mathematical set theory Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concern ...

, an Ulam matrix is an array of subsets of a

cardinal number In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. Th ...

with certain properties. Ulam matrices were introduced by

Stanislaw Ulam Stanisław Marcin Ulam (; 13 April 1909 – 13 May 1984) was a Polish-American scientist in the fields of mathematics and nuclear physics. He participated in the Manhattan Project, originated the Teller–Ulam design of thermonuclear weapon ...

in his 1930 work on measurable cardinals: they may be used, for example, to show that a real-valued measurable cardinal is weakly inaccessible.

Definition

Suppose that κ and λ are cardinal numbers, and let ''F'' be a λ-complete filter on λ. An Ulam matrix is a collection of subsets ''A''_αβ of λ indexed by α in κ, β in λ such that *If β is not γ then ''A''_αβ and ''A''_αγ are disjoint. *For each β the union of the sets ''A''_αβ is in the filter ''F''.

References

* Set theory {{settheory-stub