Measurable

Formally, a measurable cardinal is an uncountable cardinal number

Cardinal number

In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality of sets. The cardinality of a finite set is a natural number – the number of elements in the set. The transfinite cardinal numbers describe the sizes of infinite...

 κ such that there exists a κ-additive, non-trivial, 0-1-valued measure

Measure (mathematics)

In mathematical analysis, a measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset. In this sense, a measure is a generalization of the concepts of length, area, and volume...

 on the power set of κ. (Here the term κ-additive means that, for any sequence A_α, α<λ of cardinality λ<κ, A_α being pairwise disjoint sets of ordinals less than κ, the measure of the union of the A_α equals the sum of the measures of the individual A_α.)

Equivalently, κ is measurable means that it is the critical point

Critical point (set theory)

In set theory, the critical point of an elementary embedding of a transitive class into another transitive class is the smallest ordinal which is not mapped to itself....

 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

Dana Scott

Dana Stewart Scott is the emeritus Hillman University Professor of Computer Science, Philosophy, and Mathematical Logic at Carnegie Mellon University; he is now retired and lives in Berkeley, California...

, and uses the ultrapower

Ultraproduct

The ultraproduct is a mathematical construction that appears mainly in abstract algebra and in model theory, a branch of mathematical logic. An ultraproduct is a quotient of the direct product of a family of structures. All factors need to have the same signature...

 construction from model theory

Model theory

In mathematics, model theory is the study of mathematical structures using tools from mathematical logic....

. Since V is a proper class, a small technical problem that is not usually present when considering ultrapowers needs to be addressed, by what is now called Scott's trick

Scott's trick

In set theory, Scott's trick is a method for choosing sets of representatives for equivalence classes without using the axiom of choice, if the axiom of regularity is available . It can be used to define representatives for ordinal numbers in Zermelo–Fraenkel set theory...

.

Equivalently, κ is a measurable cardinal if and only if it is an uncountable cardinal with a κ-complete, non-principal ultrafilter

Ultrafilter

In the mathematical field of set theory, an ultrafilter on a set X is a collection of subsets of X that is a filter, that cannot be enlarged . An ultrafilter may be considered as a finitely additive measure. Then every subset of X is either considered "almost everything" or "almost nothing"...

. Again, this means that the intersection of any strictly less than κ-many sets in the ultrafilter, is also in the ultrafilter.

Although it follows from ZFC that every measurable cardinal is inaccessible

Inaccessible cardinal

In set theory, an uncountable regular cardinal number is called weakly inaccessible if it is a weak limit cardinal, and strongly inaccessible, or just inaccessible, if it is a strong limit cardinal. Some authors do not require weakly and strongly inaccessible cardinals to be uncountable...

 (and is ineffable, Ramsey

Ramsey cardinal

In mathematics, a Ramsey cardinal is a certain kind of large cardinal number introduced by and named after Frank P. Ramsey.With [κ]<ω denoting the set of all finite subsets of κ, a cardinal number κ such that for every function...

, etc.), it is consistent with ZF

Zermelo–Fraenkel set theory

In mathematics, Zermelo–Fraenkel set theory with the axiom of choice, named after mathematicians Ernst Zermelo and Abraham Fraenkel and commonly abbreviated ZFC, is one of several axiomatic systems that were proposed in the early twentieth century to formulate a theory of sets without the paradoxes...

 that a measurable cardinal can be a successor cardinal

Successor cardinal

In the theory of cardinal numbers, we can define a successor operation similar to that in the ordinal numbers. This coincides with the ordinal successor operation for finite cardinals, but in the infinite case they diverge because every infinite ordinal and its successor have the same cardinality...

. It follows from ZF + axiom of determinacy

Axiom of determinacy

The axiom of determinacy is a possible axiom for set theory introduced by Jan Mycielski and Hugo Steinhaus in 1962. It refers to certain two-person games of length ω with perfect information...

 that ω₁ is measurable, and that every subset of ω₁ contains or is disjoint from a closed and unbounded subset.

The concept of a measurable cardinal was introduced by , who 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 ≤ 2^λ. 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 p∈V_κ 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 ^κM⊂M, that is, every function from κ to M is in M. Consequently, V_κ+1⊂M.

Real-valued measurable

A cardinal κ is called real-valued measurable if there is an atomless

Atom (measure theory)

In mathematics, more precisely in measure theory, an atom is a measurable set which has positive measure and contains no set of smaller but positive measure...

 κ-additive measure on the power set of κ. They were introduced by . showed that the continuum hypothesis

Continuum hypothesis

In mathematics, the continuum hypothesis is a hypothesis, advanced by Georg Cantor in 1874, about the possible sizes of infinite sets. It states:Establishing the truth or falsehood of the continuum hypothesis is the first of Hilbert's 23 problems presented in the year 1900...

 implies that is not real-valued measurable. A real valued measurable cardinal less than or equal to exists if there is a countably additive extension of the Lebesgue measure

Lebesgue measure

In measure theory, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of n-dimensional Euclidean space. For n = 1, 2, or 3, it coincides with the standard measure of length, area, or volume. In general, it is also called...

 to all sets of real numbers. A real valued measurable cardinal is weakly Mahlo.

showed that existence of measurable cardinals in ZFC, real valued measurable cardinals in ZFC, and measurable cardinals in ZF, are equiconsistent

Equiconsistency

In mathematical logic, two theories are equiconsistent if, roughly speaking, they are "as consistent as each other".It is not in general possible to prove the absolute consistency of a theory T...

.