*Feferman*–

*Schütte*

*ordinal*. Bachmann–Howard ordinal. Complexity class.

For example, is a canonical notation for an ordinal which is less than the *Feferman*–*Schütte* *ordinal*: it can be written using the Veblen functions as. Concerning the order, one might point out that (the *Feferman*–*Schütte* *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 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.

The *Feferman*–*Schütte* *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.

Γ 0 is the *Feferman*–*Schütte* *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.

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 *Feferman*–*Schütte* *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".

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* 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. *

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 *Church*–*Kleene* *ordinal*). It uses a subset of the natural numbers instead of finite strings of symbols.

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.

.}, 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 (...

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 *Feferman*–*Schütte* *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.

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.

In mathematics, the *Church*–*Kleene* *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 *Church*–*Kleene* *ordinal* is a limit ordinal. It is also the first ordinal which is not hyperarithmetical, and the first admissible ordinal after ω. *

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.

The supremum of all recursive ordinals is called the *Church*–*Kleene* *ordinal* and denoted by. The *Church*–*Kleene* *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.

He worked on predicative mathematics, in particular introducing the *Feferman*–*Schütte* *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.

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.

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.

Perhaps the most important ordinal that limits a system of construction in this manner is the *Church*–*Kleene* *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 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 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.

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.