- Arithmetic
- >
- Comparison (mathematical)
- >
- Order theory
- >
- Ordinal numbers

- Binary operations
- >
- Comparison (mathematical)
- >
- Order theory
- >
- Ordinal numbers

- Elementary arithmetic
- >
- Comparison (mathematical)
- >
- Order theory
- >
- Ordinal numbers

- Fields of mathematics
- >
- Arithmetic
- >
- Numbers
- >
- Ordinal numbers

- Fields of mathematics
- >
- Mathematical logic
- >
- Set theory
- >
- Ordinal numbers

- Mathematical concepts
- >
- Mathematical objects
- >
- Numbers
- >
- Ordinal numbers

- Mathematical notation
- >
- Numeral systems
- >
- Numbers
- >
- Ordinal numbers

- Mathematics
- >
- Fields of mathematics
- >
- Order theory
- >
- Ordinal numbers

- Philosophy of mathematics
- >
- Mathematical logic
- >
- Set theory
- >
- Ordinal numbers

- Philosophy of mathematics
- >
- Mathematical objects
- >
- Numbers
- >
- Ordinal numbers

Ordinal collapsing function

In mathematical logic and set theory, an ordinal collapsing function (or projection function) is a technique for defining (notations for) certain recursive large countable ordinals, whose principle is

Cardinal and Ordinal Numbers

Cardinal and Ordinal Numbers is a book on transfinite numbers, by Polish mathematician Wacław Sierpiński. It was published in 1958 by Państwowe Wydawnictwo Naukowe, as volume 34 of the series Monograf

Theories of iterated inductive definitions

In set theory and logic, Buchholz's ID hierarchy is a hierarchy of subsystems of first-order arithmetic. The systems/theories are referred to as "the formal theories of ν-times iterated inductive defi

Even and odd ordinals

In mathematics, even and odd ordinals extend the concept of parity from the natural numbers to the ordinal numbers. They are useful in some transfinite induction proofs. The literature contains a few

Ackermann ordinal

In mathematics, the Ackermann ordinal is a certain large countable ordinal, named after Wilhelm Ackermann. The term "Ackermann ordinal" is also occasionally used for the small Veblen ordinal, a somewh

Buchholz hydra

In mathematical logic, the Buchholz hydra game is a hydra game, which is a single-player game based on the idea of chopping pieces off a mathematical tree. The hydra game can be used to generate a rap

Order topology

In mathematics, an order topology is a certain topology that can be defined on any totally ordered set. It is a natural generalization of the topology of the real numbers to arbitrary totally ordered

Primitive recursive set function

In mathematics, primitive recursive set functions or primitive recursive ordinal functions are analogs of primitive recursive functions, defined for sets or ordinals rather than natural numbers. They

Ordinal date

An ordinal date is a calendar date typically consisting of a year and a day of the year or ordinal day number (or simply ordinal day or day number), an ordinal number ranging between 1 and 366 (starti

Bachmann–Howard ordinal

In mathematics, the Bachmann–Howard ordinal (or Howard ordinal) is a large countable ordinal.It is the proof-theoretic ordinal of several mathematical theories, such as Kripke–Platek set theory (with

Ordinal analysis

In proof theory, ordinal analysis assigns ordinals (often large countable ordinals) to mathematical theories as a measure of their strength. If theories have the same proof-theoretic ordinal they are

Feferman–Schütte ordinal

In mathematics, the Feferman–Schütte ordinal Γ0 is a large countable ordinal.It is the proof-theoretic ordinal of several mathematical theories, such as arithmetical transfinite recursion.It is named

Zero-based numbering

Zero-based numbering is a way of numbering in which the initial element of a sequence is assigned the index 0, rather than the index 1 as is typical in everyday non-mathematical or non-programming cir

Regular cardinal

In set theory, a regular cardinal is a cardinal number that is equal to its own cofinality. More explicitly, this means that is a regular cardinal if and only if every unbounded subset has cardinality

Limit ordinal

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 is an ordinal less than λ, and whenever β is a

Transfinite induction

Transfinite induction is an extension of mathematical induction to well-ordered sets, for example to sets of ordinal numbers or cardinal numbers. Its correctness is a theorem of ZFC.

Transfinite number

In mathematics, transfinite numbers are numbers that are "infinite" in the sense that they are larger than all finite numbers, yet not necessarily absolutely infinite. These include the transfinite ca

Veblen function

In mathematics, the Veblen functions are a hierarchy of normal functions (continuous strictly increasing functions from ordinals to ordinals), introduced by Oswald Veblen in . If φ0 is any normal func

Ordinal number

In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, nth, etc.) aimed to extend enumeration to infinite sets. A finite set can be enumerated by success

Epsilon number

In mathematics, the epsilon numbers are a collection of transfinite numbers whose defining property is that they are fixed points of an exponential map. Consequently, they are not reachable from 0 via

Diagonal intersection

Diagonal intersection is a term used in mathematics, especially in set theory. If is an ordinal number and is a sequence of subsets of , then the diagonal intersection, denoted by is defined to be Tha

Von Neumann cardinal assignment

The von Neumann cardinal assignment is a cardinal assignment that uses ordinal numbers. For a well-orderable set U, we define its cardinal number to be the smallest ordinal number equinumerous to U, u

Burali-Forti paradox

In set theory, a field of mathematics, the Burali-Forti paradox demonstrates that constructing "the set of all ordinal numbers" leads to a contradiction and therefore shows an antinomy in a system tha

Well-order

In mathematics, a well-order (or well-ordering or well-order relation) on a set S is a total order on S with the property that every non-empty subset of S has a least element in this ordering. The set

Large countable ordinal

In the mathematical discipline of set theory, there are many ways of describing specific countable ordinals. The smallest ones can be usefully and non-circularly expressed in terms of their Cantor nor

Ordinal arithmetic

In the mathematical field of set theory, ordinal arithmetic describes the three usual operations on ordinal numbers: addition, multiplication, and exponentiation. Each can be defined in essentially tw

Admissible ordinal

In set theory, an ordinal number α is an admissible ordinal if Lα is an admissible set (that is, a transitive model of Kripke–Platek set theory); in other words, α is admissible when α is a limit ordi

Club set

In mathematics, particularly in mathematical logic and set theory, a club set is a subset of a limit ordinal that is closed under the order topology, and is unbounded (see below) relative to the limit

Continuous function (set theory)

In set theory, a continuous function is a sequence of ordinals such that the values assumed at limit stages are the limits (limit suprema and limit infima) of all values at previous stages. More forma

Order type

In mathematics, especially in set theory, two ordered sets X and Y are said to have the same order type if they are order isomorphic, that is, if there exists a bijection (each element pairs with exac

Fixed-point lemma for normal functions

The fixed-point lemma for normal functions is a basic result in axiomatic set theory stating that any normal function has arbitrarily large fixed points (Levy 1979: p. 117). It was first proved by Osw

First uncountable ordinal

In mathematics, the first uncountable ordinal, traditionally denoted by or sometimes by , is the smallest ordinal number that, considered as a set, is uncountable. It is the supremum (least upper boun

Successor ordinal

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 ordinal.

Takeuti–Feferman–Buchholz ordinal

In the mathematical fields of set theory and proof theory, the Takeuti–Feferman–Buchholz ordinal (TFBO) is a large countable ordinal, which acts as the limit of the range of Buchholz's psi function an

Normal function

In axiomatic set theory, a function f : Ord → Ord is called normal (or a normal function) if and only if it is continuous (with respect to the order topology) and strictly monotonically increasing. Th

Ordinal logic

In mathematics, ordinal logic is a logic associated with an ordinal number by recursively adding elements to a sequence of previous logics. The concept was introduced in 1938 by Alan Turing in his PhD

Buchholz's ordinal

In mathematics, ψ0(Ωω), widely known as Buchholz's ordinal, is a large countable ordinal that is used to measure the proof-theoretic strength of some mathematical systems. In particular, it is the pro

Kleene's O

In set theory and computability theory, Kleene's is a canonical subset of the natural numbers when regarded as ordinal notations. It contains ordinal notations for every computable ordinal, that is, o

Stationary set

In mathematics, specifically set theory and model theory, a stationary set is a set that is not too small in the sense that it intersects all club sets, and is analogous to a set of non-zero measure i

Additively indecomposable ordinal

In set theory, a branch of mathematics, an additively indecomposable ordinal α is any ordinal number that is not 0 such that for any , we have Additively indecomposable ordinals are also called gamma

Systems of Logic Based on Ordinals

Systems of Logic Based on Ordinals was the PhD dissertation of the mathematician Alan Turing. Turing's thesis is not about a new type of formal logic, nor was he interested in so-called ‘ranked logic’

Ordinal notation

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

Rathjen's psi function

In mathematics, Rathjen's psi function is an ordinal collapsing function developed by Michael Rathjen. It collapses weakly Mahlo cardinals to generate large countable ordinals. A weakly Mahlo cardinal

Large Veblen ordinal

In mathematics, the large Veblen ordinal is a certain large countable ordinal, named after Oswald Veblen. There is no standard notation for ordinals beyond the Feferman–Schütte ordinal Γ0. Most system

Computable ordinal

In mathematics, specifically computability and set theory, an ordinal is said to be computable or recursive if there is a computable well-ordering of a subset of the natural numbers having the order t

Nonrecursive ordinal

In mathematics, particularly set theory, non-recursive ordinals are large countable ordinals greater than all the recursive ordinals, and therefore can not be expressed using ordinal collapsing functi

Buchholz psi functions

Buchholz's psi-functions are a hierarchy of single-argument ordinal functions introduced by German mathematician Wilfried Buchholz in 1986. These functions are a simplified version of the -functions,

Nimber

In mathematics, the nimbers, also called Grundy numbers, are introduced in combinatorial game theory, where they are defined as the values of heaps in the game Nim. The nimbers are the ordinal numbers

Cofinality

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. This definition of cofinality relies on the

Small Veblen ordinal

In mathematics, the small Veblen ordinal is a certain large countable ordinal, named after Oswald Veblen. It is occasionally called the Ackermann ordinal, though the Ackermann ordinal described by is

© 2023 Useful Links.