Ordinal analysis

proof-theoretic ordinalproof-theoretic strengthproof theoretic ordinal
FefermanSchütte ordinal. Bachmann–Howard ordinal. Complexity class.

Ordinal collapsing function

collapsingcollapsing functionordinal collapsing functions
For example, is a canonical notation for an ordinal which is less than the FefermanSchütte ordinal: it can be written using the Veblen functions as. Concerning the order, one might point out that (the FefermanSchütte ordinal) is much more than (because \Omega is greater than \psi of anything), and is itself much more than (because is greater than \Omega, so any sum-product-or-exponential expression involving and smaller value will remain less than ). In fact, is already less than.

Reverse mathematics

arithmetical transfinite recursionbounded reverse mathematicsconstructive reverse mathematics
Reverse mathematics is a program in mathematical logic that seeks to determine which axioms are required to prove theorems of mathematics. Its defining method can briefly be described as "going backwards from the theorems to the axioms", in contrast to the ordinary mathematical practice of deriving theorems from axioms. It can be conceptualized as sculpting out necessary conditions from sufficient ones.

Kurt Schütte

Kurt SchützeSchütte, Kurt
The FefermanSchütte ordinal, which he showed to be the precise ordinal bound for predicativity, is named after him. He was the doctoral advisor of 16 students, including Wolfgang Bibel, Wolfgang Maaß, Wolfram Pohlers, and Martin Wirsing. * with Helmut Schwichtenberg: Mathematische Logik, in Fischer, Hirzebruch et al. (eds.) Ein Jahrhundert Mathematik 1890-1990, Vieweg 1990 * * Kurt Schütte at the Mathematics Genealogy Project Beweistheorie, Springer, Grundlehren der mathematischen Wissenschaften, 1960; new edition trans. into English as Proof Theory, Springer-Verlag 1977.

Veblen function

fundamental sequenceVeblen hierarchyFundamental sequence (ordinals)
Γ 0 is the FefermanSchütte ordinal, i.e. it is the smallest α such that φ α (0) = α. For Γ 0, a fundamental sequence could be chosen to be and For Γ β+1, let and For Γ β where is a limit, let To build the Veblen function of a finite number of arguments (finitary Veblen function), let the binary function be as defined above. Let z be an empty string or a string consisting of one or more comma-separated zeros 0,0,...,0 and s be an empty string or a string consisting of one or more comma-separated ordinals with. The binary function can be written as where both s and z are empty strings.

Small Veblen ordinal

“small” Veblen ordinalsmall“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 somewhat smaller than the small Veblen ordinal. Unfortunately there is no standard notation for ordinals beyond the FefermanSchütte ordinal Γ 0. Most systems of notation use symbols such as ψ, some of which are modifications of the Veblen functions to produce countable ordinals even for uncountable arguments, and some of which are "collapsing functions".

Large Veblen ordinal

“large” Veblen ordinallarge“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 FefermanSchütte ordinal Γ 0. Most systems of notation use symbols such as ψ, some of which are modifications of the Veblen functions to produce countable ordinals even for uncountable arguments, and some of which are ordinal collapsing functions. The large Veblen ordinal is sometimes denoted by or or. It was constructed by Veblen using an extension of Veblen functions allowing infinitely many arguments. *

Ordinal notation

Buchholz's notationFeferman's functionnotation
Define: *Ω ξ = ω ξ if ξ > 0, Ω 0 = 1 The functions ψ v (α) for α an ordinal, v an ordinal at most ω, are defined by induction on α as follows: *ψ v (α) is the smallest ordinal not in C v (α) where C v (α) is the smallest set such that This system has about the same strength as Fefermans system, as for v ≤ ω. described a system of notation for all recursive ordinals (those less than the ChurchKleene ordinal). It uses a subset of the natural numbers instead of finite strings of symbols.

Epsilon numbers (mathematics)

ε 0 ε 0 epsilon numbers
Large countable ordinal. J.H. Conway, On Numbers and Games (1976) Academic Press ISBN: 0-12-186350-6. Section XIV.20 of.

Impredicativity

impredicativepredicativepredicativism
Something that is impredicative, in mathematics, logic and philosophy of mathematics, is a self-referencing definition. Roughly speaking, a definition is impredicative if it invokes (mentions or quantifies over) the set being defined, or (more commonly) another set that contains the thing being defined. There is no generally accepted precise definition of what it means to be predicative or impredicative. Authors have given different but related definitions.

Fast-growing hierarchy

Fast-growingLöb–Wainer hierarchy
.}, and α ranges up to some large countable ordinal). A primary example is the Wainer hierarchy, or Löb–Wainer hierarchy, which is an extension to all α < ε 0. Such hierarchies provide a natural way to classify computable functions according to rate-of-growth and computational complexity. Let μ be a large countable ordinal such to every limit ordinal α < μ there is assigned a fundamental sequence (a strictly increasing sequence of ordinals whose supremum is α). A fast-growing hierarchy of functions f α : N → N, for α < μ, is then defined as follows: * if α is a limit ordinal. Here f α n (n) = f α (f α (...

Ackermann ordinal

ordinals
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. Unfortunately there is no standard notation for ordinals beyond the FefermanSchütte ordinal Γ 0. Most systems of notation use symbols such as ψ, some of which are modifications of the Veblen functions to produce countable ordinals even for uncountable arguments, and some of which are "collapsing functions". The smaller Ackermann ordinal is the limit of a system of ordinal notations invented by, and is sometimes denoted by or or.

Bachmann–Howard ordinal

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 the axiom of infinity) and the system CZF of constructive set theory. It was introduced by and. The Bachmann–Howard ordinal is defined using an ordinal collapsing function: The Bachmann–Howard ordinal can also be defined as for an extension of the Veblen functions φ α to certain functions α of ordinals; this extension is not completely straightforward. * (Slides of a talk given at Fischbachau.) ε α enumerates the epsilon numbers, the ordinals ε such that ω ε = ε.

Church–Kleene ordinal

In mathematics, the ChurchKleene ordinal, named after Alonzo Church and S. C. Kleene, is a large countable ordinal. It is the set of all recursive ordinals and consequently the smallest non-recursive ordinal. Since the successor of a recursive ordinal is recursive, the ChurchKleene ordinal is a limit ordinal. It is also the first ordinal which is not hyperarithmetical, and the first admissible ordinal after ω. *

Ordinal arithmetic

Cantor normal formordinal additiontransfinite arithmetic
Such ordinals are known as large countable ordinals. The operations of addition, multiplication and exponentiation are all examples of primitive recursive ordinal functions, and more general primitive recursive ordinal functions can be used to describe larger ordinals. The natural sum and natural product operations on ordinals were defined in 1906 by Gerhard Hessenberg, and are sometimes called the Hessenberg sum (or product). These are the same as the addition and multiplication (restricted to ordinals) of John Conway's field of surreal numbers. They have the advantage that they are associative and commutative, and natural product distributes over natural sum.

First uncountable ordinal

ω 1 first uncountable ordinal number&omega; 1
Large countable ordinal. Continuum hypothesis. Thomas Jech, Set Theory, 3rd millennium ed., 2003, Springer Monographs in Mathematics, Springer, ISBN: 3-540-44085-2. Lynn Arthur Steen and J. Arthur Seebach, Jr., Counterexamples in Topology. Springer-Verlag, New York, 1978. Reprinted by Dover Publications, New York, 1995. ISBN: 0-486-68735-X (Dover edition).

Recursive ordinal

recursiveChurch–Kleene ordinalcomputable ordinal
The supremum of all recursive ordinals is called the ChurchKleene ordinal and denoted by. The ChurchKleene ordinal is a limit ordinal. An ordinal is recursive if and only if it is smaller than. Since there are only countably many recursive relations, there are also only countably many recursive ordinals. Thus, is countable. The recursive ordinals are exactly the ordinals that have an ordinal notation in Kleene's \mathcal{O}. Arithmetical hierarchy. Large countable ordinal. Ordinal notation. Rogers, H. The Theory of Recursive Functions and Effective Computability, 1967. Reprinted 1987, MIT Press, ISBN: 0-262-68052-1 (paperback), ISBN: 0-07-053522-1. Sacks, G. Higher Recursion Theory.

Solomon Feferman

FefermanFeferman, SolomonS. Feferman
He worked on predicative mathematics, in particular introducing the FefermanSchütte ordinal as a measure of the strength of certain predicative systems. Feferman was awarded a Guggenheim Fellowship in 1972 and 1986 and the Rolf Schock Prize in logic and philosophy in 2003. In 2006 he was invited to deliver the Tarski Lectures. In 2012 he became a fellow of the American Mathematical Society. * *Criticism of non-standard analysis * Solomon Feferman official website (via Internet Archive) at Stanford University Feferman, Solomon; Vaught, Robert L. (1959), "The first order properties of products of algebraic systems", Fund. Math. 47, 57–103.

Kleene's O

Kleene's \mathcal{O}Kleene's set \mathcal{O}notation o in Kleene's sense
Large countable ordinal. Ordinal notation.

Set theory

axiomatic set theoryset-theoreticset
Set theory is a branch of mathematical logic that studies sets, which informally are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics. The language of set theory can be used to define nearly all mathematical objects.

Countable set

countablecountably infinitecountably
In mathematics, a countable set is a set with the same cardinality (number of elements) as some subset of the set of natural numbers. A countable set is either a finite set or a countably infinite set. Whether finite or infinite, the elements of a countable set can always be counted one at a time and, although the counting may never finish, every element of the set is associated with a unique natural number.

Ordinal number

ordinalordinalsordinal numbers
Perhaps the most important ordinal that limits a system of construction in this manner is the ChurchKleene ordinal, (despite the \omega_1 in the name, this ordinal is countable), which is the smallest ordinal that cannot in any way be represented by a computable function (this can be made rigorous, of course). Considerably large ordinals can be defined below, however, which measure the "proof-theoretic strength" of certain formal systems (for example, measures the strength of Peano arithmetic). Large countable ordinals such as countable admissible ordinals can also be defined above the Church-Kleene ordinal, which are of interest in various parts of logic.

Proof theory

proof-theoreticProof-theoreticallyderive
Proof theory is a major branch of mathematical logic that represents proofs as formal mathematical objects, facilitating their analysis by mathematical techniques. Proofs are typically presented as inductively-defined data structures such as plain lists, boxed lists, or trees, which are constructed according to the axioms and rules of inference of the logical system. As such, proof theory is syntactic in nature, in contrast to model theory, which is semantic in nature.

Computable function

computablerecursive functionstotal computable function
Computable functions are the basic objects of study in computability theory. Computable functions are the formalized analogue of the intuitive notion of algorithms, in the sense that a function is computable if there exists an algorithm that can do the job of the function, i.e. given an input of the function domain it can return the corresponding output. Computable functions are used to discuss computability without referring to any concrete model of computation such as Turing machines or register machines. Any definition, however, must make reference to some specific model of computation but all valid definitions yield the same class of functions.

Halting problem

always terminatesavoid the halting problemdetect non-terminating computations
In computability theory, the halting problem is the problem of determining, from a description of an arbitrary computer program and an input, whether the program will finish running, or continue to run forever.