Axiom of choice
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that the Cartesian product of a collection of non-empty sets is non-empty. Informally put, the axiom of choice says that given any collection of bins, each containing at least one object, it is possible to make a selection of one object from each bin if the collection is infinite. Formally, it states that for every indexed family i ∈ I of nonempty sets there exists an indexed family i ∈ I of elements such that x i ∈ S i for every i ∈ I; the axiom of choice was formulated in 1904 by Ernst Zermelo in order to formalize his proof of the well-ordering theorem. In many cases, such a selection can be made without invoking the axiom of choice. An illustrative example is sets picked from the natural numbers. From such sets, one may always select the smallest number, e.g. in the smallest elements are. In this case, "select the smallest number" is a choice function. If infinitely many sets were collected from the natural numbers, it will always be possible to choose the smallest element from each set to produce a set.
That is, the choice function provides the set of chosen elements. However, no choice function is known for the collection of all non-empty subsets of the real numbers. In that case, the axiom of choice must be invoked. Bertrand Russell coined an analogy: for any collection of pairs of shoes, one can pick out the left shoe from each pair to obtain an appropriate selection. For an infinite collection of pairs of socks, there is no obvious way to make a function that selects one sock from each pair, without invoking the axiom of choice. Although controversial, the axiom of choice is now used without reservation by most mathematicians, it is included in the standard form of axiomatic set theory, Zermelo–Fraenkel set theory with the axiom of choice. One motivation for this use is that a number of accepted mathematical results, such as Tychonoff's theorem, require the axiom of choice for their proofs. Contemporary set theorists study axioms that are not compatible with the axiom of choice, such as the axiom of determinacy.
The axiom of choice is avoided in some varieties of constructive mathematics, although there are varieties of constructive mathematics in which the axiom of choice is embraced. A choice function is a function f, defined on a collection X of nonempty sets, such that for every set A in X, f is an element of A. With this concept, the axiom can be stated: Formally, this may be expressed as follows: ∀ X. Thus, the negation of the axiom of choice states that there exists a collection of nonempty sets that has no choice function; each choice function on a collection X of nonempty sets is an element of the Cartesian product of the sets in X. This is not the most general situation of a Cartesian product of a family of sets, where a given set can occur more than once as a factor; the axiom of choice asserts the existence of such elements. In this article and other discussions of the Axiom of Choice the following abbreviations are common: AC – the Axiom of Choice. ZF – Zermelo–Fraenkel set theory omitting the Axiom of Choice.
ZFC – Zermelo–Fraenkel set theory, extended to include the Axiom of Choice. There are many other equivalent statements of the axiom of choice; these are equivalent in the sense that, in the presence of other basic axioms of set theory, they imply the axiom of choice and are implied by it. One variation avoids the use of choice functions by, in effect, replacing each choice function with its range. Given any set X of pairwise disjoint non-empty sets, there exists at least one set C that contains one element in common with each of the sets in X; this guarantees for any partition of a set X the existence of a subset C of X containing one element from each part of the partition. Another equivalent axiom only considers collections X that are powersets of other sets: For any set A, the power set of A has a choice function. Authors who use this formulation speak of the choice function on A, but be advised that this is a different notion of choice function, its domain is the powerset of A, and
Original proof of Gödel's completeness theorem
The proof of Gödel's completeness theorem given by Kurt Gödel in his doctoral dissertation of 1929 is not easy to read today. The version given below attempts to represent all the steps in the proof and all the important ideas faithfully, while restating the proof in the modern language of mathematical logic; this outline should not be considered a rigorous proof of the theorem. We work with first-order predicate calculus. Our languages allow constant and relation symbols. Structures consist of domains and interpretations of the relevant symbols as constant members, functions or relations over that domain. We assume classical logic. We fix some axiomatization of the predicate calculus: logical rules of inference. Any of the several well-known equivalent axiomatizations will do. Gödel's original proof assumed the Hilbert-Ackermann proof system. We assume without proof all the basic well-known results about our formalism that we need, such as the normal form theorem or the soundness theorem. We axiomatize predicate calculus without equality, i.e. there are no special axioms expressing the properties of equality as a special relation symbol.
After the basic form of the theorem has been proved, it will be easy to extend it to the case of predicate calculus with equality. In the following, we state two equivalent forms of the theorem, show their equivalence. We prove the theorem; this is done in the following steps: Reducing the theorem to sentences in prenex form, i.e. with all quantifiers at the beginning. Furthermore, we reduce it to formulas whose first quantifier is ∀; this is possible because for every sentence, there is an equivalent one in prenex form whose first quantifier is ∀. Reducing the theorem to sentences of the form ∀x1∀x2...∀xk ∃y1∃y2...∃ym φ. While we cannot do this by rearranging the quantifiers, we show that it is yet enough to prove the theorem for sentences of that form. We prove the theorem for sentences of that form; this is done by first noting that a sentence such as B = ∀x1∀x2...∀xk ∃y1∃y2...∃ym φ is either refutable or satisfiable, i.e. there is some model in which it holds. The reason for, the completeness of propositional logic, with the existential quantifiers playing no role.
We extend this result to more and more complex and lengthy sentences, Dn, built out from B, so that either any of them is refutable and therefore so is φ, or all of them are not refutable and therefore each holds in some model. We use the models in which the Dn hold in order to build a model in which φ holds; this is the most basic form of the completeness theorem. We restate it in a form more convenient for our purposes: When we say "all structures", it is important to specify that the structures involved are classical interpretations I, where I=<U,F>. "φ is refutable" means by definition "¬φ is provable". To see the equivalence, note first that if Theorem 1 holds, φ is not satisfiable in any structure ¬φ is valid in all structures and therefore provable, thus φ is refutable and Theorem 2 holds. If on the other hand Theorem 2 holds and φ is valid in all structures ¬φ is not satisfiable in any structure and therefore refutable. We approach the proof of Theorem 2 by successively restricting the class of all formulas φ for which we need to prove "φ is either refutable or satisfiable".
At the beginning we need to prove this for all possible formulas φ in our language. However, suppose that for every formula φ there is some formula ψ taken from a more restricted class of formulas C, such that "ψ is either refutable or satisfiable" → "φ is either refutable or satisfiable". Once this claim is proved, it will suffice to prove "φ is either refutable or satisfiable" only for φ's belonging to the class C. Note that if φ is provably equivalent to ψ it is indeed the case that "ψ is either refutable or satisfiable" → "φ is either refutable or satisfiable". There are standard techniques for rewriting an arbitrary formula into one that does not use function or constant symbols, at the cost of introducing additional quantifiers. Gödel's paper uses a version of first-order predicate calculus that has no function or constant symbols to begin with. Next we consider a generic formula φ and apply the prenex form theorem to find a formula ψ in normal form such that φ≡ψ, it follows now. Next, we eliminate all free variables from φ by quantifying them existentially: if, say, x1...xn are free in φ, we form ψ =
Kurt Friedrich Gödel was an Austrian, American, logician and philosopher. Considered along with Aristotle, Alfred Tarski and Gottlob Frege to be one of the most significant logicians in history, Gödel made an immense impact upon scientific and philosophical thinking in the 20th century, a time when others such as Bertrand Russell, Alfred North Whitehead, David Hilbert were analyzing the use of logic and set theory to understand the foundations of mathematics pioneered by Georg Cantor. Gödel published his two incompleteness theorems in 1931 when he was 25 years old, one year after finishing his doctorate at the University of Vienna; the first incompleteness theorem states that for any self-consistent recursive axiomatic system powerful enough to describe the arithmetic of the natural numbers, there are true propositions about the naturals that cannot be proved from the axioms. To prove this theorem, Gödel developed a technique now known as Gödel numbering, which codes formal expressions as natural numbers.
He showed that neither the axiom of choice nor the continuum hypothesis can be disproved from the accepted axioms of set theory, assuming these axioms are consistent. The former result opened the door for mathematicians to assume the axiom of choice in their proofs, he made important contributions to proof theory by clarifying the connections between classical logic, intuitionistic logic, modal logic. Gödel was born April 28, 1906, in Brünn, Austria-Hungary into the German family of Rudolf Gödel, the manager of a textile factory, Marianne Gödel. Throughout his life, Gödel would remain close to his mother. At the time of his birth the city had a German-speaking majority, his father was Catholic and his mother was Protestant and the children were raised Protestant. The ancestors of Kurt Gödel were active in Brünn's cultural life. For example, his grandfather Joseph Gödel was a famous singer of that time and for some years a member of the Brünner Männergesangverein. Gödel automatically became a Czechoslovak citizen at age 12 when the Austro-Hungarian Empire broke up at the end of World War I.
In his family, young Kurt was known as Herr Warum because of his insatiable curiosity. According to his brother Rudolf, at the age of six or seven Kurt suffered from rheumatic fever. Beginning at age four, Gödel suffered from "frequent episodes of poor health," which would continue for his entire life. Gödel attended the Evangelische Volksschule, a Lutheran school in Brünn from 1912 to 1916, was enrolled in the Deutsches Staats-Realgymnasium from 1916 to 1924, excelling with honors in all his subjects in mathematics and religion. Although Kurt had first excelled in languages, he became more interested in history and mathematics, his interest in mathematics increased when in 1920 his older brother Rudolf left for Vienna to go to medical school at the University of Vienna. During his teens, Kurt studied Gabelsberger shorthand, Goethe's Theory of Colours and criticisms of Isaac Newton, the writings of Immanuel Kant. At the age of 18, Gödel entered the University of Vienna. By that time, he had mastered university-level mathematics.
Although intending to study theoretical physics, he attended courses on mathematics and philosophy. During this time, he adopted ideas of mathematical realism, he read Kant's Metaphysische Anfangsgründe der Naturwissenschaft, participated in the Vienna Circle with Moritz Schlick, Hans Hahn, Rudolf Carnap. Gödel studied number theory, but when he took part in a seminar run by Moritz Schlick which studied Bertrand Russell's book Introduction to Mathematical Philosophy, he became interested in mathematical logic. According to Gödel, mathematical logic was "a science prior to all others, which contains the ideas and principles underlying all sciences."Attending a lecture by David Hilbert in Bologna on completeness and consistency of mathematical systems may have set Gödel's life course. In 1928, Hilbert and Wilhelm Ackermann published Grundzüge der theoretischen Logik, an introduction to first-order logic in which the problem of completeness was posed: Are the axioms of a formal system sufficient to derive every statement, true in all models of the system?
This problem became the topic. In 1929, at the age of 23, he completed his doctoral dissertation under Hans Hahn's supervision. In it, he established his eponymous completeness theorem regarding the first-order predicate calculus, he was awarded his doctorate in 1930, his thesis was published by the Vienna Academy of Science. Kurt Gödel's achievement in modern logic is singular and monumental—indeed it is more than a monument, it is a landmark which will remain visible far in space and time.... The subject of logic has completely changed its nature and possibilities with Gödel's achievement. In 1931 and while still in Vienna, Gödel published
A computer is a device that can be instructed to carry out sequences of arithmetic or logical operations automatically via computer programming. Modern computers have the ability to follow generalized sets of called programs; these programs enable computers to perform an wide range of tasks. A "complete" computer including the hardware, the operating system, peripheral equipment required and used for "full" operation can be referred to as a computer system; this term may as well be used for a group of computers that are connected and work together, in particular a computer network or computer cluster. Computers are used as control systems for a wide variety of industrial and consumer devices; this includes simple special purpose devices like microwave ovens and remote controls, factory devices such as industrial robots and computer-aided design, general purpose devices like personal computers and mobile devices such as smartphones. The Internet is run on computers and it connects hundreds of millions of other computers and their users.
Early computers were only conceived as calculating devices. Since ancient times, simple manual devices like the abacus aided people in doing calculations. Early in the Industrial Revolution, some mechanical devices were built to automate long tedious tasks, such as guiding patterns for looms. More sophisticated electrical machines did specialized analog calculations in the early 20th century; the first digital electronic calculating machines were developed during World War II. The speed and versatility of computers have been increasing ever since then. Conventionally, a modern computer consists of at least one processing element a central processing unit, some form of memory; the processing element carries out arithmetic and logical operations, a sequencing and control unit can change the order of operations in response to stored information. Peripheral devices include input devices, output devices, input/output devices that perform both functions. Peripheral devices allow information to be retrieved from an external source and they enable the result of operations to be saved and retrieved.
According to the Oxford English Dictionary, the first known use of the word "computer" was in 1613 in a book called The Yong Mans Gleanings by English writer Richard Braithwait: "I haue read the truest computer of Times, the best Arithmetician that euer breathed, he reduceth thy dayes into a short number." This usage of the term referred to a human computer, a person who carried out calculations or computations. The word continued with the same meaning until the middle of the 20th century. During the latter part of this period women were hired as computers because they could be paid less than their male counterparts. By 1943, most human computers were women. From the end of the 19th century the word began to take on its more familiar meaning, a machine that carries out computations; the Online Etymology Dictionary gives the first attested use of "computer" in the 1640s, meaning "one who calculates". The Online Etymology Dictionary states that the use of the term to mean "'calculating machine' is from 1897."
The Online Etymology Dictionary indicates that the "modern use" of the term, to mean "programmable digital electronic computer" dates from "1945 under this name. Devices have been used to aid computation for thousands of years using one-to-one correspondence with fingers; the earliest counting device was a form of tally stick. Record keeping aids throughout the Fertile Crescent included calculi which represented counts of items livestock or grains, sealed in hollow unbaked clay containers; the use of counting rods is one example. The abacus was used for arithmetic tasks; the Roman abacus was developed from devices used in Babylonia as early as 2400 BC. Since many other forms of reckoning boards or tables have been invented. In a medieval European counting house, a checkered cloth would be placed on a table, markers moved around on it according to certain rules, as an aid to calculating sums of money; the Antikythera mechanism is believed to be the earliest mechanical analog "computer", according to Derek J. de Solla Price.
It was designed to calculate astronomical positions. It was discovered in 1901 in the Antikythera wreck off the Greek island of Antikythera, between Kythera and Crete, has been dated to c. 100 BC. Devices of a level of complexity comparable to that of the Antikythera mechanism would not reappear until a thousand years later. Many mechanical aids to calculation and measurement were constructed for astronomical and navigation use; the planisphere was a star chart invented by Abū Rayhān al-Bīrūnī in the early 11th century. The astrolabe was invented in the Hellenistic world in either the 1st or 2nd centuries BC and is attributed to Hipparchus. A combination of the planisphere and dioptra, the astrolabe was an analog computer capable of working out several different kinds of problems in spherical astronomy. An astrolabe incorporating a mechanical calendar computer and gear-wheels was invented by Abi Bakr of Isfahan, Persia in 1235. Abū Rayhān al-Bīrūnī invented the first mechanical geared lunisolar calendar astrolabe, an early fixed-wired knowledge processing machine with a gear train and gear-wheels, c. 1000 AD.
The sector, a calculating instrument used for solving problems in proportion, trigonometry and division, for various functions, such as squares and cube roots, was developed in
A tree structure or tree diagram is a way of representing the hierarchical nature of a structure in a graphical form. It is named a "tree structure" because the classic representation resembles a tree though the chart is upside down compared to an actual tree, with the "root" at the top and the "leaves" at the bottom. A tree structure is conceptual, appears in several forms. For a discussion of tree structures in specific fields, see Tree for computer science: insofar as it relates to graph theory, see tree, or tree. Other related pages are listed; the tree elements are called "nodes". The lines connecting elements are called "branches". Nodes without children are called leaf nodes, "end-nodes", or "leaves"; every finite tree structure has a member. This member is called root node; the root is the starting node. But the converse is not true: infinite tree structures may or may not have a root node; the names of relationships between nodes model the kinship terminology of family relations. The gender-neutral names "parent" and "child" have displaced the older "father" and "son" terminology, although the term "uncle" is still used for other nodes at the same level as the parent.
A node's "parent". "Sibling" nodes share the same parent node. A node's "uncles" are siblings of that node's parent. A node, connected to all lower-level nodes is called an "ancestor"; the connected lower-level nodes are "descendants" of the ancestor node. In the example, "encyclopedia" is the parent of its children. "Art" and "craft" are siblings, children of "culture", their parent and thus one of their ancestors. "encyclopedia", as the root of the tree, is the ancestor of "science", "culture", "art" and "craft". "science", "art" and "craft", as leaves, are ancestors of no other node. Tree structures can depict all kinds of taxonomic knowledge, such as family trees, the biological evolutionary tree, the evolutionary tree of a language family, the grammatical structure of a language, the way web pages are logically ordered in a web site, mathematical trees of integer sets, et cetera; the Oxford English Dictionary records use of both the terms "tree structure" and "tree-diagram" from 1965 in Noam Chomsky's Aspects of the Theory of Syntax.
In a tree structure there is one and only one path from any point to any other point. Computer science uses tree structures extensively For a formal definition see set theory, for a generalization in which children are not successors, see prefix order. Internet: usenet hierarchy Vacuum tubes Document Object Model's logical structure, Yahoo! Subject index, Curlie Operating system: directory structure Information management: Dewey Decimal System, PSH, this hierarchical bulleted list Management: hierarchical organizational structures Computer Science: binary search tree Red-Black Tree AVL tree R-tree Biology: evolutionary tree Business: pyramid selling scheme Project management: work breakdown structure Linguistics: Phrase structure trees Tree model of language change Sports: business chess, playoffs brackets Mathematics: Von Neumann universe Group theory: descendant trees There are many ways of visually representing tree structures. Always, these boil down to variations, or combinations, of a few basic styles: Classical node-link diagrams, that connect nodes together with line segments: Nested sets that use enclosure/containment to show parenthood, examples include TreeMaps and fractal maps: Layered "icicle" diagrams that use alignment/adjacency.
Lists or diagrams that use indentation, sometimes called "outlines" or "tree views". An outline: A tree view: A correspondence to nested parentheses was first noticed by Sir Arthur Cayley: Trees can be represented radially: Kinds of trees B-tree Dancing tree Decision tree Left-child right-sibling binary tree Tree Tree Tree Related articles Data drilling Hierarchical model: clustering and query Tree testing Identification of some of the basic styles of tree structures can be found in: Jacques Bertin, Semiology of Graphics, 1983, University of Wisconsin Press (2nd edition 1973, ISBN 978-0299090609. Manuel Lima, The Book of Trees: Visualizing Branches of Knowledge, Princeton Architectural Press, ISBN 978-1-616-89218-0 Visualization of phylogenetic trees on the T-REX server Using a tree structure to design a business process - from the Society for Technical Communication
Gisbert F. R. Hasenjaeger was a German mathematical logician. Independently and with Leon Henkin in 1949, he developed a new proof of the completeness theorem of Kurt Gödel for predicate logic, he worked as an assistant to Heinrich Scholz at Section IVa of Oberkommando der Wehrmacht Chiffrierabteilung, was responsible for the security of the Enigma machine. Gisbert Hasenjaeger went to high school in Mülheim, where his father Edwin Renatus Hasenjaeger was a lawyer and local politician. After completing school in 1936, Gisbert volunteered for labor service, he was drafted for military service in World War II, fought as an artillerist in the Russian campaign, where he was badly wounded in January 1942. After his recovery, in October 1942 Heinrich Scholz got him an employment in the Cipher Department of the High Command of the Wehrmacht, where he was the youngest member at 24, he attended a cryptography training course by Erich Hüttenhain, was put into the founded Section IVa "Security check of own Encoding Procedures" under Karl Stein, who assigned him the security check of the Enigma machine.
At the end of the war as OKW/Chi disintegrated, Hasenjaeger managed to escape TICOM, the United States effort to roundup and seize captured German intelligence people and material. From the end of 1945, he studied mathematics and mathematical logic with Heinrich Scholz at the Westfälische Wilhelms-Universität University in Münster. In 1950 received his doctorate Topological studies on the semantics and syntax of an extended predicate calculus and completed his habilitation in 1953. In Münster, he worked as an assistant to Scholz and co-author, to write the textbook Fundamentals of Mathematical Logic in Springer's Grundlehren series, which he published in 1961 6 years after Scholz's death. In 1962 he became professor at the University of Bonn, where he was Director of the newly created Department of Logic. In 1962, Dr. Hasenjaeger left Münster University to take a full professorship at Bonn University, where he was established Director of the newly established Department of Logic and Basic Research.
In 1964/65 he spent a year at Princeton University at the Institute for Advanced Study His doctoral students at Bonn included Ronald B. Jensen, his most famous pupil, he became professor emeritus in 1984. In Oct 1942, after starting work at OKW/Chi, Hasenjaeger was trained in cryptology, given by the mathematician, Erich Hüttenhain, considered the most important German cryptologist of his time. Hasenjaeger was put into a newly formed department, whose principal responsibility was the defensive testing and security control of their own methods and devices. Hasenjaeger was ordered, by the mathematician Karl Stein, conscripted at OKW/Chi, to examine the Enigma machine for cryptologic weaknesses, while Stein was to examine the Siemens and Halske T52 and the Lorenz SZ-42; the Enigma machine that Hasenjaeger examined was a variation that worked with 3 rotors and had no plug board. Germany sold this version to neutral countries to accrue foreign exchange. Hasenjaeger was presented with a 100 character encrypted message for analysis and found a weakness which enabled the identification of the correct wiring rotors and the appropriate rotor positions, to decrypt the messages.
Further success eluded him however. He crucially failed to identify the most important weakness of the Enigma machine: the lack of fixed points due to the reflector. Hasenjaeger could take some comfort from the fact that Alan Turing missed this weakness. Instead the honour was attributed to Gordon Welchman, who used the knowledge to decrypt several hundred thousand Enigma messages during the war. In fact fixed points were earlier used by Polish codebreaker, Henryk Zygalski, as the basis for his method of attack on Enigma cipher, referred to by the Poles as "Zygalski sheets" and by the British as the "Netz method", it was while Hasenjaeger was working at Westfälische Wilhelms-Universität University in Münster in the period between 1946 and 1953 that Hasenjaeger made a most amazing discovery - a proof of Kurt Gödel's Gödel's completeness theorem for full predicate logic with identity and function symbols. Gödel's proof of 1930 for predicate logic did not automatically establish a procedure for the general case.
When he had solved the problem in late 1949, he was frustrated to find that a young American mathematician Leon Henkin, had created a proof. Both construct from extension of a term model, the model for the initial theory. Although the Henkin proof was considered by Hasenjaeger and his peers to more flexible, Hasenjaeger' is considered simpler and more transparent. Hasenjaeger continued to refine his proof through to 1953. According to the mathematicians Alfred Tarski, Stephen Cole Kleene and Andrzej Mostowski, the Arithmetical hierarchy of formulas is the set of arithmetical propositions that are true in the standard model, but not arithmetically definable. So, what does the concept of truth for the term model mean, the results for the recursively axiomatized Peano arithmetic from the Hasenjaeger method? The result was the truth predicate is well arithmetically, it is Δ 2 0. So far down in the arithmetic hierarchy, that goes for any recursively axiomatized theories. If you are true in all the natural numbers Π 1 0 formulas to the axioms.
This classic proof is a early, original application of the arithmetic hierarchy theory to a general-logical problem. It appeare