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Mathematical logic is a subfield of mathematics exploring the applications of formal logic to mathematics. It bears close connections to metamathematics, the foundations of mathematics, and theoretical computer science. The unifying themes in mathematical logic include the study of the expressive power of formal systems and the deductive power of formal proof systems.
The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers. The standard way to resolve these debates is to define the operations of calculus using epsilon–delta procedures rather than infinitesimals. Nonstandard analysis instead reformulates the calculus using a logically rigorous notion of infinitesimal numbers.
In mathematics, the system of hyperreal numbers is a way of treating infinite and infinitesimal quantities. The hyperreals, or nonstandard reals, *R, are an extension of the real numbers R that contains numbers greater than anything of the form
In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero. They do not exist in the standard real number system, but do exist in many other number systems, such as the surreal numbers and hyperreal numbers, which can be thought of as the real numbers augmented with a system of infinitesimal quantities, as well as infinite quantities, which are the reciprocals of the infinitesimals.
In programming language theory and proof theory, the Curry–Howard correspondence is the direct relationship between computer programs and mathematical proofs.
In mathematical logic, and particularly in its subfield model theory, a saturated modelM is one that realizes as many complete types as may be "reasonably expected" given its size. For example, an ultrapower model of the hyperreals is -saturated, meaning that every descending nested sequence of internal sets has a nonempty intersection, see Goldblatt (1998).
Game semantics is an approach to formal semantics that grounds the concepts of truth or validity on game-theoretic concepts, such as the existence of a winning strategy for a player, somewhat resembling Socratic dialogues or medieval theory of Obligationes.
In nonstandard analysis, a branch of mathematics, overspill is a widely used proof technique. It is based on the fact that the set of standard natural numbers N is not an internal subset of the internal set *N of hypernatural numbers.
In model theory, a transfer principle states that all statements of some language that are true for some structure are true for another structure. One of the first examples was the Lefschetz principle, which states that any sentence in the first-order language of fields that is true for the complex numbers is also true for any algebraically closed field of characteristic 0.
In mathematics, in the field of category theory, a discrete category is a category whose only morphisms are the identity morphisms:
Johannes Franciscus Abraham Karel (Johan) van Benthem is a University Professor (universiteitshoogleraar) of logic at the University of Amsterdam at the Institute for Logic, Language and Computation and professor of philosophy at Stanford University. He was awarded the Spinozapremie in 1996 and elected a Foreign Fellow of the American Academy of Arts & Sciences in 2015.
Giorgi Japaridze is a Georgian-American researcher in logic and theoretical computer science. He currently holds the title of Full Professor at the Computing Sciences Department of Villanova University. Japaridze is best known for his invention of computability logic, cirquent calculus, and Japaridze's polymodal logic.
In mathematical logic, in particular in model theory and nonstandard analysis, an internal set is a set that is a member of a model.
Steve Vickers is a British mathematician and computer scientist. In the early 1980s, he wrote ROM firmware and manuals for three home computers, the Sinclair ZX81 and ZX Spectrum and the Jupiter Ace. The latter was produced by Jupiter Cantab, a short-lived company Vickers formed together with Richard Altwasser, after the two had left Sinclair Research. Since the late 1980s, Vickers has been an academic in the field of geometric logic, writing over 30 papers in scholarly journals on mathematical aspects of computer science. His book Topology via Logic has been influential over a range of fields. In October 2018, he retired as senior lecturer at the University of Birmingham. As announced on his university homepage, he continues to supervise PhD students at the university and focus on his research.
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 , the unique real infinitely close to it, i.e. is infinitesimal. As such, it is a mathematical implementation of the historical concept of adequality introduced by Pierre de Fermat, as well as Leibniz's Transcendental law of homogeneity.
Nonstandard analysis and its offshoot, nonstandard calculus, have been criticized by several authors, notably Errett Bishop, Paul Halmos, and Alain Connes. These criticisms are analyzed below.
Elementary Calculus: An Infinitesimal approach is a textbook by H. Jerome Keisler. The subtitle alludes to the infinitesimal numbers of the hyperreal number system of Abraham Robinson and is sometimes given as An approach using infinitesimals. The book is available freely online and is currently published by Dover.
In the mathematical field of category theory, FinSet is the category whose objects are all finite sets and whose morphisms are all functions between them. FinOrd is the category whose objects are all finite ordinal numbers and whose morphisms are all functions between them.
In mathematical logic, computational complexity theory, and computer science, the existential theory of the reals is the set of all true sentences of the form
In mathematics, a Loeb space is a type of measure space introduced by Loeb (1975) using nonstandard analysis.
References
- ↑ Benjamin C. Pierce (1991). Basic category theory for computer scientists. MIT Press. p. 73. ISBN 978-0-262-66071-6.
