nLab measurable cardinal

Definition

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

Properties

Any measurable cardinal is, in ZFC, necessarily inaccessible, and in fact much larger than the smallest inaccessible. In fact, if $\kappa$ is measurable, then there is a $\kappa$-complete ultrafilter $\mathcal{U}$ on $\{\lambda | \lambda \lt \kappa\}$ which contains the set $\{\lambda | \lambda \lt \kappa$ and $\lambda$ is inaccessible $\}$. In particular, there are $\kappa$ inaccessible cardinals smaller than $\kappa$. Note that in ZF it is consistent that $\omega_1$, a successor cardinal, is measurable.

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 $V$ into some submodel $M$ (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 cardinals, and weakly compact cardinal?s) and “large” large cardinals (such as strongly compact cardinals, 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 existence of a measurable cardinal is equivalent to the existence of an exact functor $F: Set \to Set$ that is not naturally isomorphic to the identity. This was essentially proved by V. Trnková, and it was rediscovered by Blass in his paper “Exact functors and measurable cardinals” (Blass 1976).

Furthermore, the category $Set^{op}$ 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 $\lt\lambda$ is dense in $Set^{op}$ precisely when there are no measurable cardinals larger than $\lambda$. In particular, the full subcategory on $\mathbb{N}$ is dense in $Set^{op}$ precisely when there are no measurable cardinals at all.

This is theorem A.5 of Locally Presentable and Accessible Categories.

Related entries

• large cardinal

• real-valued-measurable cardinal

References

• M. Adelman, A. Blass, Exact functors, local connectedness and measurable cardinals , Rend. Sem. Mat. Fis. Milano 54 (1984) pp.9-28.

• Andreas Blass, Exact Functors and Measurable Cardinals , Pacific J. Math. 63 (1976) pp.335-346. (euclid)

• Andreas Blass, Corrections to: ‘Exact Functors and Measurable Cardinals’ , Pacific J. Math. 73 (1977) p.540. (euclid)

• John Isbell, Adequate subcategories , Illinois J. Math. 4 (1960) pp.541-552. MR0175954.

  (euclid)

• John Isbell, Subobjects, adequacy, completeness and categories of algebras , Rozprawy Mat. 36 (1964) pp.1-32. (toc)

• David P. Blecher?, Nik Weaver, Quantum measurable cardinals (arXiv:1607.08505)