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.

In constructive set theories

There are two different definitions of a measurable cardinal: Stanisław Ulam‘s definition of a measurable cardinal in terms of ultrafilters and Dana Scott’s definition of a measurable cardinal in terms of elementary embeddings.

In IZF, ultrafilters and point-set measure theory more generally are not well-behaved in constructive mathematics. As a result, Ulam’s usual definition of a measurable cardinal in terms of ultrafilters is also not well-behaved in IZF. Ultrafilters are thus typically replaced with completely prime filters on locales or some other concept in point-free measure theory. One can define a point-free version of Ulam’s measurability for cardinals, but the existence of such point-free measurable cardinals is still equivalent to the existence of Scott’s measurable cardinals defined in terms of elementary embeddings.

For IZF, the category of sets having a non-trivial exact functor $F:\mathrm{Set} \to \mathrm{Set}$ is not equivalent to a measurable cardinal in IZF. In particular, if the category of sets is not a Boolean category, then the structure of an exact functor is insufficient to preserve first-order logic in the category of sets, needed for the definition of a measurable cardinal in terms of elementary embeddings. Instead, having a measurable cardinal $\kappa$ in IZF is equivalent to having a non-trivial logical functor $F:\mathrm{Set} \to \mathrm{Set}$ that commute with $\kappa$-small colimits and defined uniformly via a class-sized formula across the structural hierarchy of sets.

In IZF, like ZFC, the axiom of constructibility is inconsistent with measurable cardinals.

Related entries

• large cardinal

• real-valued-measurable cardinal

References

Stanisław Ulam’s definition of a measurable cardinal in terms of ultrafilters first appears in:

• Stanisław Ulam: Zur Masstheorie in der allgemeinen Mengenlehre, Fundamenta Mathematicae, 16 (1930): 140–150 [doi:10.4064%2Ffm-16-1-140-150, eudml:doc/212487]

Dana Scott’s definition of a measurable cardinal in terms of elementary embeddings first appears in:

• Dana Scott: Measurable cardinals and constructible sets, Bulletin de l’ Académie Polonaise des Sciences, Série des sciences mathématiques, astronomiques et physiques, vol. 9 (1961), pp. 521–524. [pdf]

Measurable cardinals in category theory:

• 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)

• categorical characterization of large cardinals, MathOverflow (web)

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

Some discussion on measurable cardinals in the context of a polynomial function whose Lesbegue measurability is independent of ZFC occurs in:

• James E. Hanson, Any function I can actually write down is measurable, right? (arXiv:2501.02693)