Ulam matrix
Appearance
In mathematical set theory, an Ulam matrix is an array of subsets of a cardinal number with certain properties. Ulam matrices were introduced by Stanislaw Ulam in his 1930 work on measurable cardinals: they may be used, for example, to show that a real-valued measurable cardinal is weakly inaccessible.[1]
Definition
[edit]Suppose that κ and λ are cardinal numbers, and let be a -complete filter on . An Ulam matrix is a collection of subsets of indexed by such that
- If then and are disjoint.
- For each , the union over of the sets , is in the filter .
References
[edit]- ^ Jech, Thomas (2003), Set Theory, Springer Monographs in Mathematics (Third Millennium ed.), Berlin, New York: Springer-Verlag, p. 131, ISBN 978-3-540-44085-7, Zbl 1007.03002
- Ulam, Stanisław (1930), "Zur Masstheorie in der allgemeinen Mengenlehre", Fundamenta Mathematicae, 16 (1): 140–150