

A cardinal number κ\kappa is measurable if some (hence any) set of cardinality κ\kappa admits a two-valued measure which is κ\kappa-additive, or equivalently an ultrafilter which is κ\kappa-complete.

Properties

Any measurable cardinal is necessarily inaccessible, and in fact much larger than the smallest inaccessible. In fact, if κ\kappa is measurable, then there is a κ\kappa-complete ultrafilter 𝒰\mathcal{U} on {λλ<κ}\{\lambda | \lambda \lt \kappa\} which contains the set {λλ<κ\{\lambda | \lambda \lt \kappa and λ\lambda is inaccessible }\}. In particular, there are κ\kappa inaccessible cardinals smaller than κ\kappa.

It follows from this that the existence of any measurable cardinals cannot be proven in ZFC, since the existence of inaccessible cardinals cannot be so proven. Thus measurable cardinals are a kind of large cardinal. They play an especially important role in large cardinal theory, since any measurable cardinal gives rise to an elementary embedding of the universe VV into some submodel MM (such as an ultrapower by a countably-complete ultrafilter), while the “critical point” of any such embedding is necessarily measurable.

Measurable cardinals are sometimes said to mark the boundary between “small” large cardinals (such as inaccessibles, Mahlo cardinal?s, and weakly compact cardinal?s) and “large” large cardinals (such as strongly compact cardinal?s, supercompact cardinals, and so on).

In category theory

The existence or nonexistence of measurable cardinals can have noticeable impacts on category theory, notably in terms of the properties of the category Set.

For instance, the category Set opSet^{op} has a small dense subcategory if and only if there does not exist a proper class of measurable cardinals. Specifically, the subcategory of all sets of cardinality <λ\lt\lambda is dense in Set opSet^{op} precisely when there are no measurable cardinals larger than λ\lambda. In particular, the full subcategory on \mathbb{N} is dense in Set opSet^{op} precisely when there are no measurable cardinals at all.

This is theorem A.5 of LPAC.

Revised on September 5, 2011 16:14:30 by Toby Bartels (75.88.82.16)