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In linguistics, ordinal numerals or ordinal number words are words representing position or rank in a sequential order; the order may be of size, importance, chronology, and so on (e.g., "third", "tertiary"). They differ from cardinal numerals, which represent quantity (e.g., "three") and other types of numerals.
If α is any ordinal and X is a set, an α-indexed sequence of elements of X is a function from α to X. This concept, a transfinite sequence (if α is infinite) or ordinal-indexed sequence, is a generalization of the concept of a sequence. An ordinary sequence corresponds to the case α = ω, while a finite α corresponds to a tuple, a.k.a ...
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Ordinal numbers, not cardinal numbers, are commonly used to represent dates, because they are in the format of 'in the tenth year of Caesar', etc. which also carried over into the anno Domini system and Christian dating, e.g. annō post Chrīstum nātum centēsimō for AD 100.
A binary clock might use LEDs to express binary values. In this clock, each column of LEDs shows a binary-coded decimal numeral of the traditional sexagesimal time.. The common names are derived somewhat arbitrarily from a mix of Latin and Greek, in some cases including roots from both languages within a single name. [27]
To form the ordinal numerals we add -ti (m.), -ta (f.), -to (n.) to the basic numeral. Exception to this rule are the ordinal numerals first, second and third. If the basic word ends on the letter t and we add the suffixes for ordinal numerals, then a double t is generally produced.
This set is ordered lexicographically with the least significant position first: we write f < g if and only if there exists x ∈ β with f(x) < g(x) and f(y) = g(y) for all y ∈ β with x < y. This is a well-ordering and hence gives an ordinal number. The definition of exponentiation can also be given by transfinite recursion on the exponent β.
The ordinal ε 0 is still countable, as is any epsilon number whose index is countable. 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 ...