+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ### Context #### Foundations +-- {: .hide} [[!include foundations - contents]] =-- =-- =-- # Contents * automatic table of contents goes here {: toc} ## 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 cardinal|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 cardinal]]s, and [[weakly compact cardinal]]s) and "large" large cardinals (such as [[strongly compact cardinal]]s, [[supercompact cardinal]]s, 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](#Blass_76)). Furthermore, the category $Set^{op}$ has a [[small category|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, [[Andreas Blass|A. Blass]], _Exact functors, local connectedness and measurable cardinals_ , Rend. Sem. Mat. Fis. Milano **54** (1984) pp.9-28. * {#Blass_76} [[Andreas Blass]], _Exact Functors and Measurable Cardinals_ , Pacific J. Math. **63** (1976) pp.335-346. ([euclid](https://projecteuclid.org/euclid.pjm/1102867389)) * [[Andreas Blass]], _Corrections to: 'Exact Functors and Measurable Cardinals'_ , Pacific J. Math. **73** (1977) p.540. ([euclid](https://projecteuclid.org/euclid.pjm/1102810622)) * {: #MR0175954 } [[John Isbell]], _Adequate subcategories_ , Illinois J. Math. **4** (1960) pp.541-552. [MR0175954](http://www.ams.org/mathscinet-getitem?mr=0175954). ([euclid](https://projecteuclid.org/euclid.ijm/1255456274)) * [[John Isbell]], _Subobjects, adequacy, completeness and categories of algebras_ , Rozprawy Mat. **36** (1964) pp.1-32. ([toc](http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.desklight-0dbcb276-0b92-49eb-b504-a9963119ea3e)) * [[David P. Blecher]], [[Nik Weaver]], *Quantum measurable cardinals* ([arXiv:1607.08505](https://arxiv.org/abs/1607.08505)) Some discussion on measurable cardinals in the context of a [[polynomial function]] whose [[Lebesgue measure|Lesbegue measurability]] is independent of [[ZFC]] occurs in: * [[James E. Hanson]], *Any function I can actually write down is measurable, right?* ([arXiv:2501.02693](https://arxiv.org/abs/2501.02693)) [[!redirects measurable cardinal]] [[!redirects measurable cardinals]]