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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 often equiconsistent, and if one theory has a larger proof-theoretic ordinal than another it can often prove the consistency of the second theory. History The field of ordinal analysis was formed when Gerhard Gentzen in 1934 used cut elimination to prove, in modern terms, that the proof-theoretic ordinal of Peano arithmetic is ε0. See Gentzen's consistency proof. Definition Ordinal analysis concerns true, effective (recursive) theories that can interpret a sufficient portion of arithmetic to make statements about ordinal notations. The proof-theoretic ordinal of such a theory T is the supremum of the order types of all ordinal notations (necessarily recursive, see next section) that the theory can prove are well founded—the supremum of all ordinals \alpha for w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proof Theory
Proof theory is a major branchAccording to Wang (1981), pp. 3–4, proof theory is one of four domains mathematical logic, together with model theory, axiomatic set theory, and recursion theory. Barwise (1978) consists of four corresponding parts, with part D being about "Proof Theory and Constructive Mathematics". 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 lists, boxed lists, or trees, which are constructed according to the axioms and rules of inference of the logical system. Consequently, proof theory is syntactic in nature, in contrast to model theory, which is semantic in nature. Some of the major areas of proof theory include structural proof theory, ordinal analysis, provability logic, reverse mathematics, proof mining, automated theorem proving, and proof complexity. Much research also focuses ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Church–Kleene 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 functions. The Church–Kleene ordinal and variants The smallest non-recursive ordinal is the Church Kleene ordinal, \omega_1^, named after Alonzo Church and S. C. Kleene; its order type is the set of all recursive ordinals. Since the successor of a recursive ordinal is recursive, the Church–Kleene ordinal is a limit ordinal. It is also the smallest ordinal that is not hyperarithmetical, and the smallest admissible ordinal after (an ordinal ''α'' is called admissible if L_\alpha \models \mathsf.). The \omega_1^-recursive subsets of are exactly the \Delta^1_1 subsets of .D. MadoreA Zoo of Ordinals(2017). Accessed September 2021. The notation \omega_1^ is in reference to , the first uncountable ordinal, which is the set of all countable ordinals, analogously to how the Church-Klee ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arithmetical Transfinite Recursion
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 reverse mathematics program was foreshadowed by results in set theory such as the classical theorem that the axiom of choice and Zorn's lemma are equivalent over ZF set theory. The goal of reverse mathematics, however, is to study possible axioms of ordinary theorems of mathematics rather than possible axioms for set theory. Reverse mathematics is usually carried out using subsystems of second-order arithmetic,Simpson, Stephen G. (2009), Subsystems of second-order arithmetic, Perspectives in Logic (2nd ed.), Cambridge University Press, doi:10.1017/CBO9780511581007, ISBN 97 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 after Solomon Feferman and Kurt Schütte, the former of whom suggested the name Γ0. There is no standard notation for ordinals beyond the Feferman–Schütte ordinal. There are several ways of representing the Feferman–Schütte ordinal, some of which use 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 to give names to certain ordinals much larger ...s: \psi(\Omega^\Omega), \theta(\Omega), \varphi_\Omega(0), or \varphi(1,0,0). Definition The Feferman–Schütte ordinal can be defined as the smallest ordinal that cannot be obtained by starting with 0 and using the operations of ordinal addition a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arithmetical Comprehension
In mathematical logic, second-order arithmetic is a collection of axiomatic systems that formalize the natural numbers and their subsets. It is an alternative to axiomatic set theory as a foundation for much, but not all, of mathematics. A precursor to second-order arithmetic that involves third-order parameters was introduced by David Hilbert and Paul Bernays in their book ''Grundlagen der Mathematik''. The standard axiomatization of second-order arithmetic is denoted by Z2. Second-order arithmetic includes, but is significantly stronger than, its first-order counterpart Peano arithmetic. Unlike Peano arithmetic, second-order arithmetic allows quantification over sets of natural numbers as well as numbers themselves. Because real numbers can be represented as ( infinite) sets of natural numbers in well-known ways, and because second-order arithmetic allows quantification over such sets, it is possible to formalize the real numbers in second-order arithmetic. For this reaso ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gentzen
Gerhard Karl Erich Gentzen (24 November 1909 – 4 August 1945) was a German mathematician and logician. He made major contributions to the foundations of mathematics, proof theory, especially on natural deduction and sequent calculus. He died of starvation in a Soviet prison camp in Prague in 1945, having been interned as a German national after the Second World War. Life and career Gentzen was a student of Paul Bernays at the University of Göttingen. Bernays was fired as "non-Aryan" in April 1933 and therefore Hermann Weyl formally acted as his supervisor. Gentzen joined the Sturmabteilung in November 1933 although he was by no means compelled to do so. Nevertheless he kept in contact with Bernays until the beginning of the Second World War. In 1935, he corresponded with Abraham Fraenkel in Jerusalem and was implicated by the Nazi teachers' union as one who "keeps contacts to the Chosen People." In 1935 and 1936, Hermann Weyl, head of the Göttingen mathematics department in 1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Primitive Recursive Arithmetic
Primitive recursive arithmetic (PRA) is a quantifier-free formalization of the natural numbers. It was first proposed by Norwegian mathematician , reprinted in translation in as a formalization of his finitist conception of the foundations of arithmetic, and it is widely agreed that all reasoning of PRA is finitist. Many also believe that all of finitism is captured by PRA, but others believe finitism can be extended to forms of recursion beyond primitive recursion, up to ε0, which is the proof-theoretic ordinal of Peano arithmetic. PRA's proof theoretic ordinal is ωω, where ω is the smallest transfinite ordinal. PRA is sometimes called Skolem arithmetic. The language of PRA can express arithmetic propositions involving natural numbers and any primitive recursive function, including the operations of addition, multiplication, and exponentiation. PRA cannot explicitly quantify over the domain of natural numbers. PRA is often taken as the basic metamathematica ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 Necessity and sufficiency, necessary conditions from Necessity and sufficiency, sufficient ones. The reverse mathematics program was foreshadowed by results in set theory such as the classical theorem that the axiom of choice and Zorn's lemma are equivalent over ZF set theory. The goal of reverse mathematics, however, is to study possible axioms of ordinary theorems of mathematics rather than possible axioms for set theory. Reverse mathematics is usually carried out using subsystems of second-order arithmetic,Simpson, Stephen G. (2009), Subsystems of second-order arithmetic, Perspectives in Logic (2nd ed.), Cambridge Univ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Second-order Arithmetic
In mathematical logic, second-order arithmetic is a collection of axiomatic systems that formalize the natural numbers and their subsets. It is an alternative to axiomatic set theory as a foundation for much, but not all, of mathematics. A precursor to second-order arithmetic that involves third-order parameters was introduced by David Hilbert and Paul Bernays in their book '' Grundlagen der Mathematik''. The standard axiomatization of second-order arithmetic is denoted by Z2. Second-order arithmetic includes, but is significantly stronger than, its first-order counterpart Peano arithmetic. Unlike Peano arithmetic, second-order arithmetic allows quantification over sets of natural numbers as well as numbers themselves. Because real numbers can be represented as ( infinite) sets of natural numbers in well-known ways, and because second-order arithmetic allows quantification over such sets, it is possible to formalize the real numbers in second-order arithmetic. For this reason, se ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grzegorczyk Hierarchy
The Grzegorczyk hierarchy (, ), named after the Polish logician Andrzej Grzegorczyk, is a hierarchy of functions used in computability theory. Every function in the Grzegorczyk hierarchy is a primitive recursive function, and every primitive recursive function appears in the hierarchy at some level. The hierarchy deals with the rate at which the values of the functions grow; intuitively, functions in lower levels of the hierarchy grow slower than functions in the higher levels. Definition First we introduce an infinite set of functions, denoted ''Ei'' for some natural number ''i''. We define E_0(x,y)=x+y and E_1(x)=x^2+2. I.e., ''E0'' is the addition function, and ''E1'' is a unary function which squares its argument and adds two. Then, for each ''n'' greater than 2, we define E_n(x)=E^_(2), i.e. the ''x''-th iterate of E_ evaluated at 2. From these functions we define the Grzegorczyk hierarchy. \mathcal^n, the ''n''-th set in the hierarchy, contains the following functions: # ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grand Conjecture
In proof theory, a branch of mathematical logic, elementary function arithmetic (EFA), also called elementary arithmetic and exponential function arithmetic,C. Smoryński, "Nonstandard Models and Related Developments" (p. 217). From ''Harvey Friedman's Research on the Foundations of Mathematics'' (1985), Studies in Logic and the Foundations of Mathematics vol. 117. is the system of arithmetic with the usual elementary properties of 0, 1, +, ×, ''x''''y'', together with induction for formulas with bounded quantifiers. EFA is a very weak logical system, whose proof theoretic ordinal is ω3, but still seems able to prove much of ordinary mathematics that can be stated in the language of first-order arithmetic. Definition EFA is a system in first order logic (with equality). Its language contains: *two constants 0, 1, *three binary operations +, ×, exp, with exp(''x'',''y'') usually written as ''x''''y'', *a binary relation symbol < (This is not really ne ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 Necessity and sufficiency, necessary conditions from Necessity and sufficiency, sufficient ones. The reverse mathematics program was foreshadowed by results in set theory such as the classical theorem that the axiom of choice and Zorn's lemma are equivalent over ZF set theory. The goal of reverse mathematics, however, is to study possible axioms of ordinary theorems of mathematics rather than possible axioms for set theory. Reverse mathematics is usually carried out using subsystems of second-order arithmetic,Simpson, Stephen G. (2009), Subsystems of second-order arithmetic, Perspectives in Logic (2nd ed.), Cambridge Univ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |