ⓘ 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-valu ..

                                     

ⓘ 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 }, α ∈ κ are small, complements of small sets are large and vice versa. The intersection of fewer than κ large sets is again large.

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

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

                                     

1. 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 A α, α

                                     
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  • large cardinal number. In fact, by results of Moti Gitik, ZFC the negation of SCH is equiconsistent with ZFC the existence of a measurable cardinal κ
  • smallest measurable cardinal is not Σ2 1 - indescribable. However there are many totally indescribable cardinals below any measurable cardinal Totally
  • Ramsey cardinal is sufficient to prove the existence of 0 or indeed that every set with rank less than κ has a sharp. Every measurable cardinal is a Ramsey
  • all ordinals λ. By definitions, strong cardinals lie below supercompact cardinals and above measurable cardinals in the consistency strength hierarchy
  • concept. Strong compactness implies measurability and is implied by supercompactness. Given that the relevant cardinals exist, it is consistent with ZFC
  • all subsets of the reals are measurable However, Solovay s result depends on the existence of an inaccessible cardinal whose existence and consistency
  • See also strong cardinal A Woodin cardinal is preceded by a stationary set of measurable cardinals and thus it is a Mahlo cardinal However, the first
  • chosen large cardinal. For example, if there is no inner model for a measurable cardinal then the Dodd Jensen core model, KDJ is the core model and satisfies
  • Ramsey cardinals measurable cardinals 0 λ - strong, strong cardinals tall cardinals Woodin, weakly hyper - Woodin, Shelah, hyper - Woodin cardinals superstrong