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Paul Bernays
Paul Isaac Bernays ( ; ; 17 October 1888 – 18 September 1977) was a Swiss mathematician who made significant contributions to mathematical logic, axiomatic set theory, and the philosophy of mathematics. He was an assistant and close collaborator of David Hilbert. Biography Bernays was born into a distinguished German-Jewish family of scholars and businessmen. His great-grandfather, Isaac ben Jacob Bernays, served as chief rabbi of Hamburg from 1821 to 1849. Bernays spent his childhood in Berlin, and attended the Köllnische Gymnasium, 1895–1907. At the University of Berlin, he studied mathematics under Issai Schur, Edmund Landau, Ferdinand Georg Frobenius, and Friedrich Schottky; philosophy under Alois Riehl, Carl Stumpf and Ernst Cassirer; and physics under Max Planck. At the University of Göttingen, he studied mathematics under David Hilbert, Edmund Landau, Hermann Weyl, and Felix Klein; physics under Voigt and Max Born; and philosophy under Leonard Nelson. In 1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
London
London is the Capital city, capital and List of urban areas in the United Kingdom, largest city of both England and the United Kingdom, with a population of in . London metropolitan area, Its wider metropolitan area is the largest in Western Europe, with a population of 14.9 million. London stands on the River Thames in southeast England, at the head of a tidal estuary down to the North Sea, and has been a major settlement for nearly 2,000 years. Its ancient core and financial centre, the City of London, was founded by the Roman Empire, Romans as Londinium and has retained its medieval boundaries. The City of Westminster, to the west of the City of London, has been the centuries-long host of Government of the United Kingdom, the national government and Parliament of the United Kingdom, parliament. London grew rapidly 19th-century London, in the 19th century, becoming the world's List of largest cities throughout history, largest city at the time. Since the 19th cen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Axiom Of Adjunction
In mathematical set theory, the axiom of adjunction states that for any two sets ''x'', ''y'' there is a set ''w'' = ''x'' ∪ given by "adjoining" the set ''y'' to the set ''x''. It is stated as :\forall x. \forall y. \exists w. \forall z. \big( z \in w \leftrightarrow (z \in x \lor z=y) \big). introduced the axiom of adjunction as one of the axioms for a system of set theory that he introduced in about 1929. It is a weak axiom, used in some weak systems of set theory such as general set theory or finitary set theory. The adjunction operation is also used as one of the operations of primitive recursive set functions. Interpretability of arithmetic Tarski and Szmielew showed that Robinson arithmetic () can be interpreted in a weak set theory whose axioms are extensionality, the existence of the empty set, and the axiom of adjunction . In fact, empty set and adjunction alone (without extensionality) suffice to interpret . (They are mutually interpreta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Philosophy Of Mathematics
Philosophy of mathematics is the branch of philosophy that deals with the nature of mathematics and its relationship to other areas of philosophy, particularly epistemology and metaphysics. Central questions posed include whether or not mathematical objects are purely abstract entities or are in some way concrete, and in what the relationship such objects have with physical reality consists. Major themes that are dealt with in philosophy of mathematics include: *''Reality'': The question is whether mathematics is a pure product of human mind or whether it has some reality by itself. *''Logic and rigor'' *''Relationship with physical reality'' *''Relationship with science'' *''Relationship with applications'' *''Mathematical truth'' *''Nature as human activity'' (science, the arts, art, game, or all together) Major themes Reality Logic and rigor Mathematical reasoning requires Mathematical rigor, rigor. This means that the definitions must be absolutely unambiguous and th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Set Theory
Set theory is the branch of mathematical logic that studies Set (mathematics), 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 of set theory, 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 the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Axiom
An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or fit' or 'that which commends itself as evident'. The precise definition varies across fields of study. In classic philosophy, an axiom is a statement that is so evident or well-established, that it is accepted without controversy or question. In modern logic, an axiom is a premise or starting point for reasoning. In mathematics, an ''axiom'' may be a " logical axiom" or a " non-logical axiom". Logical axioms are taken to be true within the system of logic they define and are often shown in symbolic form (e.g., (''A'' and ''B'') implies ''A''), while non-logical axioms are substantive assertions about the elements of the domain of a specific mathematical theory, for example ''a'' + 0 = ''a'' in integer arithmetic. N ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Logic
Mathematical logic is the study of Logic#Formal logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory (also known as computability 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 th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Von Neumann–Bernays–Gödel Set Theory
In the foundations of mathematics, von Neumann–Bernays–Gödel set theory (NBG) is an axiomatic set theory that is a conservative extension of Zermelo–Fraenkel–choice set theory (ZFC). NBG introduces the notion of class, which is a collection of sets defined by a formula whose quantifiers range only over sets. NBG can define classes that are larger than sets, such as the class of all sets and the class of all ordinals. Morse–Kelley set theory (MK) allows classes to be defined by formulas whose quantifiers range over classes. NBG is finitely axiomatizable, while ZFC and MK are not. A key theorem of NBG is the class existence theorem, which states that for every formula whose quantifiers range only over sets, there is a class consisting of the sets satisfying the formula. This class is built by mirroring the step-by-step construction of the formula with classes. Since all set-theoretic formulas are constructed from two kinds of atomic formulas (membership and equality ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert–Bernays Paradox
The Hilbert–Bernays paradox is a distinctive paradox belonging to the family of the paradoxes of reference. It is named after David Hilbert and Paul Bernays. History The paradox appears in Hilbert and Bernays' ''Grundlagen der Mathematik'' and is used by them to show that a sufficiently strong consistent theory cannot contain its own reference functor. Although it has gone largely unnoticed in the course of the 20th century, it has recently been rediscovered and appreciated for the distinctive difficulties it presents. Formulation Just as the semantic property of truth seems to be governed by the naive schema: :(T) The sentence ′''P''′ is true if and only if ''P'' (where single quotes refer to the linguistic expression inside the quotes), the semantic property of reference seems to be governed by the naive schema: :(R) If ''a'' exists, the referent of the name ′''a''′ is identical with ''a'' Let us suppose however that, for every expression e in the language, the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert–Bernays Provability Conditions
In mathematical logic, the Hilbert–Bernays provability conditions, named after David Hilbert and Paul Bernays, are a set of requirements for formalized provability predicates in formal theories of arithmetic (Smith 2007:224). These conditions are used in many proofs of Kurt Gödel's second incompleteness theorem. They are also closely related to axioms of provability logic. The conditions Let be a formal theory of arithmetic with a formalized provability predicate , which is expressed as a formula of with one free number variable. For each formula in the theory, let be the Gödel number of . The Hilbert–Bernays provability conditions are: # If proves a sentence then proves . # For every sentence , proves # proves that and imply Note that is predicate of numbers, and it is a provability predicate in the sense that the intended interpretation of is that there exists a number that codes for a proof of . Formally what is required of is the above three condition ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Implicational Propositional Calculus
In mathematical logic, the implicational propositional calculus is a version of classical propositional calculus that uses only one connective, called implication or conditional. In formulas, this binary operation is indicated by "implies", "if ..., then ...", "→", "\rightarrow ", etc.. Functional (in)completeness Implication alone is not functionally complete as a logical operator because one cannot form all other two-valued truth functions from it. For example, the two-place truth function that always returns '' false'' is not definable from → and arbitrary propositional variables: any formula constructed from → and propositional variables must receive the value ''true'' when all of its variables are evaluated to true. It follows that is not functionally complete. However, if one adds a nullary connective ⊥ for falsity, then one can define all other truth functions. Formulas over the resulting set of connectives are called f-implicational. If ''P'' and ''Q'' are pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bernays–Schönfinkel Class
The Bernays–Schönfinkel class (also known as Bernays–Schönfinkel–Ramsey class) of formulas, named after Paul Bernays, Moses Schönfinkel and Frank P. Ramsey, is a fragment of first-order logic formulas where satisfiability is decidable. It is the set of sentences that, when written in prenex normal form, have an \exists^*\forall^* quantifier prefix and do not contain any function symbols. Ramsey proved that, if \phi is a formula in the Bernays–Schönfinkel class with one free variable, then either \ is finite, or \ is finite. This class of logic formulas is also sometimes referred as effectively propositional (EPR) since it can be effectively translated into propositional logic formulas by a process of grounding or instantiation. The satisfiability problem for this class is NEXPTIME In computational complexity theory, the complexity class NEXPTIME (sometimes called NEXP) is the set of decision problems that can be solved by a non-deterministic Turing machine using ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reflection Principle
In set theory, a branch of mathematics, a reflection principle says that it is possible to find sets that, with respect to any given property, resemble the class of all sets. There are several different forms of the reflection principle depending on exactly what is meant by "resemble". Weak forms of the reflection principle are theorems of ZF set theory due to , while stronger forms can be new and very powerful axioms for set theory. The name "reflection principle" comes from the fact that properties of the universe of all sets are "reflected" down to a smaller set. Motivation A naive version of the reflection principle states that "for any property of the universe of all sets we can find a set with the same property". This leads to an immediate contradiction: the universe of all sets contains all sets, but there is no set with the property that it contains all sets. To get useful (and non-contradictory) reflection principles we need to be more careful about what we mean by "prop ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
