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Buchholz's ID Hierarchy
In set theory and logic, Buchholz's ID hierarchy is a hierarchy of subsystems of first-order arithmetic. The systems/theories ID_\nu are referred to as "the formal theories of ν-times iterated inductive definitions". IDν extends PA by ν iterated least fixed points of monotone operators. Definition Original definition The formal theory IDω (and IDν in general) is an extension of Peano Arithmetic, formulated in the language LID, by the following axioms:W. Buchholz, "An Independence Result for (\Pi^1_1\textrm)\textrm", Annals of Pure and Applied Logic vol. 33 (1987). * \forall y \forall x (\mathfrak_y(P^\mathfrak_y, x) \rightarrow x \in P^\mathfrak_y) * \forall y (\forall x (\mathfrak_y(F, x) \rightarrow F(x)) \rightarrow \forall x (x \in P^\mathfrak_y \rightarrow F(x))) for every LID-formula F(x) * \forall y \forall x_0 \forall x_1(P^\mathfrak_x_0x_1 \leftrightarrow x_0 < y \land x_1 \in P^\mathfrak_) The theory IDν with ν ≠ ω is defined as: * [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Set Theory
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concerned with those that are relevant to mathematics as a whole. The modern study of set theory was initiated by the German mathematicians Richard Dedekind and Georg Cantor in the 1870s. In particular, Georg Cantor is commonly considered the founder of set theory. The non-formalized systems investigated during this early stage go under the name of '' naive set theory''. After the discovery of paradoxes within naive set theory (such as Russell's paradox, Cantor's paradox and the Burali-Forti paradox) various axiomatic systems were proposed in the early twentieth century, of which Zermelo–Fraenkel set theory (with or without the axiom of choice) is still the best-known and most studied. Set theory is commonly employed as a foundational ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Logic
Mathematical logic is the study of logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal systems of logic such as their expressive or deductive power. However, it can also include uses of logic to characterize correct mathematical reasoning or to establish foundations of mathematics. Since its inception, mathematical logic has both contributed to and been motivated by the study of foundations of mathematics. This study began in the late 19th century with the development of axiomatic frameworks for geometry, arithmetic, and Mathematical analysis, analysis. In the early 20th century it was shaped by David Hilbert's Hilbert's program, program to prove the consistency of foundational theories. Results of Kurt Gödel, Gerhard Gentzen, and others provided partial resolution to the program, and clarified the issues involved in pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Peano Axioms
In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano. These axioms have been used nearly unchanged in a number of metamathematical investigations, including research into fundamental questions of whether number theory is consistent and complete. The need to formalize arithmetic was not well appreciated until the work of Hermann Grassmann, who showed in the 1860s that many facts in arithmetic could be derived from more basic facts about the successor operation and induction. In 1881, Charles Sanders Peirce provided an axiomatization of natural-number arithmetic. In 1888, Richard Dedekind proposed another axiomatization of natural-number arithmetic, and in 1889, Peano published a simplified version of them as a collection of axioms in his book, ''The principles of arithmetic presented by a new method'' ( la, Arithmet ... [...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, secon ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bachmann–Howard Ordinal
In mathematics, the Bachmann–Howard ordinal (also known as the Howard ordinal, or Howard-Bachmann ordinal) is a large countable ordinal. It is the ordinal analysis, proof-theoretic ordinal of several mathematical theory (logic), 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 . Definition The Bachmann–Howard ordinal is defined using an ordinal collapsing function: *''ε''''α'' enumerates the Epsilon numbers (mathematics), epsilon numbers, the ordinals ''ε'' such that ω''ε'' = ''ε''. *Ω = ω1 is the first uncountable ordinal. *''ε''Ω+1 is the first epsilon number after Ω = ''ε''Ω. *''ψ''(''α'') is defined to be the smallest ordinal that cannot be constructed by starting with 0, 1, ω and Ω, and repeatedly applying ordinal addition, multiplication and exponentiation, and ''ψ'' to previously constructed ordinals (except that ''ψ'' can only be applied to arguments les ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 and Feferman's theta function. It was named by David Madore, after Gaisi Takeuti, Solomon Feferman and Wilfried Buchholz. It is written as \psi_0(\varepsilon_) using Buchholz's psi function, an ordinal collapsing function invented by Wilfried Buchholz, and \theta_(0) in Feferman's theta function, an ordinal collapsing function invented by Solomon Feferman. It is the proof-theoretic ordinal of several formal theories: * \Pi_1^1 -CA + BI, a subsystem of second-order arithmetic * \Pi_1^1-comprehension + transfinite induction * IDω, the system of ω-times iterated inductive definitions Despite being one of the largest large countable ordinals and recursive ordinals, it is still vastly smaller than the proof-theoretic ordinal of ZFC. Definition * Let \Omega_\alpha represent the smal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 which the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ordinal Numbers
In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets. A finite set can be enumerated by successively labeling each element with the least natural number that has not been previously used. To extend this process to various infinite sets, ordinal numbers are defined more generally as linearly ordered labels that include the natural numbers and have the property that every set of ordinals has a least element (this is needed for giving a meaning to "the least unused element"). This more general definition allows us to define an ordinal number \omega that is greater than every natural number, along with ordinal numbers \omega + 1, \omega + 2, etc., which are even greater than \omega. A linear order such that every subset has a least element is called a well-order. The axiom of choice implies that every set can be well-ordered, and given two well-ordered sets, one is isomorphic to ... [...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. Jon Barwise, Barwise (1978) consists of four corresponding parts, with part D being about "Proof Theory and Constructive Mathematics". of mathematical logic that represents Mathematical proof, proofs as formal mathematical objects, facilitating their analysis by mathematical techniques. Proofs are typically presented as Recursive data type, inductively-defined data structures such as list (computer science), lists, boxed lists, or Tree (data structure), trees, which are constructed according to the axioms and rule of inference, rules of inference of the logical system. Consequently, proof theory is syntax (logic), syntactic in nature, in contrast to model theory, which is Formal semantics (logic), semantic in nature. Some of the major areas of proof theory include structural proof theory, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Set Theory
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concerned with those that are relevant to mathematics as a whole. The modern study of set theory was initiated by the German mathematicians Richard Dedekind and Georg Cantor in the 1870s. In particular, Georg Cantor is commonly considered the founder of set theory. The non-formalized systems investigated during this early stage go under the name of '' naive set theory''. After the discovery of paradoxes within naive set theory (such as Russell's paradox, Cantor's paradox and the Burali-Forti paradox) various axiomatic systems were proposed in the early twentieth century, of which Zermelo–Fraenkel set theory (with or without the axiom of choice) is still the best-known and most studied. Set theory is commonly employed as a foundational ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
