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Ordinal indicator – Character(s) following an ordinal number (used when writing ordinal numbers, such as a super-script) Ordinal number – Generalization of "n-th" to infinite cases (the related, but more formal and abstract, usage in mathematics) Ordinal data, in statistics; Ordinal date – Date written as number of days since first day of ...
A simple application of ordinal parity is the idempotence law for cardinal addition (given the well-ordering theorem). Given an infinite cardinal κ, or generally any limit ordinal κ, κ is order-isomorphic to both its subset of even ordinals and its subset of odd ordinals. Hence one has the cardinal sum κ + κ = κ. [2] [7]
In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, n th, etc.) aimed to extend enumeration to infinite sets. [1] A finite set can be enumerated by successively labeling each element with the least natural number that has not been previously used.
That such an ordinal exists and is unique is guaranteed by the fact that U is well-orderable and that the class of ordinals is well-ordered, using the axiom of replacement. With the full axiom of choice , every set is well-orderable , so every set has a cardinal; we order the cardinals using the inherited ordering from the ordinal numbers.
The augmented simplex category, denoted by + is the category of all finite ordinals and order-preserving maps, thus + = [], where [] =. Accordingly, this category might also be denoted FinOrd. The augmented simplex category is occasionally referred to as algebraists' simplex category and the above version is called topologists' simplex category.
Uncountable ordinals also exist, along with uncountable epsilon numbers whose index is an uncountable ordinal. The smallest epsilon number ε 0 appears in many induction proofs, because for many purposes transfinite induction is only required up to ε 0 (as in Gentzen's consistency proof and the proof of Goodstein's theorem ).
In human developmental psychology or non-human primate experiments, ordinal numerical competence or ordinal numerical knowledge is the ability to count objects in order and to understand the greater than and less than relationships between numbers. It has been shown that children as young as two can make some ordinal numerical decisions.
When considered as a set, the elements of are the countable ordinals (including finite ordinals), [1] of which there are uncountably many. Like any ordinal number (in von Neumann's approach ), ω 1 {\displaystyle \omega _{1}} is a well-ordered set , with set membership serving as the order relation.