In mathematics, a measurable cardinal is a certain kind of large cardinal number.

However there are many totally indescribable cardinals below any measurable cardinal.

This result is important in the proof that the existence of a measurable cardinal implies that sets are determined.

If is a set and is a measurable cardinal, then is -homogeneously Suslin.

A real valued measurable cardinal is weakly Mahlo.

Many such properties are studied, including inaccessible cardinals, measurable cardinals, and many more.

Notice that 0-huge is the same as measurable cardinal; and 1-huge is the same as huge.

The consistency of Bernays's reflection principle is implied by the existence of a measurable cardinal.

Solovay showed that the existence of 0 follows from the existence of two measurable cardinals.

The completeness of a countably complete nonprincipal ultrafilter on a set is always a measurable cardinal.

For example, measurable cardinals cease to be measurable but remain Mahlo in L.

(See measurable cardinal for the prototypical example.)

For example, the Löwenheim number of second-order logic is already larger than the first measurable cardinal, if such a cardinal exists.

Martin, Donald A.: Measurable cardinals and analytic games.

The existence of measurable cardinals cannot be proved in ZF set theory but (as of 2013) is thought to be consistent with it.

If there is a Woodin cardinal with a measurable cardinal above it, then Π determinacy holds.

From the existence of more measurable cardinals, one can prove the determinacy of more levels of the difference hierarchy over .

The negation of the singular cardinals hypothesis is intimately related to violating the GCH at a measurable cardinal.

But the existence of a singular Jónsson cardinal is equiconsistent to the existence of a measurable cardinal.

It is obvious from the definitions that strong cardinals lie below supercompact cardinals and above measurable cardinals in the consistency strength hierarchy.

Actually an apparently stronger result follows: If there is a measurable cardinal, then every game in the first ω levels of the difference hierarchy over is determined.

A Woodin cardinal is preceded by a stationary set of measurable cardinals, and thus it is a Mahlo cardinal.

The existence of a measurable cardinal is enough to imply over ZFC that all analytic subsets of Polish spaces are determined.

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.

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