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Extension By Definition
In mathematical logic, more specifically in the proof theory of first-order theories, extensions by definitions formalize the introduction of new symbols by means of a definition. For example, it is common in naive set theory to introduce a symbol \emptyset for the set that has no member. In the formal setting of first-order theories, this can be done by adding to the theory a new constant \emptyset and the new axiom \forall x(x\notin\emptyset), meaning "for all ''x'', ''x'' is not a member of \emptyset". It can then be proved that doing so adds essentially nothing to the old theory, as should be expected from a definition. More precisely, the new theory is a conservative extension of the old one. Definition of relation symbols ''Let'' T be a first-order theory and \phi(x_1,\dots,x_n) a formula of T such that x_1, ..., x_n are distinct and include the variables free in \phi(x_1,\dots,x_n). Form a new first-order theory T' from T by adding a new n-ary relation symbol R, the lo ... [...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] |
First-order Logic
First-order logic, also called predicate logic, predicate calculus, or quantificational logic, is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantified variables over non-logical objects, and allows the use of sentences that contain variables. Rather than propositions such as "all humans are mortal", in first-order logic one can have expressions in the form "for all ''x'', if ''x'' is a human, then ''x'' is mortal", where "for all ''x"'' is a quantifier, ''x'' is a variable, and "... ''is a human''" and "... ''is mortal''" are predicates. This distinguishes it from propositional logic, which does not use quantifiers or relations; in this sense, propositional logic is the foundation of first-order logic. A theory about a topic, such as set theory, a theory for groups,A. Tarski, ''Undecidable Theories'' (1953), p. 77. Studies in Logic and the Foundation of Mathematics, North-Holland or a formal theory o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Joseph R
Joseph is a common male name, derived from the Hebrew (). "Joseph" is used, along with " Josef", mostly in English, French and partially German languages. This spelling is also found as a variant in the languages of the modern-day Nordic countries. In Portuguese and Spanish, the name is "José". In Arabic, including in the Quran, the name is spelled , . In Kurdish (''Kurdî''), the name is , Persian, the name is , and in Turkish it is . In Pashto the name is spelled ''Esaf'' (ايسپ) and in Malayalam it is spelled ''Ousep'' (ഔസേപ്പ്). In Tamil, it is spelled as ''Yosepu'' (யோசேப்பு). The name has enjoyed significant popularity in its many forms in numerous countries, and ''Joseph'' was one of the two names, along with '' Robert'', to have remained in the top 10 boys' names list in the US from 1925 to 1972. It is especially common in contemporary Israel, as either "Yossi" or "Yossef", and in Italy, where the name "Giuseppe" was the most co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elliott Mendelson
Elliott Mendelson (May 24, 1931 – May 7, 2020) was an American logician. He was a professor of mathematics at Queens College of the City University of New York, and the Graduate Center, CUNY. He was Jr. Fellow, Society of Fellows, Harvard University, 1956–58. Career Mendelson earned his BA from Columbia University and PhD from Cornell University. Mendelson taught mathematics at the college level for more than 30 years, and is the author of books on logic, philosophy of mathematics, calculus, game theory and mathematical analysis. His ''Introduction to Mathematical Logic'', first published in 1964, was reviewed by Dirk van Dalen who noted that it included "a large variety of subjects that should be part of the education of any mathematics student with an interest in foundational matters." Dirk van Dalen (1969Review: Introduction to Mathematical Logic Journal of Symbolic Logic 34(1): 110,1 Books Sole author * * * * * * * Co-author * * * * Editor * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Extension By New Constant And Function Names
In mathematical logic, a theory can be extended with new constants or function names under certain conditions with assurance that the extension will introduce no contradiction. Extension by definitions is perhaps the best-known approach, but it requires unique existence of an object with the desired property. Addition of new names can also be done safely without uniqueness. Suppose that a ''closed'' formula :\exists x_1\ldots\exists x_m\,\varphi(x_1,\ldots,x_m) is a theorem of a first-order theory T. Let T_1 be a theory obtained from T by extending its language with new constants :a_1,\ldots,a_m and adding a new axiom :\varphi(a_1,\ldots,a_m). Then T_1 is a conservative extension of T, which means that the theory T_1 has the same set of theorems in the original language (i.e., without constants a_i) as the theory T. Such a theory can also be conservatively extended by introducing a new functional symbol: Suppose that a ''closed'' formula \forall \vec\,\exists y\,\!\, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Definite Description
In formal semantics and philosophy of language, a definite description is a denoting phrase in the form of "the X" where X is a noun-phrase or a singular common noun. The definite description is ''proper'' if X applies to a unique individual or object. For example: " the first person in space" and " the 42nd President of the United States of America" are proper. The definite descriptions "the person in space" and "the Senator from Ohio" are ''improper'' because the noun phrase X applies to more than one thing, and the definite descriptions "the first man on Mars" and "the Senator from Washington D.C." are ''improper'' because X applies to nothing. Improper descriptions raise some difficult questions about the law of excluded middle, denotation, modality, and mental content. Russell's analysis As France is currently a republic, it has no king. Bertrand Russell pointed out that this raises a puzzle about the truth value of the sentence "The present King of France is bald." The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conservative Extension
In mathematical logic, a conservative extension is a supertheory of a theory which is often convenient for proving theorems, but proves no new theorems about the language of the original theory. Similarly, a non-conservative extension is a supertheory which is not conservative, and can prove more theorems than the original. More formally stated, a theory T_2 is a ( proof theoretic) conservative extension of a theory T_1 if every theorem of T_1 is a theorem of T_2, and any theorem of T_2 in the language of T_1 is already a theorem of T_1. More generally, if \Gamma is a set of formulas in the common language of T_1 and T_2, then T_2 is \Gamma-conservative over T_1 if every formula from \Gamma provable in T_2 is also provable in T_1. Note that a conservative extension of a consistent theory is consistent. If it were not, then by the principle of explosion, every formula in the language of T_2 would be a theorem of T_2, so every formula in the language of T_1 would be a theorem of T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group (mathematics)
In mathematics, a group is a Set (mathematics), set with an Binary operation, operation that combines any two elements of the set to produce a third element within the same set and the following conditions must hold: the operation is Associative property, associative, it has an identity element, and every element of the set has an inverse element. For example, the integers with the addition, addition operation form a group. The concept of a group was elaborated for handling, in a unified way, many mathematical structures such as numbers, geometric shapes and polynomial roots. Because the concept of groups is ubiquitous in numerous areas both within and outside mathematics, some authors consider it as a central organizing principle of contemporary mathematics. In geometry, groups arise naturally in the study of symmetries and geometric transformations: The symmetries of an object form a group, called the symmetry group of the object, and the transformations of a given type form a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Power Set
In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is postulated by the axiom of power set. The powerset of is variously denoted as , , , \mathbb(S), or . Any subset of is called a ''family of sets'' over . Example If is the set , then all the subsets of are * (also denoted \varnothing or \empty, the empty set or the null set) * * * * * * * and hence the power set of is . Properties If is a finite set with the cardinality (i.e., the number of all elements in the set is ), then the number of all the subsets of is . This fact as well as the reason of the notation denoting the power set are demonstrated in the below. : An indicator function or a characteristic function of a subset of a set with the cardinality is a function from to the two-element set , denoted as , ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bound Variable
In mathematics, and in other disciplines involving formal languages, including mathematical logic and computer science, a variable may be said to be either free or bound. Some older books use the terms real variable and apparent variable for free variable and bound variable, respectively. A ''free variable'' is a notation (symbol) that specifies places in an expression where substitution may take place and is not a parameter of this or any container expression. The idea is related to a ''placeholder'' (a symbol that will later be replaced by some value), or a wildcard character that stands for an unspecified symbol. In computer programming, the term free variable refers to variables used in a function that are neither local variables nor parameters of that function. The term non-local variable is often a synonym in this context. An instance of a variable symbol is ''bound'', in contrast, if the value of that variable symbol has been bound to a specific value or range of v ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proof Theory
Proof theory is a major branchAccording to , proof theory is one of four domains mathematical logic, together with model theory, axiomatic set theory, and recursion theory. consists of four corresponding parts, with part D being about "Proof Theory and Constructive Mathematics". of mathematical logic and theoretical computer science within which proofs are treated 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 a given 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
