Milner–Rado paradox
Encyclopedia
In set theory
, a branch of mathematics, the Milner – Rado paradox, found by , states that every ordinal number
α less than the successor
κ+ of some cardinal number
κ can be written as the union of sets X1,X2,... where Xn is of order type
at most κn for n a positive integer.
Set theory
Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
, a branch of mathematics, the Milner – Rado paradox, found by , states that every ordinal number
Ordinal number
In set theory, an ordinal number, or just ordinal, is the order type of a well-ordered set. They are usually identified with hereditarily transitive sets. Ordinals are an extension of the natural numbers different from integers and from cardinals...
α less than the successor
Successor cardinal
In the theory of cardinal numbers, we can define a successor operation similar to that in the ordinal numbers. This coincides with the ordinal successor operation for finite cardinals, but in the infinite case they diverge because every infinite ordinal and its successor have the same cardinality...
κ+ of some cardinal number
Cardinal number
In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality of sets. The cardinality of a finite set is a natural number – the number of elements in the set. The transfinite cardinal numbers describe the sizes of infinite...
κ can be written as the union of sets X1,X2,... where Xn is of order type
Order type
In mathematics, especially in set theory, two ordered sets X,Y are said to have the same order type just when they are order isomorphic, that is, when there exists a bijection f: X → Y such that both f and its inverse are monotone...
at most κn for n a positive integer.