mathematical set theory, a set S is said to be ordinal definable if, informally, it can be defined in terms of a finite number of ordinals by a first-order...

In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, nth, etc.) aimed to extend enumeration to infinite...

have only countably many formulas, every notion of definable numbers has at most countably many definable real numbers. However, by Cantor's diagonal argument...

well ordered by ∈. 2.  An ordinal definable set is a set that can be defined by a first-order formula with ordinals as parameters ot Abbreviation for...

In the mathematical field of set theory, ordinal arithmetic describes the three usual operations on ordinal numbers: addition, multiplication, and exponentiation...

examples of ordinal data include socioeconomic status, military ranks, and letter grades for coursework. Ordinal data analysis requires a different set of analyses...

countable ordinals, and thus the first ordinal at which all the Borel sets are obtained is ω1, the first uncountable ordinal. The resulting sequence of sets is...

organizational change and performance Ordinal definable set, a set requiring only finitely-many ordinals to define under first-order logic. OD (video game)...

different way of introducing the ordinals, in which an ordinal is equated with the set of all smaller ordinals. This form of ordinal number is thus a canonical...

the set of all natural numbers less than it. This definition, can be extended to the von Neumann definition of ordinals for defining all ordinal numbers...

that, considered as a set, is uncountable. It is the supremum (least upper bound) of all countable ordinals. When considered as a set, the elements of ω...

In set theory, a limit ordinal is an ordinal number that is neither zero nor a successor ordinal. Alternatively, an ordinal λ is a limit ordinal if there...

In set theory, the successor of an ordinal number α is the smallest ordinal number greater than α. An ordinal number that is a successor is called a successor...

numbers definable by language. Curry's paradox Grelling–Nelson paradox Kleene–Rosser paradox List of paradoxes Löb's theorem Ordinal definable set, a set-theoretic...

In proof theory, ordinal analysis assigns ordinals (often large countable ordinals) to mathematical theories as a measure of their strength. If theories...

set is defined inductively as the smallest ordinal number greater than the ranks of all members of the set. In particular, the rank of the empty set is...

In statistics, ordinal regression, also called ordinal classification, is a type of regression analysis used for predicting an ordinal variable, i.e....

non-empty set of cardinal numbers has a least member. The cofinality of a partially ordered set A can alternatively be defined as the least ordinal x such...

In mathematical logic and set theory, an ordinal collapsing function (or projection function) is a technique for defining (notations for) certain recursive...

Statements true in L Reflection principle Axiomatic set theory Transitive set L(R) Ordinal definable Gödel 1938. K. J. Devlin, "An introduction to the fine...

image of any set under any definable mapping is also a set. It is necessary for the construction of certain infinite sets in ZF. The axiom schema is motivated...

prove the existence of a definable (by a formula) well order of the reals. However it is consistent with ZFC that a definable well ordering of the reals...

is definable over a countable sequence of ordinals is Lebesgue measurable, and has the Baire and perfect set properties. (This includes all definable and...

instance, the class of all ordinal numbers, and the class of all sets, are proper classes in many formal systems. In Quine's set-theoretical writing, the...

well-ordered set is aleph-one ℵ1, then ℵ2 and so on. Continuing in this manner, it is possible to define a cardinal number ℵα for every ordinal number α,...

In mathematical logic and set theory, an ordinal notation is a partial function mapping the set of all finite sequences of symbols, themselves members...

set has a function to the empty set. In the von Neumann construction of the ordinals, 0 is defined as the empty set, and the successor of an ordinal is...

closed sets in the sense that we have already defined, namely, those that contain a limit ordinal whenever they contain all sufficiently large ordinals below...

hierarchy if it is definable by a formula of second-order arithmetic with only existential set quantifiers and no other set quantifiers. A set is classified...

cases: The empty set is computable. The entire set of natural numbers is computable. Each natural number (as defined in standard set theory) is computable;...