86-506: The Baroque Cycle is a series of novels by American writer Neal Stephenson . It was published in three volumes containing eight books in 2003 and 2004. The story follows the adventures of a sizable cast of characters living amidst some of the central events of the late 17th and early 18th centuries in Europe, Africa, Asia, and Central America. Despite featuring a literary treatment consistent with historical fiction, Stephenson has characterized
172-538: A B.A. in geography and a minor in physics. Since 1984, Stephenson has lived mostly in the Pacific Northwest and currently lives in Seattle with his family. Stephenson's first novel, The Big U , published in 1984, is a satirical take on life at American Megaversity, a vast, bland, and alienating research university beset by chaotic riots. His next novel, Zodiac (1988), is a thriller following
258-444: A biochemistry laboratory, and her father was a biochemistry professor. Stephenson's family moved to Champaign-Urbana, Illinois , in 1960, and then in 1966 to Ames, Iowa . He graduated from Ames High School in 1977. Stephenson studied at Boston University , first specializing in physics, then switching to geography after he found that it would allow him to spend more time on the university mainframe. He graduated in 1981 with
344-561: A blockchain . Stephenson's writing is influential in technology circles. Bill Gates , Sergey Brin , John Carmack , and Peter Thiel are all fans of his work. In Snow Crash Stephenson coined the term Metaverse and popularized the term avatar in a computing context. The Metaverse inspired the inventors of Google Earth and Snow Crash was required reading on the Xbox development team under Microsoft executive J Allard . According to academic Paul Youngquist, Snow Crash also dealt
430-415: 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,
516-480: A cofounder of Subutai Corporation, whose first offering is the interactive fiction project The Mongoliad . He was Magic Leap 's Chief Futurist from 2014 to 2020. Born on October 31, 1959, in Fort Meade , Maryland , Stephenson came from a family of engineers and scientists ; his father is a professor of electrical engineering while his paternal grandfather was a physics professor. His mother worked in
602-434: 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
688-430: 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
774-561: 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
860-400: 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
946-415: 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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#17327722722081032-430: 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',
1118-524: 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
1204-494: 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
1290-561: 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
1376-547: A more standard aerospace company. In 2012, Stephenson launched a Kickstarter campaign for Clang , a realistic sword-fighting fantasy game. The concept was to use motion control to provide an immersive experience. The campaign's funding goal of $ 500,000 was reached by the target date of July 9, 2012, on Kickstarter, but funding options remained open and the project continued to accept contributions on its official site. The project ran out of money in September 2013. This, and
1462-403: 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
1548-434: A novel that will keep scholars and critics occupied for the next 100 years". Neal Stephenson Neal Town Stephenson (born October 31, 1959) is an American writer known for his works of speculative fiction . His novels have been categorized as science fiction , historical fiction , cyberpunk and baroque . Stephenson's work explores mathematics , cryptography , linguistics , philosophy , currency , and
1634-489: 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
1720-422: A political thriller, Interface , under the pen name "Stephen Bury"; they followed this in 1996 with The Cobweb . Stephenson's next solo novel, published in 1995, was The Diamond Age: Or, A Young Lady's Illustrated Primer . The plot involves a weapon implanted in a character's skull, near-limitless replicators for everything from mattresses to foods, smartpaper , and air and blood-sanitizing nanobots. It
1806-422: 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
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#17327722722081892-564: A publishing standpoint"; Stephenson felt calling the works a trilogy would be "bogus". Appearing in print in 2003 and 2004 , the cycle contains eight books originally published in three volumes: The books travel throughout early modern Europe between the Restoration of the Stuart monarchy and the beginning of the 18th century. Though most of the focus is in Europe, the adventures of one character, Jack Shaftoe, do take him throughout
1978-433: A radical environmentalist in his struggle against corporate polluters. Neither novel attracted much critical attention on first publication, but both showcased concerns that Stephenson would further develop in his later work. Stephenson's breakthrough came in 1992 with Snow Crash , a cyberpunk or postcyberpunk novel fusing memetics , computer viruses , and other high-tech themes with Sumerian mythology , along with
2064-491: 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
2150-423: 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
2236-404: 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
2322-457: 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,
2408-401: A sociological extrapolation of extreme laissez-faire capitalism and collectivism . Stephenson at this time would later be described by Mike Godwin as "a slight, unassuming grad-student type whose soft-spoken demeanor gave no obvious indication that he had written the manic apotheosis of cyberpunk science fiction." In 1994, Stephenson joined with his uncle, J. Frederick George , to publish
2494-571: 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
2580-443: 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
2666-581: 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
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2752-517: 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
2838-461: 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
2924-453: 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
3010-470: 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
3096-468: 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
3182-446: 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
3268-547: Is set in a world with a neo-Victorian social structure. This was followed by Cryptonomicon in 1999, a novel including concepts ranging from Alan Turing 's research into codebreaking and cryptography during the Second World War , to a modern attempt to set up a data haven . In 2013, Cryptonomicon won the Prometheus Hall of Fame Award . The Baroque Cycle is a series of historical novels set in
3354-642: 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
3440-500: 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
3526-749: The Philippines , Japan and Mexico . The System of the World takes place principally in London in 1714, about ten years after the events of The Confusion . A central theme in the series is Europe's transformation away from feudal rule and control toward the rational, scientific, and more merit-based systems of government, finance, and social development that define what is now considered "western" and "modern". Characters include Sir Isaac Newton , Gottfried Leibniz , Nicolas Fatio de Duillier , William of Orange , Louis XIV of France , Oliver Cromwell , Peter
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3612-443: The cyberpunk genre a "killer blow". According to Publishers Weekly , Cryptonomicon is "often credited with sketching the basis for cryptocurrency ." 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
3698-406: The history of science . He also writes non-fiction articles about technology in publications such as Wired . He has written novels with his uncle, George Jewsbury ("J. Frederick George"), under the collective pseudonym Stephen Bury. Stephenson has worked part-time as an advisor for Blue Origin , a company (founded by Jeff Bezos ) developing a spacecraft and a space launch system, and is also
3784-409: 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
3870-466: 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
3956-483: The 17th and 18th centuries, and is in some respects a prequel to Cryptonomicon . It was originally published in three volumes of two or three books each – Quicksilver (2003), The Confusion , (2004) and The System of the World (2004) – but was subsequently republished as eight separate books: Quicksilver , King of the Vagabonds , Odalisque , Bonanza , Juncto , Solomon's Gold , Currency , and System of
4042-462: The Great , John Churchill, 1st Duke of Marlborough and many other people of note of that time. The fictional characters of Eliza, Jack and Daniel collectively cause real historic effects. The books feature considerable sections concerning alchemy . The principal alchemist of the tale is the mysterious Enoch Root, who, along with the descendants of several characters in this series, is also featured in
4128-609: The Machines that suggests Leibniz was "arguably the founder of symbolic logic and he worked with computing machines". He also had heard considerable discussion of the Leibniz–Newton calculus controversy and Newton's work at the treasury during the last 30 years of his life, and in particular the case against Leibniz as summed up in the Commercium Epistolicum of 1712 was a huge inspiration which went on to inform
4214-574: The Stephenson novels Cryptonomicon and Fall . Mercury provides a unifying theme, both in the form of the common name "quicksilver" for the element Mercury , long associated with alchemy and the title of the first volume of the cycle, and the Roman god Mercury , especially the god's various patronages: financial gain, commerce, eloquence, messages, communication, travelers, boundaries, luck, trickery, and thieves, all of which are central themes in
4300-471: The Subutai Corporation, of which Stephenson was named chairman, announced the production of an experimental multimedia fiction project called The Mongoliad , which centered upon a narrative written by Stephenson and other speculative fiction authors. Stephenson's novel Reamde was released on September 20, 2011. The title is a play on the common filename README . This thriller, set in
4386-562: The World . (The titles and exact breakdown vary in different markets.) The System of the World won the Prometheus Award in 2005. Following this, Stephenson wrote Anathem (2008), a long and detailed novel of speculative fiction . It is set in an Earthlike world, deals with metaphysics, and refers heavily to Ancient Greek philosophy . Anathem won the Locus Award for Best Science Fiction Novel in 2009. In May 2010,
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#17327722722084472-402: The circumstances around it, angered some backers with some threatening a class action lawsuit. The Clang project ended in September 2014 without being completed. Stephenson took part of the responsibility for the project's failure, stating, "I probably focused too much on historical accuracy and not enough on making it sufficiently fun to attract additional investment". In 2014, Stephenson
4558-469: 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 ,
4644-608: 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
4730-412: 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
4816-679: 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
4902-581: 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
4988-407: 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
5074-450: 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
5160-441: 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
5246-628: 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
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#17327722722085332-423: 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
5418-443: 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
5504-491: 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
5590-601: 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
5676-575: 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
5762-411: 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
5848-445: 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
5934-482: 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 ,
6020-486: 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
6106-418: 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
6192-548: The plot. Astronomy is also a significant (although secondary) theme in the cycle; a transit of Mercury was notably observed in London on day of the coronation of King Charles II of England , whose Restoration marks, chronologically, the earliest key historical event in the cycle. Stephenson was inspired to write The Baroque Cycle when, while working on Cryptonomicon , he encountered a statement by George Dyson in Darwin among
6278-442: The present, centers around a group of MMORPG developers caught in the middle of Chinese cyber-criminals, Islamic terrorists, and Russian mafia. On August 7, 2012, Stephenson released a collection of essays and other previously published fiction entitled Some Remarks: Essays and Other Writing . This collection also includes a new essay and a short story created specifically for this volume. In late 2013, Stephenson stated that he
6364-406: The project. He found "this information striking when [he] was already working on a book about money and a book about computers". Further research into the period excited Stephenson and he embarked on writing the historical piece that became The Baroque Cycle . Robert Wiersem of The Toronto Star called The Baroque Cycle a "sublime, immersive, brain-throttlingly complex marvel of
6450-499: The same time. The discursive nature of his writing, together with significant plot and character complexity and an abundance of detail suggests a baroque writing style, which Stephenson brought fully to bear in the three-volume Baroque Cycle . Stephenson worked at Blue Origin , Jeff Bezos ' spaceflight company, for seven years in the early 2000s while its focus was on "novel alternate approaches to space , alternate propulsion systems , and business models." He left after Blue became
6536-530: 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,
6622-463: 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
6708-411: The work as science fiction , because of the presence of some anomalous occurrences and the work's particular emphasis on themes relating to science and technology. The sciences of cryptology and numismatics feature heavily in the series, as they do in some of Stephenson's other works. The Baroque Cycle consists of several novels "lumped together into three volumes because it is more convenient from
6794-399: 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
6880-661: The world, and the fledgling British colonies in North America are important to another (Daniel Waterhouse). Quicksilver takes place mainly in the years between the Restoration of the Stuart monarchy in England (1660) and the Glorious Revolution of 1688. The Confusion follows Quicksilver without temporal interruption, but ranges geographically from Europe and the Mediterranean through India to
6966-609: Was completed about a year later and was published in May 2015. On June 8, 2016, plans were announced to adapt Seveneves for the screen. In May 2016, as part of a video discussion with Bill Gates , Stephenson revealed that he had just submitted the manuscript for a new historical novel—"a time travel book"—co-written with Nicole Galland , one of his Mongoliad coauthors. This was released as The Rise and Fall of D.O.D.O. on June 13, 2017. In June 2019, his novel Fall; or, Dodge in Hell
7052-486: 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
7138-652: Was hired as Chief Futurist by the Florida-based augmented reality company Magic Leap . Stephenson left the company in April 2020 as part of a layoff. In June 2021, Stephenson and colleagues Sean Stewart and Austin Grossman released New Found Land: The Long Haul , an Audible audio drama based on the intellectual property they developed at Magic Leap. In 2022, Stephenson launched Lamina1 to build an open source metaverse that would use smart contracts on
7224-407: 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
7310-458: Was published. It is a near-future novel that explores mind uploading into the cloud, and contains characters from 2011's Reamde , 1999's Cryptonomicon , and other books. Termination Shock , published in November 2021, is a climate fiction novel about solar geoengineering . Stephenson's books tend to have elaborate plots drawing on numerous technological and sociological ideas at
7396-401: Was working on a multi-volume work of historical novels that would "have a lot to do with scientific and technological themes and how those interact with the characters and civilisation during a particular span of history". He expected the first two volumes to be released in mid-to-late 2014. However, at about the same time, he shifted his attention to a science fiction novel, Seveneves , which
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