In order to define the concept, one introduces a two-valued measure on a cardinal κ, or more generally on any set.
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.
If this were the case, we could identify κ with some collection of 0-1 sequences of length λ.
Thus, assuming the Axiom of Choice, we can infer that κ is a strong limit cardinal, which completes the proof of its inaccessibility.
This property can be used to show that κ is a limit of most types of large cardinals that are weaker than measurable.
Banach & Kuratowski (1929) showed that the continuum hypothesis implies that 𝔠 is not real-valued measurable.
The class of Ulam numbers is closed under the cardinal successor operation.
By property (2) of μ, the set is countable, and hence Thus there is a b0 such that implying, since α is an Ulam number and using the second definition (with ν = μ and conditions (1)–(4) fulfilled), If b0 < x < β and fx(b0) = ax then x ∈ U(b0, ax).
Together with the previous result, this implies that a cardinal that is not an Ulam number is weakly inaccessible.