128-541: The Jubjub bird is a dangerous creature mentioned in Lewis Carroll 's nonsense poems " Jabberwocky " (1871) and " The Hunting of the Snark " (1876). In "Jabberwocky," the only detail given about the bird is that the protagonist should "beware" it. In The Hunting of the Snark , however, the creature is described in much greater depth. It is found in a narrow, dark, depressing and isolated valley. Its voice when heard
256-415: A cumulative hierarchy of sets. New Foundations takes a different approach; it allows objects such as the set of all sets at the cost of restrictions on its set-existence axioms. The system of Kripke–Platek set theory is closely related to generalized recursion theory. Two famous statements in set theory are the axiom of choice and the continuum hypothesis . The axiom of choice, first stated by Zermelo,
384-596: A stammer – a condition shared by most of his siblings – that often inhibited his social life throughout his years. At the age of twelve he was sent to Richmond Grammar School (now part of Richmond School ) in Richmond, North Yorkshire . In 1846, Dodgson entered Rugby School , where he was evidently unhappy, as he wrote some years after leaving: "I cannot say ... that any earthly considerations would induce me to go through my three years again ... I can honestly say that if I could have been ... secure from annoyance at night,
512-434: A correspondence between syntax and semantics in first-order logic. Gödel used the completeness theorem to prove the compactness theorem , demonstrating the finitary nature of first-order logical consequence . These results helped establish first-order logic as the dominant logic used by mathematicians. In 1931, Gödel published On Formally Undecidable Propositions of Principia Mathematica and Related Systems , which proved
640-430: A definition of the real numbers in terms of Dedekind cuts of rational numbers, a definition still employed in contemporary texts. Georg Cantor developed the fundamental concepts of infinite set theory. His early results developed the theory of cardinality and proved that the reals and the natural numbers have different cardinalities. Over the next twenty years, Cantor developed a theory of transfinite numbers in
768-400: A depression that lasted some years. In 1876, Dodgson produced his next great work, The Hunting of the Snark , a fantastical "nonsense" poem, with illustrations by Henry Holiday , exploring the adventures of a bizarre crew of nine tradesmen and one beaver, who set off to find the snark. It received largely mixed reviews from Carroll's contemporary reviewers, but was enormously popular with
896-561: A different characterization, which lacked the formal logical character of Peano's axioms. Dedekind's work, however, proved theorems inaccessible in Peano's system, including the uniqueness of the set of natural numbers (up to isomorphism) and the recursive definitions of addition and multiplication from the successor function and mathematical induction. In the mid-19th century, flaws in Euclid's axioms for geometry became known. In addition to
1024-463: A dozen books under his real name. Dodgson also developed new ideas in linear algebra (e.g., the first printed proof of the Rouché–Capelli theorem ), probability, and the study of elections (e.g., Dodgson's method ) and committees ; some of this work was not published until well after his death. His occupation as Mathematical Lecturer at Christ Church gave him some financial security. His work in
1152-555: A family of high-church Anglicans , and pursued his clerical training at Oxford University's Christ Church constituent college , where he lived for most of his life as a scholar , teacher and Anglican deacon . Alice Liddell – a daughter of Henry Liddell , the Dean of Christ Church – is widely identified as the original inspiration for Alice in Wonderland , though Carroll always denied this. An avid puzzler, Carroll created
1280-400: A finitistic system together with a principle of transfinite induction . Gentzen's result introduced the ideas of cut elimination and proof-theoretic ordinals , which became key tools in proof theory. Gödel gave a different consistency proof, which reduces the consistency of classical arithmetic to that of intuitionistic arithmetic in higher types. The first textbook on symbolic logic for
1408-415: A formalized mathematical statement, whether the statement is true or false. Ernst Zermelo gave a proof that every set could be well-ordered , a result Georg Cantor had been unable to obtain. To achieve the proof, Zermelo introduced the axiom of choice , which drew heated debate and research among mathematicians and the pioneers of set theory. The immediate criticism of the method led Zermelo to publish
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#17327836836921536-430: A foundational system for mathematics, independent of set theory. These foundations use toposes , which resemble generalized models of set theory that may employ classical or nonclassical logic. Mathematical logic emerged in the mid-19th century as a subfield of mathematics, reflecting the confluence of two traditions: formal philosophical logic and mathematics. Mathematical logic, also called 'logistic', 'symbolic logic',
1664-524: A foundational theory for mathematics. Fraenkel proved that the axiom of choice cannot be proved from the axioms of Zermelo's set theory with urelements . Later work by Paul Cohen showed that the addition of urelements is not needed, and the axiom of choice is unprovable in ZF. Cohen's proof developed the method of forcing , which is now an important tool for establishing independence results in set theory. Leopold Löwenheim and Thoralf Skolem obtained
1792-494: A function as a rule for computation, or a smooth graph, were no longer adequate. Weierstrass began to advocate the arithmetization of analysis , which sought to axiomatize analysis using properties of the natural numbers. The modern (ε, δ)-definition of limit and continuous functions was already developed by Bolzano in 1817, but remained relatively unknown. Cauchy in 1821 defined continuity in terms of infinitesimals (see Cours d'Analyse, page 34). In 1858, Dedekind proposed
1920-573: A giant black bird resembling a cross between a vulture and a speckled chicken with a red head crest, a long yellow beak and a blue tongue. It is first seen when it captures both Tweedledum and Tweedledee while trying to escape with Alice . It is not seen again until the Red Queen releases the Jubjub Bird onto a rebellious crowd. During the final battle, after the Mad Hatter 's interference
2048-492: A glass, ensured the right amount of liqueur for the price paid; a double-sided adhesive strip to fasten envelopes or mount things in books; a device for helping a bedridden invalid to read from a book placed sideways; and at least two ciphers for cryptography . He also proposed alternative systems of parliamentary representation. He proposed the so-called Dodgson's method , using the Condorcet method . In 1884, he proposed
2176-407: A knee injury sustained in middle age. As a very young child, he suffered a fever that left him deaf in one ear. At the age of 17, he suffered a severe attack of whooping cough , which was probably responsible for his chronically weak chest in later life. In early childhood, he acquired a stammer , which he referred to as his "hesitation"; it remained throughout his life. The stammer has always been
2304-502: A member of his father's old college, Christ Church . After waiting for rooms in college to become available, he went into residence in January 1851. He had been at Oxford only two days when he received a summons home. His mother had died of "inflammation of the brain" – perhaps meningitis or a stroke – at the age of 47. His early academic career veered between high promise and irresistible distraction. He did not always work hard, but
2432-561: A model, or in other words that an inconsistent set of formulas must have a finite inconsistent subset. The completeness and compactness theorems allow for sophisticated analysis of logical consequence in first-order logic and the development of model theory , and they are a key reason for the prominence of first-order logic in mathematics. Gödel's incompleteness theorems establish additional limits on first-order axiomatizations. The first incompleteness theorem states that for any consistent, effectively given (defined below) logical system that
2560-403: A new concept – the computable function – had been discovered, and that this definition was robust enough to admit numerous independent characterizations. In his work on the incompleteness theorems in 1931, Gödel lacked a rigorous concept of an effective formal system; he immediately realized that the new definitions of computability could be used for this purpose, allowing him to state
2688-489: A particular sentence is true in every model that satisfies a particular set of axioms, then there must be a finite deduction of the sentence from the axioms. The compactness theorem first appeared as a lemma in Gödel's proof of the completeness theorem, and it took many years before logicians grasped its significance and began to apply it routinely. It says that a set of sentences has a model if and only if every finite subset has
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#17327836836922816-417: A pocket or purse, as the most common individual stamps could easily be carried on their own. The pack included a copy of a pamphlet version of this lecture. Another invention was a writing tablet called the nyctograph that allowed note-taking in the dark, thus eliminating the need to get out of bed and strike a light when one woke with an idea. The device consisted of a gridded card with sixteen squares and
2944-422: A portion of set theory directly in their semantics. The most well studied infinitary logic is L ω 1 , ω {\displaystyle L_{\omega _{1},\omega }} . In this logic, quantifiers may only be nested to finite depths, as in first-order logic, but formulas may have finite or countably infinite conjunctions and disjunctions within them. Thus, for example, it
3072-456: A practice new to the nineteenth century. He exerted his agency of this craft by literally rewriting the text created by the image to produce a new dialogue about childhood. However, popular taste changed with the advent of Modernism , affecting the types of photographs that he produced. To promote letter writing, Dodgson invented "The Wonderland Postage-Stamp Case" in 1889. This was a cloth-backed folder with twelve slots, two marked for inserting
3200-442: A proportional representation system based on multi-member districts, each voter casting only a single vote, quotas as minimum requirements to take seats, and votes transferable by candidates through what is now called Liquid democracy . Within the academic discipline of mathematics, Dodgson worked primarily in the fields of geometry , linear and matrix algebra , mathematical logic , and recreational mathematics , producing nearly
3328-503: A rule for finding the day of the week for any date; a means for justifying right margins on a typewriter; a steering device for a velociman (a type of tricycle); fairer elimination rules for tennis tournaments; a new sort of postal money order; rules for reckoning postage; rules for a win in betting; rules for dividing a number by various divisors; a cardboard scale for the Senior Common Room at Christ Church which, held next to
3456-491: A second exposition of his result, directly addressing criticisms of his proof. This paper led to the general acceptance of the axiom of choice in the mathematics community. Skepticism about the axiom of choice was reinforced by recently discovered paradoxes in naive set theory . Cesare Burali-Forti was the first to state a paradox: the Burali-Forti paradox shows that the collection of all ordinal numbers cannot form
3584-423: A separate domain for each higher-type quantifier to range over, the quantifiers instead range over all objects of the appropriate type. The logics studied before the development of first-order logic, for example Frege's logic, had similar set-theoretic aspects. Although higher-order logics are more expressive, allowing complete axiomatizations of structures such as the natural numbers, they do not satisfy analogues of
3712-404: A series of publications. In 1891, he published a new proof of the uncountability of the real numbers that introduced the diagonal argument , and used this method to prove Cantor's theorem that no set can have the same cardinality as its powerset . Cantor believed that every set could be well-ordered , but was unable to produce a proof for this result, leaving it as an open problem in 1895. In
3840-457: A set. Very soon thereafter, Bertrand Russell discovered Russell's paradox in 1901, and Jules Richard discovered Richard's paradox . Zermelo provided the first set of axioms for set theory. These axioms, together with the additional axiom of replacement proposed by Abraham Fraenkel , are now called Zermelo–Fraenkel set theory (ZF). Zermelo's axioms incorporated the principle of limitation of size to avoid Russell's paradox. In 1910,
3968-431: A significant part of the image of Dodgson. While one apocryphal story says that he stammered only in adult company and was free and fluent with children, there is no evidence to support this idea. Many children of his acquaintance remembered the stammer, while many adults failed to notice it. Dodgson himself seems to have been far more acutely aware of it than most people whom he met; it is said that he caricatured himself as
Jubjub bird - Misplaced Pages Continue
4096-590: A song named "Beware the Jubjub Bird and Shun the Frumious Bandersnatch" on their 2006 album Wonderland , which features other songs inspired by the works of Lewis Carroll . This article about a character in children's literature is a stub . You can help Misplaced Pages by expanding it . Lewis Carroll Charles Lutwidge Dodgson ( / ˈ l ʌ t w ɪ dʒ ˈ d ɒ d s ən / LUT -wij DOD -sən ; 27 January 1832 – 14 January 1898), better known by his pen name Lewis Carroll ,
4224-464: A special letter register which he devised. He documented his advice about how to write more satisfying letters in a missive entitled " Eight or Nine Wise Words about Letter-Writing ", published in 1890. Dodgson's existence remained little changed over the last twenty years of his life, despite his growing wealth and fame. He continued to teach at Christ Church until 1881 and remained in residence there until his death. Public appearances included attending
4352-571: A stronger limitation than the one established by the Löwenheim–Skolem theorem. The second incompleteness theorem states that no sufficiently strong, consistent, effective axiom system for arithmetic can prove its own consistency, which has been interpreted to show that Hilbert's program cannot be reached. Many logics besides first-order logic are studied. These include infinitary logics , which allow for formulas to provide an infinite amount of information, and higher-order logics , which include
4480-486: A system of symbols representing an alphabet of Dodgson's design, using letter shapes similar to the Graffiti writing system on a Palm device. He also devised a number of games, including an early version of what today is known as Scrabble . Devised sometime in 1878, he invented the "doublet" (see word ladder ), a form of brain-teaser that is still popular today, changing one word into another by altering one letter at
4608-427: A time when people commonly devised their own amusements and when singing and recitation were required social skills, and the young Dodgson was well equipped to be an engaging entertainer. He could reportedly sing at a passable level and was not afraid to do so before an audience. He was also adept at mimicry and storytelling, and reputedly quite good at charades . In the interim between his early published writings and
4736-615: A time, each successive change always resulting in a genuine word. For instance, CAT is transformed into DOG by the following steps: CAT, COT, DOT, DOG. It first appeared in the 29 March 1879 issue of Vanity Fair , with Carroll writing a weekly column for the magazine for two years; the final column dated 9 April 1881. The games and puzzles of Lewis Carroll were the subject of Martin Gardner's March 1960 Mathematical Games column in Scientific American . Other items include
4864-424: A useful entrée into higher social circles. During the most productive part of his career, he made portraits of notable sitters such as John Everett Millais , Ellen Terry , Maggie Spearman , Dante Gabriel Rossetti , Julia Margaret Cameron , Michael Faraday , Lord Salisbury , and Alfred Tennyson . By the time that Dodgson abruptly ceased photography (1880, after 24 years), he had established his own studio on
4992-443: A work generally considered as marking a turning point in the history of logic. Frege's work remained obscure, however, until Bertrand Russell began to promote it near the turn of the century. The two-dimensional notation Frege developed was never widely adopted and is unused in contemporary texts. From 1890 to 1905, Ernst Schröder published Vorlesungen über die Algebra der Logik in three volumes. This work summarized and extended
5120-581: Is computable ; this is not true in classical theories of arithmetic such as Peano arithmetic . Algebraic logic uses the methods of abstract algebra to study the semantics of formal logics. A fundamental example is the use of Boolean algebras to represent truth values in classical propositional logic, and the use of Heyting algebras to represent truth values in intuitionistic propositional logic. Stronger logics, such as first-order logic and higher-order logic, are studied using more complicated algebraic structures such as cylindric algebras . Set theory
5248-517: Is a particular formal system of logic . Its syntax involves only finite expressions as well-formed formulas , while its semantics are characterized by the limitation of all quantifiers to a fixed domain of discourse . Early results from formal logic established limitations of first-order logic. The Löwenheim–Skolem theorem (1919) showed that if a set of sentences in a countable first-order language has an infinite model then it has at least one model of each infinite cardinality. This shows that it
Jubjub bird - Misplaced Pages Continue
5376-461: Is also included as part of mathematical logic. Each area has a distinct focus, although many techniques and results are shared among multiple areas. The borderlines amongst these fields, and the lines separating mathematical logic and other fields of mathematics, are not always sharp. Gödel's incompleteness theorem marks not only a milestone in recursion theory and proof theory, but has also led to Löb's theorem in modal logic. The method of forcing
5504-465: Is also missing the personal catalogue number that Dodgson meticulously catalogued his photos under. "[Dodgson's] usual practice was to add a number on the back of any prints which he had developed". Wakeling also points out that Dodgson never made "full frontal studies...particularly a girl as mature as this.. There's no way the Liddells would have allowed a picture of this kind to have been taken." It
5632-453: Is capable of interpreting arithmetic, there exists a statement that is true (in the sense that it holds for the natural numbers) but not provable within that logical system (and which indeed may fail in some non-standard models of arithmetic which may be consistent with the logical system). For example, in every logical system capable of expressing the Peano axioms , the Gödel sentence holds for
5760-652: Is currently unknown whether this photo is by Dodgson, nor who wrote the pencil inscription on the back of it and for what reason. The photo was not included in Wakeling's catalogue raisonné of Dodgson's complete surviving photographs and has remained unused by other subsequent documentaries on Dodgson. Mathematical logic Mathematical logic is the study of 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
5888-551: Is described as "a scream, shrill and high" like a pencil squeaking on a slate , and significantly scares those who hear it, including the Beaver, who "turn[s] pale to the tip of its tail." Its character traits include that it is "desperate" and "lives in perpetual passion;" it "knows any friend it has met once before" and will not "look at a bribe." The Jubjub bird appears in Tim Burton 's 2010 version of Alice in Wonderland as
6016-470: Is employed in set theory, model theory, and recursion theory, as well as in the study of intuitionistic mathematics. The mathematical field of category theory uses many formal axiomatic methods, and includes the study of categorical logic , but category theory is not ordinarily considered a subfield of mathematical logic. Because of its applicability in diverse fields of mathematics, mathematicians including Saunders Mac Lane have proposed category theory as
6144-468: Is impossible for a set of first-order axioms to characterize the natural numbers, the real numbers, or any other infinite structure up to isomorphism . As the goal of early foundational studies was to produce axiomatic theories for all parts of mathematics, this limitation was particularly stark. Gödel's completeness theorem established the equivalence between semantic and syntactic definitions of logical consequence in first-order logic. It shows that if
6272-446: Is possible to say that an object is a whole number using a formula of L ω 1 , ω {\displaystyle L_{\omega _{1},\omega }} such as Higher-order logics allow for quantification not only of elements of the domain of discourse , but subsets of the domain of discourse, sets of such subsets, and other objects of higher type. The semantics are defined so that, rather than having
6400-642: Is the study of sets , which are abstract collections of objects. Many of the basic notions, such as ordinal and cardinal numbers, were developed informally by Cantor before formal axiomatizations of set theory were developed. The first such axiomatization , due to Zermelo, was extended slightly to become Zermelo–Fraenkel set theory (ZF), which is now the most widely used foundational theory for mathematics. Other formalizations of set theory have been proposed, including von Neumann–Bernays–Gödel set theory (NBG), Morse–Kelley set theory (MK), and New Foundations (NF). Of these, ZF, NBG, and MK are similar in describing
6528-863: The Whitby Gazette and the Oxford Critic . Most of this output was humorous, sometimes satirical, but his standards and ambitions were exacting. "I do not think I have yet written anything worthy of real publication (in which I do not include the Whitby Gazette or the Oxonian Advertiser ), but I do not despair of doing so someday," he wrote in July 1855. Sometime after 1850, he did write puppet plays for his siblings' entertainment, of which one has survived: La Guida di Bragia . In March 1856, he published his first piece of work under
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#17327836836926656-597: The Dodo in Alice's Adventures in Wonderland , referring to his difficulty in pronouncing his last name, but this is one of the many supposed facts often repeated for which no first-hand evidence remains. He did indeed refer to himself as a dodo, but whether or not this reference was to his stammer is simply speculation. Dodgson's stammer did trouble him, but it was never so debilitating that it prevented him from applying his other personal qualities to do well in society. He lived in
6784-500: The Löwenheim–Skolem theorem , which says that first-order logic cannot control the cardinalities of infinite structures. Skolem realized that this theorem would apply to first-order formalizations of set theory, and that it implies any such formalization has a countable model . This counterintuitive fact became known as Skolem's paradox . In his doctoral thesis, Kurt Gödel proved the completeness theorem , which establishes
6912-557: The Tractarian movement , and did his best to instil such views in his children. However, Charles developed an ambivalent relationship with his father's values and with the Church of England as a whole. During his early youth, Dodgson was educated at home. His "reading lists" preserved in the family archives testify to a precocious intellect: at the age of seven, he was reading books such as The Pilgrim's Progress . He also spoke with
7040-616: The West End musical Alice in Wonderland (the first major live production of his Alice books) at the Prince of Wales Theatre on 30 December 1886. The two volumes of his last novel, Sylvie and Bruno , were published in 1889 and 1893, but the intricacy of this work was apparently not appreciated by contemporary readers; it achieved nothing like the success of the Alice books, with disappointing reviews and sales of only 13,000 copies. The only known occasion on which he travelled abroad
7168-409: The natural numbers . Giuseppe Peano published a set of axioms for arithmetic that came to bear his name ( Peano axioms ), using a variation of the logical system of Boole and Schröder but adding quantifiers. Peano was unaware of Frege's work at the time. Around the same time Richard Dedekind showed that the natural numbers are uniquely characterized by their induction properties. Dedekind proposed
7296-657: The word ladder puzzle (which he then called "Doublets"), which he published in his weekly column for Vanity Fair magazine between 1879 and 1881. In 1982 a memorial stone to Carroll was unveiled at Poets' Corner in Westminster Abbey . There are societies in many parts of the world dedicated to the enjoyment and promotion of his works. Dodgson's family background was predominantly northern English , conservative , and high-church Anglican . Most of his male ancestors were army officers or Anglican clergymen. His great-grandfather, Charles Dodgson , had risen through
7424-466: The ' algebra of logic ', and, more recently, simply 'formal logic', is the set of logical theories elaborated in the course of the nineteenth century with the aid of an artificial notation and a rigorously deductive method. Before this emergence, logic was studied with rhetoric , with calculationes , through the syllogism , and with philosophy . The first half of the 20th century saw an explosion of fundamental results, accompanied by vigorous debate over
7552-507: The Cantini photo's authenticity, the BBC's failure to tell participants of the found photo, and several factual errors. Wakeling draws attention to the irregular "trimmed" nature of the photo itself, and no trace of Dodgson's writing. The inscription on the back of the photo, attributed "lewis Carroll" in pencil, "is an unknown hand... so it could have been written by anybody" . The photo negative
7680-526: The Christ Church library, where his office was close to the Deanery, where Alice Liddell lived. The young adult Charles Dodgson was about 6 feet (1.83 m) tall and slender, and he had curly brown hair and blue or grey eyes (depending on the account). He was described in later life as somewhat asymmetrical , and as carrying himself rather stiffly and awkwardly, although this might be on account of
7808-616: The Jubjub bird joins to fight for the Red Queen, only to later have its head crushed by a giant boulder from a catapult. The Bluetones recorded a song titled "The Jub-Jub Bird", which was released in 1998 on their second album Return to the Last Chance Saloon . The song "Alpenglow" by Nightwish on the 2015 album Endless Forms Most Beautiful includes the line "together we slay another fright, every Jubjub bird, spooks of
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#17327836836927936-482: The Lewis Carroll pen name, which Dodgson had first used some nine years earlier. The illustrations this time were by Sir John Tenniel ; Dodgson evidently thought that a published book would need the skills of a professional artist. Annotated versions provide insights into many of the ideas and hidden meanings that are prevalent in these books. Critical literature has often proposed Freudian interpretations of
8064-580: The alleged photo until editing of the documentary was underway. Edward Wakeling's paper/review "Eight or nine wise words on documentary making" [1] appeared in March 2015 as part of the Lewis Carroll society newsletter Bandersnatch . Wakeling also echoed Woolf's assertions that he was not given time to talk about the alleged photo. Wakeling claimed, "The documentary knew I could authenticate [the photo] or not, but they chose to keep it from me as they anticipated my response." Wakeling further criticises in his paper
8192-455: The art and became a well-known gentleman-photographer, and he seems even to have toyed with the idea of making a living out of it in his very early years. A study by Roger Taylor and Edward Wakeling exhaustively lists every surviving print, and Taylor calculates that just over half of Dodgson's surviving work depicts young girls. Thirty surviving photographs depict nude or semi-nude children. About 60% of Dodgson's original photographic portfolio
8320-583: The author of his mathematical works". He also began earning quite substantial sums of money but continued with his seemingly disliked post at Christ Church. Late in 1871, he published the sequel Through the Looking-Glass, and What Alice Found There . (The title page of the first edition erroneously gives "1872" as the date of publication. ) Its somewhat darker mood possibly reflects changes in Dodgson's life. His father's death in 1868 plunged him into
8448-551: The book as "a descent into the dark world of the subconscious ", as well as seeing it as a satire upon contemporary mathematical advances. The overwhelming commercial success of the first Alice book changed Dodgson's life in many ways. The fame of his alter ego "Lewis Carroll" soon spread around the world. He was inundated with fan mail and with sometimes unwanted attention. Indeed, according to one popular story, Queen Victoria herself enjoyed Alice in Wonderland so much that she commanded that he dedicate his next book to her, and
8576-500: The characters in the narrative are based on her. Information is scarce (Dodgson's diaries for the years 1858–1862 are missing), but it seems clear that his friendship with the Liddell family was an important part of his life in the late 1850s, and he grew into the habit of taking the children on rowing trips (first the boy, Harry, and later the three girls) accompanied by an adult friend to nearby Nuneham Courtenay or Godstow . It
8704-469: The collection is nonempty, the lack of a general, concrete rule by which the choice can be made renders the axiom nonconstructive. Stefan Banach and Alfred Tarski showed that the axiom of choice can be used to decompose a solid ball into a finite number of pieces which can then be rearranged, with no scaling, to make two solid balls of the original size. This theorem, known as the Banach–Tarski paradox ,
8832-608: The completeness and compactness theorems from first-order logic, and are thus less amenable to proof-theoretic analysis. Another type of logics are fixed-point logic s that allow inductive definitions , like one writes for primitive recursive functions . One can formally define an extension of first-order logic — a notion which encompasses all logics in this section because they behave like first-order logic in certain fundamental ways, but does not encompass all logics in general, e.g. it does not encompass intuitionistic, modal or fuzzy logic . Lindström's theorem implies that
8960-412: The consistency of elementary arithmetic, respectively; the tenth was to produce a method that could decide whether a multivariate polynomial equation over the integers has a solution. Subsequent work to resolve these problems shaped the direction of mathematical logic, as did the effort to resolve Hilbert's Entscheidungsproblem , posed in 1928. This problem asked for a procedure that would decide, given
9088-679: The context of proof theory. At its core, mathematical logic deals with mathematical concepts expressed using formal logical systems . These systems, though they differ in many details, share the common property of considering only expressions in a fixed formal language . The systems of propositional logic and first-order logic are the most widely studied today, because of their applicability to foundations of mathematics and because of their desirable proof-theoretic properties. Stronger classical logics such as second-order logic or infinitary logic are also studied, along with Non-classical logics such as intuitionistic logic . First-order logic
9216-681: The county of Surrey, just four days before the death of Henry Liddell. He was two weeks away from turning 66 years old. His funeral was held at the nearby St Mary's Church . His body was buried at the Mount Cemetery in Guildford. He is commemorated at All Saints' Church, Daresbury , in its stained glass windows depicting characters from Alice's Adventures in Wonderland , erected in 1935. A BBC documentary from 2015, The Secret World of Lewis Carroll , critically examined Dodgson's relationship with Alice Liddell and her sisters. It explored
9344-581: The development of axiomatic frameworks for geometry , arithmetic , and analysis . In the early 20th century it was shaped by David Hilbert 's 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 proving consistency. Work in set theory showed that almost all ordinary mathematics can be formalized in terms of sets, although there are some theorems that cannot be proven in common axiom systems for set theory. Contemporary work in
9472-489: The earliest modern use of a truth tree . Robbins' and Rumsey's investigation of Dodgson condensation , a method of evaluating determinants , led them to the alternating sign matrix conjecture, now a theorem. The discovery in the 1990s of additional ciphers that Dodgson had constructed, in addition to his "Memoria Technica", showed that he had employed sophisticated mathematical ideas in their creation. Dodgson wrote and received as many as 98,721 letters, according to
9600-407: The early decades of the 20th century, the main areas of study were set theory and formal logic. The discovery of paradoxes in informal set theory caused some to wonder whether mathematics itself is inconsistent, and to look for proofs of consistency. In 1900, Hilbert posed a famous list of 23 problems for the next century. The first two of these were to resolve the continuum hypothesis and prove
9728-582: The family of friend and mentor George MacDonald read Dodgson's incomplete manuscript, and the enthusiasm of the MacDonald children encouraged Dodgson to seek publication. In 1863, he had taken the unfinished manuscript to Macmillan the publisher , who liked it immediately. After the possible alternative titles were rejected – Alice Among the Fairies and Alice's Golden Hour – the work was finally published as Alice's Adventures in Wonderland in 1865 under
9856-466: The field of mathematical logic attracted renewed interest in the late 20th century. Martin Gardner's book on logic machines and diagrams and William Warren Bartley's posthumous publication of the second part of Dodgson's symbolic logic book have sparked a reevaluation of Dodgson's contributions to symbolic logic. It is recognised that in his Symbolic Logic Part II , Dodgson introduced the Method of Trees,
9984-450: The first volume of Principia Mathematica by Russell and Alfred North Whitehead was published. This seminal work developed the theory of functions and cardinality in a completely formal framework of type theory , which Russell and Whitehead developed in an effort to avoid the paradoxes. Principia Mathematica is considered one of the most influential works of the 20th century, although the framework of type theory did not prove popular as
10112-457: The following years, and would greatly influence his writing career. Dodgson became close friends with Liddell's wife, Lorina, and their children, particularly the three sisters Lorina, Edith, and Alice Liddell. He was widely assumed for many years to have derived his own "Alice" from Alice Liddell ; the acrostic poem at the end of Through the Looking-Glass spells out her name in full, and there are also many superficial references to her hidden in
10240-441: The foundations of mathematics often focuses on establishing which parts of mathematics can be formalized in particular formal systems (as in reverse mathematics ) rather than trying to find theories in which all of mathematics can be developed. The Handbook of Mathematical Logic in 1977 makes a rough division of contemporary mathematical logic into four areas: Additionally, sometimes the field of computational complexity theory
10368-628: The foundations of mathematics. Theories of logic were developed in many cultures in history, including China , India , Greece and the Islamic world . Greek methods, particularly Aristotelian logic (or term logic) as found in the Organon , found wide application and acceptance in Western science and mathematics for millennia. The Stoics , especially Chrysippus , began the development of predicate logic . In 18th-century Europe, attempts to treat
10496-418: The hardships of the daily life would have been comparative trifles to bear." He did not claim he suffered from bullying, but cited little boys as the main targets of older bullies at Rugby. Stuart Dodgson Collingwood, Dodgson's nephew, wrote that "even though it is hard for those who have only known him as the gentle and retiring don to believe it, it is nevertheless true that long after he left school, his name
10624-423: The importance of the incompleteness theorem for some time. Gödel's theorem shows that a consistency proof of any sufficiently strong, effective axiom system cannot be obtained in the system itself, if the system is consistent, nor in any weaker system. This leaves open the possibility of consistency proofs that cannot be formalized within the system they consider. Gentzen proved the consistency of arithmetic using
10752-443: The incompleteness (in a different meaning of the word) of all sufficiently strong, effective first-order theories. This result, known as Gödel's incompleteness theorem , establishes severe limitations on axiomatic foundations for mathematics, striking a strong blow to Hilbert's program. It showed the impossibility of providing a consistency proof of arithmetic within any formal theory of arithmetic. Hilbert, however, did not acknowledge
10880-491: The incompleteness theorems in generality that could only be implied in the original paper. Numerous results in recursion theory were obtained in the 1940s by Stephen Cole Kleene and Emil Leon Post . Kleene introduced the concepts of relative computability, foreshadowed by Turing, and the arithmetical hierarchy . Kleene later generalized recursion theory to higher-order functionals. Kleene and Georg Kreisel studied formal versions of intuitionistic mathematics, particularly in
11008-601: The independence of the parallel postulate , established by Nikolai Lobachevsky in 1826, mathematicians discovered that certain theorems taken for granted by Euclid were not in fact provable from his axioms. Among these is the theorem that a line contains at least two points, or that circles of the same radius whose centers are separated by that radius must intersect. Hilbert developed a complete set of axioms for geometry , building on previous work by Pasch. The success in axiomatizing geometry motivated Hilbert to seek complete axiomatizations of other areas of mathematics, such as
11136-516: The layman was written by Lewis Carroll , author of Alice's Adventures in Wonderland , in 1896. Alfred Tarski developed the basics of model theory . Beginning in 1935, a group of prominent mathematicians collaborated under the pseudonym Nicolas Bourbaki to publish Éléments de mathématique , a series of encyclopedic mathematics texts. These texts, written in an austere and axiomatic style, emphasized rigorous presentation and set-theoretic foundations. Terminology coined by these texts, such as
11264-411: 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
11392-597: The mathematics textbook that the young Dodgson used – still survives and it contained an inscription in Latin , which translates to: "This book belongs to Charles Lutwidge Dodgson: hands off!" Some pages also included annotations such as the one found on p. 129, where he wrote "Not a fair question in decimals" next to a question. He left Rugby at the end of 1849 and matriculated at the University of Oxford in May 1850 as
11520-450: The media's reactions to the UK's early 2010 Yewtree investigations. When problems about the documentary's conduct and research surfaced, The Times and The Telegraph reported it. The material in the documentary has come under intense scrutiny by Carroll scholars, including those such as Jenny Woolf and Edward Wakeling, who appeared in it. Woolf claimed that she was not told of the use of
11648-527: The most commonly used penny stamp, and one each for the other current denominations up to one shilling. The folder was then put into a slipcase decorated with a picture of Alice on the front and the Cheshire Cat on the back. It intended to organise stamps wherever one stored their writing implements; Carroll expressly notes in Eight or Nine Wise Words about Letter-Writing it is not intended to be carried in
11776-579: The name that would make him famous. A romantic poem called "Solitude" appeared in The Train under the authorship of "Lewis Carroll". This pseudonym was a play on his real name: Lewis was the anglicised form of Ludovicus , which was the Latin for Lutwidge , and Carroll an Irish surname similar to the Latin name Carolus , from which comes the name Charles . The transition went as follows: "Charles Lutwidge" translated into Latin as "Carolus Ludovicus". This
11904-445: The natural numbers and the real line . This would prove to be a major area of research in the first half of the 20th century. The 19th century saw great advances in the theory of real analysis , including theories of convergence of functions and Fourier series . Mathematicians such as Karl Weierstrass began to construct functions that stretched intuition, such as nowhere-differentiable continuous functions . Previous conceptions of
12032-482: The natural numbers but cannot be proved. Here a logical system is said to be effectively given if it is possible to decide, given any formula in the language of the system, whether the formula is an axiom, and one which can express the Peano axioms is called "sufficiently strong." When applied to first-order logic, the first incompleteness theorem implies that any sufficiently strong, consistent, effective first-order theory has models that are not elementarily equivalent ,
12160-503: The next year he failed an important scholarship exam through his self-confessed inability to apply himself to study. Even so, his talent as a mathematician won him the Christ Church Mathematical Lectureship in 1855, which he continued to hold for the next 26 years. Despite early unhappiness, Dodgson remained at Christ Church, in various capacities, until his death, including that of Sub-Librarian of
12288-486: The only extension of first-order logic satisfying both the compactness theorem and the downward Löwenheim–Skolem theorem is first-order logic. Modal logics include additional modal operators, such as an operator which states that a particular formula is not only true, but necessarily true. Although modal logic is not often used to axiomatize mathematics, it has been used to study the properties of first-order provability and set-theoretic forcing. Intuitionistic logic
12416-418: The operations of formal logic in a symbolic or algebraic way had been made by philosophical mathematicians including Leibniz and Lambert , but their labors remained isolated and little known. In the middle of the nineteenth century, George Boole and then Augustus De Morgan presented systematic mathematical treatments of logic. Their work, building on work by algebraists such as George Peacock , extended
12544-485: The other in the fairytale kingdoms of Elfland, Outland, and others. The fairytale world satirises English society and, more specifically, the world of academia. Sylvie and Bruno came out in two volumes and is considered a lesser work, although it has remained in print for over a century. In 1856, Dodgson took up the new art form of photography under the influence first of his uncle Skeffington Lutwidge , and later of his Oxford friend Reginald Southey . He soon excelled at
12672-465: The past". Saydisc Records, in 1978, on SDL294, released 'Parlour Poetry'. All the verse was spoken by Kenneth Williams . Jabberwocky was one of four Lewis Carroll pieces included. 'The Walrus and the Carpenter'; 'You are Old, Father William'; and 'Hiawatha's Photographing', were the others. The minor planet 9781 Jubjubbird is named for the Jubjub bird. Indie rock band Forgive Durden released
12800-410: The possibility that Dodgson's rift with the Liddell family (and his temporary suspension from the college) might have been caused by improper relations with their children, including Alice. The research for the documentary found a "disturbing" full frontal nude of Alice's adolescent sister Lorina during filming, and speculated on the "likelihood" of Dodgson taking the photo. However, it was later revealed
12928-495: The public, having been reprinted seventeen times between 1876 and 1908, and has seen various adaptations into musicals, opera, theatre, plays and music. Painter Dante Gabriel Rossetti reputedly became convinced that the poem was about him. In 1895, 30 years after the publication of his masterpieces, Carroll attempted a comeback, producing a two-volume tale of the fairy siblings Sylvie and Bruno . Carroll entwines two plots set in two alternative worlds, one set in rural England and
13056-547: The ranks of the church to become the Bishop of Elphin in rural Ireland. His paternal grandfather, also named Charles, was an army captain fatality of the Irish rebellion of 1803 , when his two sons were hardly more than babies. The older of these sons, yet another Charles Dodgson , was Carroll's father. He went to Rugby School and then to Christ Church, Oxford . He reverted to the other family tradition and took holy orders . He
13184-497: The roof of Tom Quad , created around 3,000 images, and become an amateur master of the medium, though fewer than 1,000 images have survived time and deliberate destruction. He stopped taking photographs because keeping his studio working was too time-consuming. He used the wet collodion process ; commercial photographers who started using the dry-plate process in the 1870s took pictures more quickly. He often altered his photographs through blurring techniques or by painting over them,
13312-531: The shadows. Most assuredly I accept to the full the doctrines you refer to—that Christ died to save us, that we have no other way of salvation open to us but through His death, and that it is by faith in Him, and through no merit of ours, that we are reconciled to God; and most assuredly I can cordially say, "I owe all to Him who loved me, and died on the Cross of Calvary." Dodgson also expressed interest in other fields. He
13440-472: The spacious rectory. This remained their home for the next 25 years. Charles' father was an active and highly conservative cleric of the Church of England who later became the Archdeacon of Richmond and involved himself, sometimes influentially, in the intense religious disputes that were dividing the church. He was high-church, inclining toward Anglo-Catholicism , an admirer of John Henry Newman and
13568-622: The success of the Alice books, Dodgson began to move in the pre-Raphaelite social circle. He first met John Ruskin in 1857 and became friendly with him. Around 1863, he developed a close relationship with Dante Gabriel Rossetti and his family. He would often take pictures of the family in the garden of the Rossetti's house in Chelsea, London . He also knew William Holman Hunt , John Everett Millais , and Arthur Hughes , among other artists. He knew fairy-tale author George MacDonald well – it
13696-436: The text of both books. It has been noted that Dodgson himself repeatedly denied in later life that his "little heroine" was based on any real child, and he frequently dedicated his works to girls of his acquaintance, adding their names in acrostic poems at the beginning of the text. Gertrude Chataway 's name appears in this form at the beginning of The Hunting of the Snark , and it is not suggested that this means that any of
13824-616: The timeline for this research had more than met the eye. The photo currently exists in the archives of the Musée Cantini in Marseille , and was attributed to Dodgson by a currently unknown hand. It was subsequently revealed in early 2015 by the Carroll scholar Edward Wakeling that the photo first appeared in the 1970s, when it was owned by Parisian photo collectors. The provenance of the photo's link to Dodgson could be questioned. It
13952-530: The traditional Aristotelian doctrine of logic into a sufficient framework for the study of foundations of mathematics . In 1847, Vatroslav Bertić made substantial work on algebraization of logic, independently from Boole. Charles Sanders Peirce later built upon the work of Boole to develop a logical system for relations and quantifiers, which he published in several papers from 1870 to 1885. Gottlob Frege presented an independent development of logic with quantifiers in his Begriffsschrift , published in 1879,
14080-463: The words bijection , injection , and surjection , and the set-theoretic foundations the texts employed, were widely adopted throughout mathematics. The study of computability came to be known as recursion theory or computability theory , because early formalizations by Gödel and Kleene relied on recursive definitions of functions. When these definitions were shown equivalent to Turing's formalization involving Turing machines , it became clear that
14208-399: The work of Boole, De Morgan, and Peirce, and was a comprehensive reference to symbolic logic as it was understood at the end of the 19th century. Concerns that mathematics had not been built on a proper foundation led to the development of axiomatic systems for fundamental areas of mathematics such as arithmetic, analysis, and geometry. In logic, the term arithmetic refers to the theory of
14336-445: Was a member of the Church of England , but "doubt[ed] if he was fully a 'High Churchman ' ". He added: I believe that when you and I come to lie down for the last time, if only we can keep firm hold of the great truths Christ taught us—our own utter worthlessness and His infinite worth; and that He has brought us back to our one Father, and made us His brethren, and so brethren to one another—we shall have all we need to guide us through
14464-785: Was a trip to Russia in 1867 as an ecclesiastic, together with the Reverend Henry Liddon . He recounts the travel in his "Russian Journal", which was first commercially published in 1935. On his way to Russia and back, he also saw different cities in Belgium, Germany, partitioned Poland and Lithuania, and France. In his early sixties, Dodgson increasingly suffered from synovitis which eventually prevented him walking and sometimes left him bed-ridden for months. Dodgson died of pneumonia following influenza on 14 January 1898 at his sisters' home, "The Chestnuts", in Guildford in
14592-465: Was accordingly presented with his next work, a scholarly mathematical volume entitled An Elementary Treatise on Determinants . Dodgson himself vehemently denied this story, commenting "... It is utterly false in every particular: nothing even resembling it has occurred"; and it is unlikely for other reasons. As T. B. Strong comments in a Times article, "It would have been clean contrary to all his practice to identify [the] author of Alice with
14720-524: Was an English author , poet , mathematician , photographer and Anglican deacon . His most notable works are Alice's Adventures in Wonderland (1865) and its sequel Through the Looking-Glass (1871). He was noted for his facility with word play , logic, and fantasy. His poems Jabberwocky (1871) and The Hunting of the Snark (1876) are classified in the genre of literary nonsense . Some of Alice's nonsensical wonderland logic reflects his published work on mathematical logic . Carroll came from
14848-542: Was an early member of the Society for Psychical Research , and one of his letters suggests that he accepted as real what was then called "thought reading". Dodgson wrote some studies of various philosophical arguments. In 1895, he developed a philosophical regressus-argument on deductive reasoning in his article " What the Tortoise Said to Achilles ", which appeared in one of the early volumes of Mind . The article
14976-403: Was deliberately destroyed. Dodgson also made many studies of men, women, boys, and landscapes; his subjects also include skeletons, dolls, dogs, statues, paintings, and trees. His pictures of children were taken with a parent in attendance and many of the pictures were taken in the Liddell garden because natural sunlight was required for good exposures. Dodgson also found photography to be
15104-486: Was developed by Heyting to study Brouwer's program of intuitionism, in which Brouwer himself avoided formalization. Intuitionistic logic specifically does not include the law of the excluded middle , which states that each sentence is either true or its negation is true. Kleene's work with the proof theory of intuitionistic logic showed that constructive information can be recovered from intuitionistic proofs. For example, any provably total function in intuitionistic arithmetic
15232-596: Was exceptionally gifted, and achievement came easily to him. In 1852, he obtained first-class honours in Mathematics Moderations and was soon afterwards nominated to a Studentship by his father's old friend Canon Edward Pusey . In 1854, he obtained first-class honours in the Final Honours School of Mathematics, standing first on the list, and thus graduated as Bachelor of Arts. He remained at Christ Church studying and teaching, but
15360-614: Was left to the Musée de Cantini. There was no link to Dodgson, and no link to the Liddell family. This was not explained in the documentary. The documentary raised suspicions about Dodgson being a "repressed paedophile", as one of the interviewees, Will Self , put it. This aspect was leaked to The Telegraph a week in advance. When reviewing the documentary, papers sought to link the 19th-century Carroll with 21st-century sexual conduct revelations about recent paedophiles. This attempted link could be considered an act of scapegoating inspired by
15488-475: Was mathematically gifted and won a double first degree, which could have been the prelude to a brilliant academic career. Instead, he became a country parson . Dodgson was born on 27 January 1832 at All Saints' Vicarage in Daresbury , Cheshire , the oldest boy and the third oldest of 11 children. When he was 11, his father was given the living of Croft-on-Tees , Yorkshire, and the whole family moved to
15616-458: Was on one such expedition on 4 July 1862 that Dodgson invented the outline of the story that eventually became his first and greatest commercial success. He told the story to Alice Liddell and she begged him to write it down, and Dodgson eventually (after much delay) presented her with a handwritten, illustrated manuscript entitled Alice's Adventures Under Ground in November 1864. Before this,
15744-518: Was ordained a deacon in the Church of England on 22 December 1861. In The Life and Letters of Lewis Carroll , the editor states that "his Diary is full of such modest depreciations of himself and his work, interspersed with earnest prayers (too sacred and private to be reproduced here) that God would forgive him the past, and help him to perform His holy will in the future." When a friend asked him about his religious views, Dodgson wrote in response that he
15872-407: Was proved independent of ZF by Fraenkel, but has come to be widely accepted by mathematicians. It states that given a collection of nonempty sets there is a single set C that contains exactly one element from each set in the collection. The set C is said to "choose" one element from each set in the collection. While the ability to make such a choice is considered obvious by some, since each set in
16000-399: Was remembered as that of a boy who knew well how to use his fists in defence of a righteous cause", which is the protection of the smaller boys. Scholastically, though, he excelled with apparent ease. "I have not had a more promising boy at his age since I came to Rugby", observed mathematics master R. B. Mayor. Francis Walkingame's The Tutor's Assistant; Being a Compendium of Arithmetic –
16128-482: Was reprinted in the same journal a hundred years later in 1995, with a subsequent article by Simon Blackburn titled "Practical Tortoise Raising". From a young age, Dodgson wrote poetry and short stories, contributing heavily to the family magazine Mischmasch and later sending them to various magazines, enjoying moderate success. Between 1854 and 1856, his work appeared in the national publications The Comic Times and The Train , as well as smaller magazines such as
16256-662: Was the enthusiastic reception of Alice by the young MacDonald children that persuaded him to submit the work for publication. In broad terms, Dodgson has traditionally been regarded as politically, religiously, and personally conservative. Martin Gardner labels Dodgson as a Tory who was "awed by lords and inclined to be snobbish towards inferiors". William Tuckwell , in his Reminiscences of Oxford (1900), regarded him as "austere, shy, precise, absorbed in mathematical reverie, watchfully tenacious of his dignity, stiffly conservative in political, theological, social theory, his life mapped out in squares like Alice's landscape". Dodgson
16384-445: Was then translated back into English as "Carroll Lewis" and then reversed to make "Lewis Carroll". This pseudonym was chosen by editor Edmund Yates from a list of four submitted by Dodgson, the others being Edgar Cuthwellis, Edgar U. C. Westhill, and Louis Carroll. In 1856, Dean Henry Liddell arrived at Christ Church at Oxford University , bringing with him his young family, all of whom would figure largely in Dodgson's life over
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