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Principles Of Mathematical Logic
''Principles of Mathematical Logic'' is the 1950 American translation of the 1938 second edition of David Hilbert's and Wilhelm Ackermann's classic text ''Grundzüge der theoretischen Logik'', on elementary mathematical logic. The 1928 first edition thereof is considered the first elementary text clearly grounded in the formalism now known as first-order logic (FOL). Hilbert and Ackermann also formalized FOL in a way that subsequently achieved canonical status. FOL is now a core formalism of mathematical logic, and is presupposed by contemporary treatments of Peano arithmetic and nearly all treatments of axiomatic set theory. The 1928 edition included a clear statement of the Entscheidungsproblem (decision problem) for FOL, and also asked whether that logic was complete (i.e., whether all semantic truths of FOL were theorems derivable from the FOL axioms and rules). The former problem was answered in the negative first by Alonzo Church and independently by Alan Turing in 1936. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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David Hilbert
David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician and philosopher of mathematics and one of the most influential mathematicians of his time. Hilbert discovered and developed a broad range of fundamental ideas including invariant theory, the calculus of variations, commutative algebra, algebraic number theory, the foundations of geometry, spectral theory of operators and its application to integral equations, mathematical physics, and the foundations of mathematics (particularly proof theory). He adopted and defended Georg Cantor's set theory and transfinite numbers. In 1900, he presented a collection of problems that set a course for mathematical research of the 20th century. Hilbert and his students contributed to establishing rigor and developed important tools used in modern mathematical physics. He was a cofounder of proof theory and mathematical logic. Life Early life and education Hilbert, the first of two children and only son of O ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Kurt Gödel
Kurt Friedrich Gödel ( ; ; April 28, 1906 – January 14, 1978) was a logician, mathematician, and philosopher. Considered along with Aristotle and Gottlob Frege to be one of the most significant logicians in history, Gödel profoundly influenced scientific and philosophical thinking in the 20th century (at a time when Bertrand Russell,For instance, in their "Principia Mathematica' (''Stanford Encyclopedia of Philosophy'' edition). Alfred North Whitehead, and David Hilbert were using logic and set theory to investigate the foundations of mathematics), building on earlier work by Frege, Richard Dedekind, and Georg Cantor. Gödel's discoveries in the foundations of mathematics led to the proof of his completeness theorem in 1929 as part of his dissertation to earn a doctorate at the University of Vienna, and the publication of Gödel's incompleteness theorems two years later, in 1931. The incompleteness theorems address limitations of formal axiomatic systems. In parti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Logic Books
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the study of deductively valid inferences or logical truths. It examines how conclusions follow from premises based on the structure of arguments alone, independent of their topic and content. Informal logic is associated with informal fallacies, critical thinking, and argumentation theory. Informal logic examines arguments expressed in natural language whereas formal logic uses formal language. When used as a countable noun, the term "a logic" refers to a specific logical formal system that articulates a proof system. Logic plays a central role in many fields, such as philosophy, mathematics, computer science, and linguistics. Logic studies arguments, which consist of a set of premises that leads to a conclusion. An example is the argument from the premises "it's Sunday" and "if it's Sunday then I don't have to work" leading to the conclusion "I don't have to work." Premis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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1938 Non-fiction Books
Events January * January 1 – state-owned enterprise, State-owned railway networks are created by merger, in France (SNCF) and the Netherlands (Nederlandse Spoorwegen – NS). * January 20 – King Farouk of Egypt marries Safinaz Zulficar, who becomes Farida of Egypt, Queen Farida, in Cairo. * January 27 – The Honeymoon Bridge (Niagara Falls), Honeymoon Bridge at Niagara Falls, New York, collapses as a result of an ice jam. February * February 4 ** Adolf Hitler abolishes the War Ministry and creates the Oberkommando der Wehrmacht (High Command of the Armed Forces), giving him direct control of the German military. In addition, he dismisses political and military leaders considered unsympathetic to his philosophy or policies. General Werner von Fritsch is forced to resign as Commander of Chief of the German Army following accusations of homosexuality, and replaced by General Walther von Brauchitsch. Foreign Minister Baron Konstantin von Neurath is dismi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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1928 Non-fiction Books
Nineteen or 19 may refer to: * 19 (number) * One of the years 19 BC, AD 19, 1919, 2019 Films * 19 (film), ''19'' (film), a 2001 Japanese film * Nineteen (1987 film), ''Nineteen'' (1987 film), a 1987 science fiction film * ''19-Nineteen'', a 2009 South Korean film * ''Diciannove'', a 2024 Italian drama film informally referred to as "Nineteen" in some sources Science * Potassium, an alkali metal * 19 Fortuna, an asteroid Music * 19 (band), a Japanese pop music duo Albums * 19 (Adele album), ''19'' (Adele album), 2008 * ''19'', a 2003 album by Alsou * ''19'', a 2006 album by Evan Yo * ''19'', a 2018 album by MHD (rapper), MHD * ''19'', one half of the double album ''63/19'' by Kool A.D. * ''Number Nineteen'', a 1971 album by American jazz pianist Mal Waldron * XIX (EP), ''XIX'' (EP), a 2019 EP by 1the9 Songs * 19 (song), "19" (song), a 1985 song by British musician Paul Hardcastle * "Stone in Focus", officially "#19", a composition by Aphex Twin * "Nineteen", a song fr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Principia Mathematica
The ''Principia Mathematica'' (often abbreviated ''PM'') is a three-volume work on the foundations of mathematics written by the mathematician–philosophers Alfred North Whitehead and Bertrand Russell and published in 1910, 1912, and 1913. In 1925–1927, it appeared in a second edition with an important ''Introduction to the Second Edition'', an ''Appendix A'' that replaced ✱9 with a new ''Appendix B'' and ''Appendix C''. ''PM'' was conceived as a sequel to Russell's 1903 '' The Principles of Mathematics'', but as ''PM'' states, this became an unworkable suggestion for practical and philosophical reasons: "The present work was originally intended by us to be comprised in a second volume of ''Principles of Mathematics''... But as we advanced, it became increasingly evident that the subject is a very much larger one than we had supposed; moreover on many fundamental questions which had been left obscure and doubtful in the former work, we have now arrived at what we bel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Liar Paradox
In philosophy and logic, the classical liar paradox or liar's paradox or antinomy of the liar is the statement of a liar that they are lying: for instance, declaring that "I am lying". If the liar is indeed lying, then the liar is telling the truth, which means the liar just lied. In "this sentence is a lie", the paradox is strengthened in order to make it amenable to more rigorous logical analysis. It is still generally called the "liar paradox" although abstraction is made precisely from the liar making the statement. Trying to assign to this statement, the strengthened liar, a classical binary truth value leads to a contradiction. Assume that "this sentence is false" is true, then we can trust its content, which states the opposite and thus causes a contradition. Similarly, we get a contradiction when we assume the opposite. History The Epimenides paradox (c. 600 BC) has been suggested as an example of the liar paradox, but they are not logically equivalent. The semi-mythica ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Russell's Paradox
In mathematical logic, Russell's paradox (also known as Russell's antinomy) is a set-theoretic paradox published by the British philosopher and mathematician, Bertrand Russell, in 1901. Russell's paradox shows that every set theory that contains an unrestricted comprehension principle leads to contradictions. According to the unrestricted comprehension principle, for any sufficiently well-defined property, there is the set of all and only the objects that have that property. Let ''R'' be the set of all sets that are not members of themselves. (This set is sometimes called "the Russell set".) If ''R'' is not a member of itself, then its definition entails that it is a member of itself; yet, if it is a member of itself, then it is not a member of itself, since it is the set of all sets that are not members of themselves. The resulting contradiction is Russell's paradox. In symbols: : Let R = \. Then R \in R \iff R \not \in R. Russell also showed that a version of the paradox co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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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] |
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Alan Turing
Alan Mathison Turing (; 23 June 1912 – 7 June 1954) was an English mathematician, computer scientist, logician, cryptanalyst, philosopher and theoretical biologist. He was highly influential in the development of theoretical computer science, providing a formalisation of the concepts of algorithm and computation with the Turing machine, which can be considered a model of a general-purpose computer. Turing is widely considered to be the father of theoretical computer science. Born in London, Turing was raised in southern England. He graduated from University of Cambridge, King's College, Cambridge, and in 1938, earned a doctorate degree from Princeton University. During World War II, Turing worked for the Government Code and Cypher School at Bletchley Park, Britain's codebreaking centre that produced Ultra (cryptography), Ultra intelligence. He led Hut 8, the section responsible for German naval cryptanalysis. Turing devised techniques for speeding the breaking of Germ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Wilhelm Ackermann
Wilhelm Friedrich Ackermann (; ; 29 March 1896 – 24 December 1962) was a German mathematician and logician best known for his work in mathematical logic and the Ackermann function, an important example in the theory of computation. Biography Ackermann was born in Herscheid, Germany, and was awarded a Ph.D. by the University of Göttingen in 1925 for his thesis ''Begründung des "tertium non datur" mittels der Hilbertschen Theorie der Widerspruchsfreiheit'', which was a consistency proof of arithmetic apparently without Peano induction (although it did use e.g. induction over the length of proofs). This was one of two major works in proof theory in the 1920s and the only one following Hilbert's school of thought. From 1929 until 1948, he taught at the Arnoldinum Gymnasium in Burgsteinfurt, and then at Lüdenscheid until 1961. He was also a corresponding member of the Akademie der Wissenschaften (''Academy of Sciences'') in Göttingen, and was an honorary professor at the U ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Alonzo Church
Alonzo Church (June 14, 1903 – August 11, 1995) was an American computer scientist, mathematician, logician, and philosopher who made major contributions to mathematical logic and the foundations of theoretical computer science. He is best known for the lambda calculus, the Church–Turing thesis, proving the unsolvability of the ''Entscheidungsproblem'' ("decision problem"), the Frege–Church ontology, and the Church–Rosser theorem. Alongside his doctoral student Alan Turing, Church is considered one of the founders of computer science. Life Alonzo Church was born on June 14, 1903, in Washington, D.C., where his father, Samuel Robbins Church, was a justice of the peace and the judge of the Municipal Court for the District of Columbia. He was the grandson of Alonzo Webster Church (1829–1909), United States Senate Librarian from 1881 to 1901, and great-grandson of Alonzo Church, a professor of Mathematics and Astronomy and 6th President of the University of Ge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |