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Harvey Friedman
__NOTOC__ Harvey Friedman (born 23 September 1948)Handbook of Philosophical Logic, , p. 38 is an American mathematical logician at Ohio State University in Columbus, Ohio. He has worked on reverse mathematics, a project intended to derive the axioms of mathematics from the theorems considered to be necessary. In recent years this has advanced to a study of Boolean relation theory, which attempts to justify large cardinal axioms by demonstrating their necessity for deriving certain propositions considered "concrete". Friedman earned his Ph.D. from the Massachusetts Institute of Technology in 1967, with a dissertation on ''Subsystems of Analysis''. His advisor was Gerald Sacks. Friedman received the Alan T. Waterman Award in 1984. He also assumed the title of Vising Scientist at IBM. He delivered the Tarski Lectures in 2007. In 1967, Friedman was listed in the ''Guinness Book of World Records'' for being the world's youngest professor when he taught at Stanford University at age ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harvey Friedman
__NOTOC__ Harvey Friedman (born 23 September 1948)Handbook of Philosophical Logic, , p. 38 is an American mathematical logician at Ohio State University in Columbus, Ohio. He has worked on reverse mathematics, a project intended to derive the axioms of mathematics from the theorems considered to be necessary. In recent years this has advanced to a study of Boolean relation theory, which attempts to justify large cardinal axioms by demonstrating their necessity for deriving certain propositions considered "concrete". Friedman earned his Ph.D. from the Massachusetts Institute of Technology in 1967, with a dissertation on ''Subsystems of Analysis''. His advisor was Gerald Sacks. Friedman received the Alan T. Waterman Award in 1984. He also assumed the title of Vising Scientist at IBM. He delivered the Tarski Lectures in 2007. In 1967, Friedman was listed in the ''Guinness Book of World Records'' for being the world's youngest professor when he taught at Stanford University at age ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Set Theorists
Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electronics and computing *Set (abstract data type), a data type in computer science that is a collection of unique values ** Set (C++), a set implementation in the C++ Standard Library * Set (command), a command for setting values of environment variables in Unix and Microsoft operating-systems * Secure Electronic Transaction, a standard protocol for securing credit card transactions over insecure networks * Single-electron transistor, a device to amplify currents in nanoelectronics * Single-ended triode, a type of electronic amplifier * Set!, a programming syntax in the scheme programming language Biology and psychology * Set (psychology), a set of expectations which shapes perception or thought *Set or sett, a badger's den *Set, a small tuber ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Logicians
American(s) may refer to: * American, something of, from, or related to the United States of America, commonly known as the "United States" or "America" ** Americans, citizens and nationals of the United States of America ** American ancestry, people who self-identify their ancestry as "American" ** American English, the set of varieties of the English language native to the United States ** Native Americans in the United States, indigenous peoples of the United States * American, something of, from, or related to the Americas, also known as "America" ** Indigenous peoples of the Americas * American (word), for analysis and history of the meanings in various contexts Organizations * American Airlines, U.S.-based airline headquartered in Fort Worth, Texas * American Athletic Conference, an American college athletic conference * American Recordings (record label), a record label previously known as Def American * American University, in Washington, D.C. Sports teams Soccer * B ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
21st-century American Mathematicians
The 1st century was the century spanning AD 1 ( I) through AD 100 ( C) according to the Julian calendar. It is often written as the or to distinguish it from the 1st century BC (or BCE) which preceded it. The 1st century is considered part of the Classical era, epoch, or historical period. The 1st century also saw the appearance of Christianity. During this period, Europe, North Africa and the Near East fell under increasing domination by the Roman Empire, which continued expanding, most notably conquering Britain under the emperor Claudius ( AD 43). The reforms introduced by Augustus during his long reign stabilized the empire after the turmoil of the previous century's civil wars. Later in the century the Julio-Claudian dynasty, which had been founded by Augustus, came to an end with the suicide of Nero in AD 68. There followed the famous Year of Four Emperors, a brief period of civil war and instability, which was finally brought to an end by Vespasian, ninth Roman empero ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Leo Harrington
Leo Anthony Harrington (born May 17, 1946) is a professor of mathematics at the University of California, Berkeley who works in recursion theory, model theory, and set theory. Having retired from being a Mathematician, Professor Leo Harrington is now a Philosopher. His notable results include proving the Paris–Harrington theorem along with Jeff Paris, showing that if the axiom of determinacy holds for all analytic sets then ''x''# exists for all reals ''x'', and proving with Saharon Shelah that the first-order theory of the partially ordered set of recursively enumerable Turing degrees In computer science and mathematical logic the Turing degree (named after Alan Turing) or degree of unsolvability of a set of natural numbers measures the level of algorithmic unsolvability of the set. Overview The concept of Turing degree is fund ... is undecidable. References External linksHome page * Living people American logicians 20th-century American mathematicians 21st-cent ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Friedman Translation
In mathematical logic, the Friedman translation is a certain transformation of intuitionistic formulas. Among other things it can be used to show that the Π02-theorems of various first-order theories of classical mathematics are also theorems of intuitionistic mathematics. It is named after its discoverer, Harvey Friedman. Definition Let ''A'' and ''B'' be intuitionistic formulas, where no free variable of ''B'' is quantified in ''A''. The translation ''AB'' is defined by replacing each atomic subformula ''C'' of ''A'' by . For purposes of the translation, ⊥ is considered to be an atomic formula as well, hence it is replaced with (which is equivalent to ''B''). Note that ¬''A'' is defined as an abbreviation for hence Application The Friedman translation can be used to show the closure of many intuitionistic theories under the Markov rule, and to obtain partial conservativity results. The key condition is that the \Delta^0_0 sentences of the logic be decidable, allowing ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Friedman's Grand Conjecture
In proof theory, a branch of mathematical logic, elementary function arithmetic (EFA), also called elementary arithmetic and exponential function arithmetic,C. Smoryński, "Nonstandard Models and Related Developments" (p. 217). From ''Harvey Friedman's Research on the Foundations of Mathematics'' (1985), Studies in Logic and the Foundations of Mathematics vol. 117. is the system of arithmetic with the usual elementary properties of 0, 1, +, ×, ''x''''y'', together with induction for formulas with bounded quantifiers. EFA is a very weak logical system, whose proof theoretic ordinal is ω3, but still seems able to prove much of ordinary mathematics that can be stated in the language of first-order arithmetic. Definition EFA is a system in first order logic (with equality). Its language contains: *two constants 0, 1, *three binary operations +, ×, exp, with exp(''x'',''y'') usually written as ''x''''y'', *a binary relation symbol < (This is not really neces ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sy Friedman
Sy-David Friedman (born May 23, 1953 in Chicago) is an American and Austrian mathematician and a (retired) professor of mathematics at the University of Vienna and the former director of the Kurt Gödel Research Center for Mathematical Logic. His main research interest lies in mathematical logic, in particular in set theory and recursion theory. Friedman is the brother of Ilene Friedman and the brother of mathematician Harvey Friedman. Biography He studied at Northwestern University and, from 1970, at the Massachusetts Institute of Technology. He received his Ph.D. in 1976 from MIT (his thesis ''Recursion on Inadmissible Ordinals'' was written under the supervision of Gerald E. Sacks). In 1979 Sy Friedman accepted a position at MIT, and in 1990 he became a full professor there. Since 1999 he has been a professor of mathematical logic at the University of Vienna (since 2018 retired). He is a Fellow of Collegium Invisibile. Selected publications and results He has authored ab ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zermelo–Fraenkel Set Theory
In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such as Russell's paradox. Today, Zermelo–Fraenkel set theory, with the historically controversial axiom of choice (AC) included, is the standard form of axiomatic set theory and as such is the most common foundation of mathematics. Zermelo–Fraenkel set theory with the axiom of choice included is abbreviated ZFC, where C stands for "choice", and ZF refers to the axioms of Zermelo–Fraenkel set theory with the axiom of choice excluded. Informally, Zermelo–Fraenkel set theory is intended to formalize a single primitive notion, that of a hereditary well-founded set, so that all entities in the universe of discourse are such sets. Thus the axioms of Zermelo–Fraenkel set theory refer only to pure sets and prevent its models from containing u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gödel's Ontological Proof
Gödel's ontological proof is a formal argument by the mathematician Kurt Gödel (1906–1978) for the existence of God. The argument is in a line of development that goes back to Anselm of Canterbury (1033–1109). St. Anselm's ontological argument, in its most succinct form, is as follows: "God, by definition, is that for which no greater can be conceived. God exists in the understanding. If God exists in the understanding, we could imagine Him to be greater by existing in reality. Therefore, God must exist." A more elaborate version was given by Gottfried Leibniz (1646–1716); this is the version that Gödel studied and attempted to clarify with his ontological argument. Gödel left a fourteen-point outline of his philosophical beliefs in his papers. Points relevant to the ontological proof include :4. There are other worlds and rational beings of a different and higher kind. :5. The world in which we live is not the only one in which we shall live or have lived. :13. There is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
