Measurable cardinal

Measurable cardinal
Name	Measurable cardinal
Field	Set theory
Introduced by	Ulam, Mahlo
First proposed	1930s
Related concepts	Ultrafilter, Elementary embedding, Inaccessible cardinal, Strongly compact cardinal

Measurable cardinal

A measurable cardinal is an uncountable cardinal κ carrying a nontrivial, κ-complete, {0,1}-valued measure; it is a central notion in higher infinite combinatorics and large-cardinal theory. It connects with elementary embeddings, ultrafilters, and consistency strengths studied by Gödel, Cohen, and later researchers, and it yields strong combinatorial and structural consequences for models such as Gödel's constructible universe and inner model candidates like those studied by Mitchell and Steel. The concept has driven work by Ulam, Kunen, Solovay, Silver, and Woodin on independence, inner models, and determinacy.

Definition

A cardinal κ is measurable if there exists a κ-complete nonprincipal ultrafilter on κ providing a two-valued measure that assigns measure one to κ and measure zero to every singletons set. The original formulation by Ulam used a {0,1}-measure μ on P(κ) with μ(κ)=1 and μ({α})=0 for each α<κ, while later treatments by Tarski and Solovay rephrased this via κ-complete ultrafilters and elementary embeddings j:V→M with critical point κ as in Kunen's characterization. Notable contributors include Ulam, Tarski, and Mahlo, and later expositors such as Jech, Kanamori, and Kunen developed equivalences involving ultrapowers, Los's theorem, and ultrapower embeddings.

Large-cardinal properties and relationships

Measurable cardinals sit strictly above inaccessible and Mahlo cardinals in the large-cardinal hierarchy and below stronger notions like supercompact, huge, and Woodin cardinals as investigated by Solovay, Silver, Magidor, and Gitik. Relations studied by Kunen and Mitchell compare measurability with weak compactness, strong compactness, and ineffability, and preservation under forcing has been explored by Levy, Easton, and Gitik. Results by Menas and Reinhardt connect measurable cardinals to normal ultrafilters and to the Mitchell order, while Woodin and Neeman analyzed interactions between measurability and determinacy hypotheses like AD in models constructed by Steel and Sargsyan.

Ultrafilters and measures

Existence of a κ-complete nonprincipal ultrafilter U on κ is equivalent to the existence of a two-valued κ-additive measure μ; Los's theorem applied to the ultrapower V^κ/U yields an elementary embedding j:V→M with critical point κ, a construction used extensively by Kunen and Magidor. Normal ultrafilters, introduced by Ulam and studied by Fichtenholz and Jech, yield measures closed under diagonal intersections and are tied to regressive function theorems from Fodor and stationary set combinatorics as developed by Solovay and Foreman. The Mitchell order classifies measures by iterability and strength, with Mitchell and Steel analyzing higher-order measures in inner model theory, while Bukovský and Ketonen investigated precipitousness and Rudin–Keisler orderings among ultrafilters.

Consistency and independence results

Consistency results trace back to Ulam's observation and to relative consistency proofs using forcing and inner models by Cohen, Gödel, and Lévy, with Kunen proving limits like the Kunen inconsistency under certain large-cardinal hypotheses and Mitchell establishing fine-structure lower bounds. Relative consistency of the existence of measurable cardinals usually requires assuming consistency of stronger large cardinals or models with measurable cardinals, a theme in work by Solovay, Silver, and Kunen, while independence of many combinatorial statements at measurable cardinals has been shown using Prikry and Magidor forcing techniques developed by Prikry, Magidor, and Gitik. Core model theory by Dodd, Jensen, Mitchell, and Steel addresses how the existence of measurable cardinals affects core models like K and mice constructed by Woodin and Sargsyan.

Examples and nontrivial consequences

Concrete examples of measurable cardinals do not exist in ZFC alone; any model containing a measurable cardinal is obtained via relative consistency as shown by Gödel-style inner model constructions and forcing extensions used by Lévy, Kunen, and Solovay. Nontrivial consequences include failure of the Generalized Continuum Hypothesis at many cardinals in the presence of measurable cardinals as demonstrated by Easton and Silver, saturation properties for ideals studied by Jech and Foreman, and strong compactness-like combinatorics observed by Magidor and Menas. Measurables imply nontrivial partition properties and the existence of 0#-like objects in certain contexts as explored by Kunen, Silver, and Jensen, with further structural consequences in models studied by Mitchell and Steel.

Applications and connections in set theory

Measurable cardinals drive developments in inner model theory, determinacy, descriptive set theory, and large-cardinal embedding techniques used by Woodin, Martin, Steel, and Sargsyan, influencing proofs about projective determinacy and scales as in Martin and Steel's work. They interact with forcing axioms and combinatorial principles analyzed by Foreman, Magidor, and Shelah, and inform the study of stationary reflection and saturation phenomena investigated by Gitik and Jech. Connections to theoretical frameworks like HOD, mice, and extender models studied by Mitchell, Steel, and Jensen make measurable cardinals foundational to modern research programs bridging canonical inner models and high-consistency hypotheses pursued by Woodin and others.

Category:Set theory