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Internal Set
In mathematical logic, in particular in model theory and nonstandard analysis, an internal set is a set that is a member of a model. The concept of internal sets is a tool in formulating the transfer principle, which concerns the logical relation between the properties of the real numbers R, and the properties of a larger field denoted *R called the hyperreal numbers. The field *R includes, in particular, infinitesimal ("infinitely small") numbers, providing a rigorous mathematical justification for their use. Roughly speaking, the idea is to express analysis over R in a suitable language of mathematical logic, and then point out that this language applies equally well to *R. This turns out to be possible because at the set-theoretic level, the propositions in such a language are interpreted to apply only to internal sets rather than to all sets (note that the term "language" is used in a loose sense in the above). Edward Nelson's internal set theory is an axiomatic approach to ... [...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] |
Internal Set Theory
Internal set theory (IST) is a mathematical theory of sets developed by Edward Nelson that provides an axiomatic basis for a portion of the nonstandard analysis introduced by Abraham Robinson. Instead of adding new elements to the real numbers, Nelson's approach modifies the axiomatic foundations through syntactic enrichment. Thus, the axioms introduce a new term, "standard", which can be used to make discriminations not possible under the conventional ZFC axioms for sets. Thus, IST is an enrichment of ZFC: all axioms of ZFC are satisfied for all classical predicates, while the new unary predicate "standard" satisfies three additional axioms I, S, and T. In particular, suitable nonstandard elements within the set of real numbers can be shown to have properties that correspond to the properties of infinitesimal and unlimited elements. Nelson's formulation is made more accessible for the lay-mathematician by leaving out many of the complexities of meta-mathematical logic that w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperreal Number
In mathematics, the system of hyperreal numbers is a way of treating infinite and infinitesimal (infinitely small but non-zero) quantities. The hyperreals, or nonstandard reals, *R, are an extension of the real numbers R that contains numbers greater than anything of the form :1 + 1 + \cdots + 1 (for any finite number of terms). Such numbers are infinite, and their reciprocals are infinitesimals. The term "hyper-real" was introduced by Edwin Hewitt in 1948. The hyperreal numbers satisfy the transfer principle, a rigorous version of Leibniz's heuristic law of continuity. The transfer principle states that true first-order statements about R are also valid in *R. For example, the commutative law of addition, , holds for the hyperreals just as it does for the reals; since R is a real closed field, so is *R. Since \sin()=0 for all integers ''n'', one also has \sin()=0 for all hyperintegers H. The transfer principle for ultrapowers is a consequence of Łoś' theorem of 1955. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Robert Goldblatt
__notoc__ Robert Ian Goldblatt (born 1949) is a mathematical logician who is Emeritus Professor in the School of Mathematics and Statistics at Victoria University, Wellington, New Zealand. His most popular books are ''Logics of Time and Computation'' and ''Topoi: the Categorial Analysis of Logic''. He has also written a graduate level textbook on hyperreal numbers which is an introduction to nonstandard analysis. He has been Coordinating Editor of The ''Journal of Symbolic Logic'' and a Managing Editor of ''Studia Logica''. He was elected Fellow and Councillor of the Royal Society of New Zealand, President of the New Zealand Mathematical Society, and represented New Zealand to the International Mathematical Union. In 2012 he was awarded theJones Medalfor lifetime achievement in mathematics. Books and handbook chapters * 1979: ''Topoi: The Categorial Analysis of Logic'', North-Holland. Revised edition 1984. Dover Publications edition 2006Internet edition Project Euclid. :: Benjami ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Superstructure (mathematics)
In mathematics, and particularly in set theory, category theory, type theory, and the foundations of mathematics, a universe is a collection that contains all the entities one wishes to consider in a given situation. In set theory, universes are often classes that contain (as elements) all sets for which one hopes to prove a particular theorem. These classes can serve as inner models for various axiomatic systems such as ZFC or Morse–Kelley set theory. Universes are of critical importance to formalizing concepts in category theory inside set-theoretical foundations. For instance, the canonical motivating example of a category is Set, the category of all sets, which cannot be formalized in a set theory without some notion of a universe. In type theory, a universe is a type whose elements are types. In a specific context Perhaps the simplest version is that ''any'' set can be a universe, so long as the object of study is confined to that particular set. If the object of stu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Standard Part Function
In nonstandard analysis, the standard part function is a function from the limited (finite) hyperreal numbers to the real numbers. Briefly, the standard part function "rounds off" a finite hyperreal to the nearest real. It associates to every such hyperreal x, the unique real x_0 infinitely close to it, i.e. x-x_0 is infinitesimal. As such, it is a mathematical implementation of the historical concept of adequality introduced by Pierre de Fermat,Karin Usadi Katz and Mikhail G. Katz (2011) A Burgessian Critique of Nominalistic Tendencies in Contemporary Mathematics and its Historiography. Foundations of Science.Searxiv The authors refer to the Fermat-Robinson standard part. as well as Leibniz's Transcendental law of homogeneity. The standard part function was first defined by Abraham Robinson who used the notation ^x for the standard part of a hyperreal x (see Robinson 1974). This concept plays a key role in defining the concepts of the calculus, such as continuity, the derivati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ultrafilter
In the mathematical field of order theory, an ultrafilter on a given partially ordered set (or "poset") P is a certain subset of P, namely a maximal filter on P; that is, a proper filter on P that cannot be enlarged to a bigger proper filter on P. If X is an arbitrary set, its power set \wp(X), ordered by set inclusion, is always a Boolean algebra and hence a poset, and ultrafilters on \wp(X) are usually called X.If X happens to be partially ordered, too, particular care is needed to understand from the context whether an (ultra)filter on \wp(X) or an (ultra)filter just on X is meant; both kinds of (ultra)filters are quite different. Some authors use "(ultra)filter" ''of'' a partial ordered set" vs. "''on'' an arbitrary set"; i.e. they write "(ultra)filter on X" to abbreviate "(ultra)filter of \wp(X)". An ultrafilter on a set X may be considered as a finitely additive measure on X. In this view, every subset of X is either considered "almost everything" (has measure 1) or "almos ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ultrapower
The ultraproduct is a mathematical construction that appears mainly in abstract algebra and mathematical logic, in particular in model theory and set theory. An ultraproduct is a quotient of the direct product of a family of structures. All factors need to have the same signature. The ultrapower is the special case of this construction in which all factors are equal. For example, ultrapowers can be used to construct new fields from given ones. The hyperreal numbers, an ultrapower of the real numbers, are a special case of this. Some striking applications of ultraproducts include very elegant proofs of the compactness theorem and the completeness theorem, Keisler's ultrapower theorem, which gives an algebraic characterization of the semantic notion of elementary equivalence, and the Robinson–Zakon presentation of the use of superstructures and their monomorphisms to construct nonstandard models of analysis, leading to the growth of the area of nonstandard analysis, which was pion ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Constructive Nonstandard Analysis
In mathematics, constructive nonstandard analysis is a version of Abraham Robinson's nonstandard analysis, developed by Moerdijk (1995), Palmgren (1998), Ruokolainen (2004). Ruokolainen wrote: :The possibility of constructivization of nonstandard analysis was studied by Palmgren (1997, 1998, 2001). The model of constructive nonstandard analysis studied there is an extension of Moerdijk’s (1995) model for constructive nonstandard arithmetic. See also *Smooth infinitesimal analysis *John Lane Bell References *Ieke Moerdijk, ''A model for intuitionistic nonstandard arithmetic'', Annals of Pure and Applied Logic, vol. 73 (1995), pp. 37–51. : "Abstract: This paper provides an explicit description of a model for intuitionistic nonstandard arithmetic, which can be formalized in a constructive metatheory without the axiom of choice* Erik Palmgren, ''Developments in Constructive Nonstandard Analysis'', Bulletin of Symbolic Logic Volume 4, Number 3 (1998), 233–272. : "Abst ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Edward Nelson
Edward Nelson (May 4, 1932 – September 10, 2014) was an American mathematician. He was professor in the Mathematics Department at Princeton University. He was known for his work on mathematical physics and mathematical logic. In mathematical logic, he was noted especially for his internal set theory, and views on ultrafinitism and the consistency of arithmetic. In philosophy of mathematics he advocated the view of formalism rather than platonism or intuitionism. He also wrote on the relationship between religion and mathematics. Biography Edward Nelson was born in Decatur, Georgia in 1932. He spent his early childhood in Rome where his father worked for the Italian YMCA. At the advent of World War II, Nelson moved with his mother to New York City, where he attended high school at the Bronx High School of Science. His father, who spoke fluent Russian, stayed in St. Petersburg in connection with issues related to prisoners of war. After the war, his family returned to Italy a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Model Theory
In mathematical logic, model theory is the study of the relationship between formal theories (a collection of sentences in a formal language expressing statements about a mathematical structure), and their models (those structures in which the statements of the theory hold). The aspects investigated include the number and size of models of a theory, the relationship of different models to each other, and their interaction with the formal language itself. In particular, model theorists also investigate the sets that can be defined in a model of a theory, and the relationship of such definable sets to each other. As a separate discipline, model theory goes back to Alfred Tarski, who first used the term "Theory of Models" in publication in 1954. Since the 1970s, the subject has been shaped decisively by Saharon Shelah's stability theory. Compared to other areas of mathematical logic such as proof theory, model theory is often less concerned with formal rigour and closer in spirit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Set-theoretic
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 system fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
