Supraordinal

Supraordinal is a term used in the context of ordinal numbers, which are used to describe the position of items in a sequence. Ordinal numbers themselves are well-ordered sets, and supraordinals are a class of ordinals that are not well-ordered. This concept is often encountered in set theory and the study of large cardinals.

The term "supraordinal" was introduced by Paul Cohen in his work on the independence of the continuum hypothesis. He used it to describe ordinals that are not well-ordered under the usual ordering of ordinals. Supraordinals are typically defined using the concept of a proper class, which is a collection of sets that is not itself a set.

One of the key properties of supraordinals is that they are not accessible from below by any well-ordered sequence of ordinals. This means that there is no way to reach a supraordinal from below using a well-ordered sequence of ordinals. This property makes supraordinals useful in the study of large cardinals, as they provide a way to describe the size of certain sets that are not well-ordered.

Supraordinals are also used in the study of the axiom of choice and the continuum hypothesis. They provide a way to describe the size of certain sets that are not well-ordered, and this can be used to prove the independence of the continuum hypothesis from the axioms of set theory.

In summary, supraordinals are a class of ordinals that are not well-ordered. They are defined using the concept of a proper class and have the property that they are not accessible from below by any well-ordered sequence of ordinals. They are used in the study of large cardinals, the axiom of choice, and the continuum hypothesis.