Measurable cardinal

In mathematics, a measurable cardinal is a certain kind of large cardinal number. In order to define the concept, one introduces a two-valued measure on a cardinal κ, or more generally on any set. For a cardinal κ, it can be described as a subdivision of all of its subsets into large and small sets such that κ itself is large, and all singletons {α}, ακ are small, complements of small sets are large and vice versa. The intersection of fewer than κ large sets is again large.[1]

It turns out that uncountable cardinals endowed with a two-valued measure are large cardinals whose existence cannot be proved from ZFC.[2]

The concept of a measurable cardinal was introduced by Stanislaw Ulam in 1930.[3]

Definition

Formally, a measurable cardinal is an uncountable cardinal number κ such that there exists a κ-additive, non-trivial, 0-1-valued measure on the power set of κ. (Here the term κ-additive means that, for any sequence Aα, α<λ of cardinality λ<κ, Aα being pairwise disjoint sets of ordinals less than κ, the measure of the union of the Aα equals the sum of the measures of the individual Aα.)

Equivalently, κ is measurable means that it is the critical point of a non-trivial elementary embedding of the universe V into a transitive class M. This equivalence is due to Jerome Keisler and Dana Scott, and uses the ultrapower construction from model theory. Since V is a proper class, a technical problem that is not usually present when considering ultrapowers needs to be addressed, by what is now called Scott's trick.

Equivalently, κ is a measurable cardinal if and only if it is an uncountable cardinal with a κ-complete, non-principal ultrafilter. Again, this means that the intersection of any strictly less than κ-many sets in the ultrafilter, is also in the ultrafilter.

Properties

Although it follows from ZFC that every measurable cardinal is inaccessible (and is ineffable, Ramsey, etc.), it is consistent with ZF that a measurable cardinal can be a successor cardinal. It follows from ZF + axiom of determinacy that ω1 is measurable, and that every subset of ω1 contains or is disjoint from a closed and unbounded subset.

Ulam showed that the smallest cardinal κ that admits a non-trivial countably-additive two-valued measure must in fact admit a κ-additive measure. (If there were some collection of fewer than κ measure-0 subsets whose union was κ, then the induced measure on this collection would be a counterexample to the minimality of κ.) From there, one can prove (with the Axiom of Choice) that the least such cardinal must be inaccessible.

It is trivial to note that if κ admits a non-trivial κ-additive measure, then κ must be regular. (By non-triviality and κ-additivity, any subset of cardinality less than κ must have measure 0, and then by κ-additivity again, this means that the entire set must not be a union of fewer than κ sets of cardinality less than κ.) Finally, if λ < κ, then it can't be the case that κ ≤ 2λ. If this were the case, then we could identify κ with some collection of 0-1 sequences of length λ. For each position in the sequence, either the subset of sequences with 1 in that position or the subset with 0 in that position would have to have measure 1. The intersection of these λ-many measure 1 subsets would thus also have to have measure 1, but it would contain exactly one sequence, which would contradict the non-triviality of the measure. Thus, assuming the Axiom of Choice, we can infer that κ is a strong limit cardinal, which completes the proof of its inaccessibility.

If κ is measurable and pVκ and M (the ultrapower of V) satisfies ψ(κ,p), then the set of α<κ such that V satisfies ψ(α,p) is stationary in κ (actually a set of measure 1). In particular if ψ is a Π1 formula and V satisfies ψ(κ,p), then M satisfies it and thus V satisfies ψ(α,p) for a stationary set of α<κ. This property can be used to show that κ is a limit of most types of large cardinals which are weaker than measurable. Notice that the ultrafilter or measure which witnesses that κ is measurable cannot be in M since the smallest such measurable cardinal would have to have another such below it which is impossible.

Every measurable cardinal κ is a 0-huge cardinal because κMM, that is, every function from κ to M is in M. Consequently, Vκ+1M.

Real-valued measurable

A cardinal κ is called real-valued measurable if there is a κ-additive probability measure on the power set of κ which vanishes on singletons. Real-valued measurable cardinals were introduced by Stefan Banach (1930). Banach & Kuratowski (1929) showed that the continuum hypothesis implies that ${\mathfrak c}$ is not real-valued measurable. Stanislaw Ulam (1930) showed that real valued measurable cardinals are weakly inaccessible (they are in fact weakly Mahlo). All measurable cardinals are real-valued measurable, and a real-valued measurable cardinal κ is measurable if and only if κ is greater than ${\mathfrak c}$. Thus a cardinal is measurable if and only if it is real-valued measurable and strongly inaccessible. A real valued measurable cardinal less than or equal to ${\mathfrak c}$ exists if and only if there is a countably additive extension of the Lebesgue measure to all sets of real numbers if and only if there is an atomless probability measure on the power set of some non-empty set.

Solovay (1971) showed that existence of measurable cardinals in ZFC, real valued measurable cardinals in ZFC, and measurable cardinals in ZF, are equiconsistent.