Ordinal numbers

mathematical concept generalizing ordinal numerals to extend enumeration to infinite sets

number larger than all finite numbers

In mathematics, a well-order (or well-ordering or well-order relation) on a set is a total ordering on with the property that every non-empty subset of has a least element in this ordering. The set together with the ordering is then called a well-ordered set (or woset). In some academic articles and textbooks these terms are instead written as wellorder, wellordered, and wellordering or well order, well ordered, and well ordering.

method of proving that a certain property applies for all elements in a well-founded set

paradox demonstrating that the class of all ordinal numbers Ω cannot be a set, since if it were, it would be an ordinal, thus an element of itself, and thus less than itself, which is a contradiction

In mathematics, especially in order theory, the cofinality cf(A) of a partially ordered set A is the least of the cardinalities of the cofinal subsets of A. Formally, \operatorname{cf}(A) = \inf \{|B| : B \subseteq A, (\forall x \in A) (\exists y \in B) (x \leq y)\}

infinite ordinal number class

smallest ordinal following a given ordinal

certain topology on totally ordered sets

cardinal number that equals its own cofinality

describes the three usual operations on ordinal numbers: addition, multiplication, and exponentiation

In mathematics, the nimbers, also called Grundy numbers (not to be confused with Grundy chromatic numbers), are introduced in combinatorial game theory, where they are defined as the values of heaps in the game Nim. The nimbers are the same proper class as the ordinal numbers but endowed with nimber addition and nimber multiplication, which are distinct from ordinal addition and ordinal multiplication.

in set theory, a subset of a cardinal that intersects every club set

ordinal number that is the fixed point of the exponentiation map

numbering scheme using non-negative integers, where the first item is numbered 0

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 strictly increasing

smallest ordinal number that, considered as a set, is uncountable

large countable ordinal; the proof-theoretic ordinal of arithmetical transfinite recursion

closed cofinal subset of an ordinal