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 sets.
In mathematics, particularly in set theory, the aleph numbers are a sequence of numbers used to represent the cardinality of infinite sets that can be well-ordered. They were introduced by the mathematician Georg Cantor and are named after the symbol he used to denote them, the Hebrew letter aleph.
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 two different ways: either by constructing an explicit well-ordered set that represents the result of the operation or by using transfinite recursion. Cantor normal form provides a standardized way of writing ordinals. In addition to these usual ordinal operations, there are also the "natural" arithmetic of ordinals and the nimber operations.
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 a finite series of applications of the chosen exponential map and of "weaker" operations like addition and multiplication. The original epsilon numbers were introduced by Georg Cantor in the context of ordinal arithmetic; they are the ordinal numbers ε that satisfy the equation
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 normal forms. Beyond that, many ordinals of relevance to proof theory still have computable ordinal notations. However, it is not possible to decide effectively whether a given putative ordinal notation is a notation or not ; various more-concrete ways of defining ordinals that definitely have notations are available.
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 computable subset of the natural numbers having the order type .
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 ordinals. A Gödel numbering is a function mapping the set of well-formed formulae of some formal language to the natural numbers. This associates each well-formed formula with a unique natural number, called its Gödel number. If a Gödel numbering is fixed, then the subset relation on the ordinals induces an ordering on well-formed formulae which in turn induces a well-ordering on the subset of natural numbers. A recursive ordinal notation must satisfy the following two additional properties:
- the subset of natural numbers is a recursive set
- the induced well-ordering on the subset of natural numbers is a recursive relation
In proof theory, ordinal analysis assigns ordinals to mathematical theories as a measure of their strength. If theories have the same proof-theoretic ordinal they are often equiconsistent, and if one theory has a larger proof-theoretic ordinal than another it can often prove the consistency of the second theory.
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 recursive ordinal notations.
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 after Solomon Feferman and Kurt Schütte, the former of whom suggested the name Γ0.
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 Ackermann (1951) is somewhat smaller than the small Veblen ordinal.
In mathematics, the Veblen functions are a hierarchy of normal functions, introduced by Oswald Veblen in Veblen (1908). If φ0 is any normal function, then for any non-zero ordinal α, φα is the function enumerating the common fixed points of φβ for β<α. These functions are all normal.
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 somewhat larger ordinal.
In mathematical logic and set theory, an ordinal collapsing function is a technique for defining certain recursive large countable ordinals, whose principle is to give names to certain ordinals much larger than the one being defined, perhaps even large cardinals, and then "collapse" them down to a system of notations for the sought-after ordinal. For this reason, ordinal collapsing functions are described as an impredicative manner of naming ordinals.
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 proof theoretic ordinal of the subsystem -CA0 of second-order arithmetic; this is one of the "big five" subsystems studied in reverse mathematics (Simpson 1999). It is also the proof-theoretic ordinal of , the theory of finitely iterated inductive definitions, and of , a fragment of Kripke-Platek set theory extended by an axiom stating every set is contained in an admissible set. Buchholz's ordinal is also the order type of the segment bounded by in Buchholz's ordinal notation . Lastly, it can be expressed as the limit of the sequence: , , , ...
In computability theory, computational complexity theory and proof theory, a fast-growing hierarchy is an ordinal-indexed family of rapidly increasing functions fα: N → N. A primary example is the Wainer hierarchy, or Löb–Wainer hierarchy, which is an extension to all α < ε0. Such hierarchies provide a natural way to classify computable functions according to rate-of-growth and computational complexity.
In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals aimed to extend enumeration to infinite sets.
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, but nevertheless have the same strength as those. Later on this approach was extended by Jäger and Schütte.
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 and Feferman's theta function. It was named by David Madore, after Gaisi Takeuti, Solomon Feferman and Wilfried Buchholz. It is written as using Buchholz's psi function, an ordinal collapsing function invented by Wilfried Buchholz, and in Feferman's theta function, an ordinal collapsing function invented by Solomon Feferman. It is the proof-theoretic ordinal of several formal theories:
In set theory, a mathematical discipline, a fundamental sequence is a cofinal sequence of ordinals all below a given limit ordinal. Depending on author, fundamental sequences may be restricted to ω-sequences only or permit fundamental sequences of length . The nth element of the fundamental sequence of is commonly denoted , although it may be denoted or . Additionally, some authors may allow fundamental sequences to be defined on successor ordinals. The term dates back to Veblen's construction of normal functions , while the concept dates back to Hardy's 1904 attempt to construct a set of cardinality .
