Set theory is a branch of mathematical logic that studies sets, which informally are collections of objects. Although any type of object can be collected into a set, set theory is applied most to objects that are relevant to mathematics; the language of set theory can be used to define nearly all mathematical objects. The modern study of set theory was initiated by Richard Dedekind in the 1870s. After the discovery of paradoxes in naive set theory, such as Russell's paradox, numerous axiom systems were proposed in the early twentieth century, of which the Zermelo–Fraenkel axioms, with or without the axiom of choice, are the best-known. Set theory is employed as a foundational system for mathematics in the form of Zermelo–Fraenkel set theory with the axiom of choice. Beyond its foundational role, set theory is a branch of mathematics in its own right, with an active research community. Contemporary research into set theory includes a diverse collection of topics, ranging from the structure of the real number line to the study of the consistency of large cardinals.
Mathematical topics emerge and evolve through interactions among many researchers. Set theory, was founded by a single paper in 1874 by Georg Cantor: "On a Property of the Collection of All Real Algebraic Numbers". Since the 5th century BC, beginning with Greek mathematician Zeno of Elea in the West and early Indian mathematicians in the East, mathematicians had struggled with the concept of infinity. Notable is the work of Bernard Bolzano in the first half of the 19th century. Modern understanding of infinity began in 1870–1874 and was motivated by Cantor's work in real analysis. An 1872 meeting between Cantor and Richard Dedekind influenced Cantor's thinking and culminated in Cantor's 1874 paper. Cantor's work polarized the mathematicians of his day. While Karl Weierstrass and Dedekind supported Cantor, Leopold Kronecker, now seen as a founder of mathematical constructivism, did not. Cantorian set theory became widespread, due to the utility of Cantorian concepts, such as one-to-one correspondence among sets, his proof that there are more real numbers than integers, the "infinity of infinities" resulting from the power set operation.
This utility of set theory led to the article "Mengenlehre" contributed in 1898 by Arthur Schoenflies to Klein's encyclopedia. The next wave of excitement in set theory came around 1900, when it was discovered that some interpretations of Cantorian set theory gave rise to several contradictions, called antinomies or paradoxes. Bertrand Russell and Ernst Zermelo independently found the simplest and best known paradox, now called Russell's paradox: consider "the set of all sets that are not members of themselves", which leads to a contradiction since it must be a member of itself and not a member of itself. In 1899 Cantor had himself posed the question "What is the cardinal number of the set of all sets?", obtained a related paradox. Russell used his paradox as a theme in his 1903 review of continental mathematics in his The Principles of Mathematics. In 1906 English readers gained the book Theory of Sets of Points by husband and wife William Henry Young and Grace Chisholm Young, published by Cambridge University Press.
The momentum of set theory was such. The work of Zermelo in 1908 and the work of Abraham Fraenkel and Thoralf Skolem in 1922 resulted in the set of axioms ZFC, which became the most used set of axioms for set theory; the work of analysts such as Henri Lebesgue demonstrated the great mathematical utility of set theory, which has since become woven into the fabric of modern mathematics. Set theory is used as a foundational system, although in some areas—such as algebraic geometry and algebraic topology—category theory is thought to be a preferred foundation. Set theory begins with a fundamental binary relation between an object o and a set A. If o is a member of A, the notation o. Since sets are objects, the membership relation can relate sets as well. A derived binary relation between two sets is the subset relation called set inclusion. If all the members of set A are members of set B A is a subset of B, denoted A ⊆ B. For example, is a subset of, so is but is not; as insinuated from this definition, a set is a subset of itself.
For cases where this possibility is unsuitable or would make sense to be rejected, the term proper subset is defined. A is called a proper subset of B if and only if A is a subset of B, but A is not equal to B. Note that 1, 2, 3 are members of the set but are not subsets of it. Just as arithmetic features binary operations on numbers, set theory features binary operations on sets. The: Union of the sets A and B, denoted A ∪ B, is the set of all objects that are a member of A, or B, or both; the union of and is the set. Intersection of the sets A and B, denoted A ∩ B, is the set of all objects that are members of both A and B; the intersection of and is the set. Set difference of U and A, denoted U \ A, is the set of all members of U that are not members of A; the set difference \ is, conversely, the set difference \ is. When A is a subset of U, the set difference U \ A is called the complement of A in U. In this case, if the choice of U is clear from the context, the notation Ac is sometimes used instead of U \ A if U is a universal set as in the study of Venn diagrams.
Symmetric difference of sets A and B, denoted A △ B or A ⊖ B, is
Lambda is the 11th letter of the Greek alphabet, representing the sound /l/. In the system of Greek numerals lambda has a value of 30. Lambda is related to the Phoenician letter Lamed. Letters in other alphabets that stemmed from lambda include the Cyrillic letter El; the ancient grammarians and dramatists give evidence to the pronunciation as in Classical Greek times. In Modern Greek the name of the letter, Λάμδα, is pronounced. In early Greek alphabets, the shape and orientation of lambda varied. Most variants consisted one longer than the other, connected at their ends; the angle might be in lower-left, or top. Other variants had a vertical line with a sloped stroke running to the right. With the general adoption of the Ionic alphabet, Greek settled on an angle at the top; the HTML 4 character entity references for the Greek capital and small letter lambda are "Λ. The Unicode code points for lambda are U+039B and U+03BB. Examples of the symbolic use of uppercase lambda include: The lambda particle is a type of subatomic particle in subatomic particle physics.
Lambda is the set of logical axioms in the axiomatic method of logical deduction in first-order logic. Lambda was used as a shield pattern by the Spartan army; this stood for Lacedaemon, the name of the polis of the Spartans, as opposed to the city itself. Lambda is the von Mangoldt function in mathematical number theory. In statistics, Wilks's lambda is used in multivariate analysis of variance to compare group means on a combination of dependent variables. In the spectral decomposition of matrices, lambda indicates the diagonal matrix of the eigenvalues of the matrix. In computer science, lambda is the time window over which a process is observed for determining the working memory set for a digital computer's virtual memory management. In astrophysics, lambda represents the likelihood that a small body will encounter a planet or a dwarf planet leading to a deflection of a significant magnitude. An object with a large value of lambda is expected to have cleared its neighborhood, satisfying the current definition of a planet.
In crystal optics, lambda is used to represent the period of a lattice. In NATO military operations, a chevron is painted on the vehicles of this military alliance for identification. In chemistry there are Δ and Λ isomers, see: coordination complex In electrochemistry, lambda denotes the "equivalent conductance" of an electrolyte solution. In cosmology, lambda is the symbol for the cosmological constant, a term added to some dynamical equations to account for the acceleration of the universe. In optics, lambda denotes the grating pitch of a Bragg reflector. In block-handwritten Russian, this letter represents Л in both lowercase. In politics the lambda is the symbol of Identitarianism a white nationalist movement that originated in France before spreading out to the rest of Europe and on to North America and New Zealand; the Identitarian lambda represents the Battle of Thermopylae. Examples of the symbolic use of lowercase lambda include: In evolutionary algorithms, λ indicates the number of offspring that would be generated from μ current population in each generation.
The terms μ and λ are originated from Evolution strategy notation. Lambda indicates the wavelength of any wave in physics, electronics engineering, mathematics. Lambda indicates the radioactivity decay constant in nuclear radioactivity; this constant is simply related to the half-life of any radioactive material. In probability theory, lambda represents the density of occurrences within a time interval, as modeled by the Poisson distribution. In mathematical logic and computer science, lambda is used to introduce anonymous functions expressed with the concepts of lambda calculus. Lambda is a unit of volume, synonymous with one microliter; this use is deprecated. Lambda indicates an eigenvalue in the mathematics of linear algebra. In the physics of electric fields, lambda sometimes indicates the linear charge density of a uniform line of electric charge. Lambda denotes a Lagrange multiplier in multi-dimensional calculus. In solid-state electronics, lambda indicates the channel length modulation parameter of a MOSFET.
In ecology, lambda denotes the long-term intrinsic growth rate of a population. This value is calculated as the dominant eigenvalue of the age/size class matrix. In formal language theory and in computer science, lambda denotes the empty string. Lambda is a nonstandard symbol in the International Phonetic Alphabet. Lambda denotes the Lebesgue measure in mathematical set theory; the Goodman and Kruskal's lambda in statistics indicates the proportional reduction in error when one variable's values are used to predict the values of another variable. Lambda denotes the oxygen sensor in a vehicle that measures the air-to-fuel ratio in the exhaust gases of an internal-combustion engine. A Lambda 4S solid-fuel rocket was used to launch Japan's first orbital satellite in 1970. Lambda denotes the failure rate of devices and systems in reliability theory, it is measured in failure events per hour. Numerically, this lambda is the reciprocal of the mean time between failures. In criminology, lambda denotes an individual's frequency of offenses.
In cartography and navigation, lambda denotes the longitude of a location. In electrochemistry, lambda denotes the ionic cond
Jan Łukasiewicz was a Polish logician and philosopher born in Lemberg, a city in the Galician kingdom of Austria-Hungary. His work centred on philosophical logic, mathematical logic, history of logic, he thought innovatively about traditional propositional logic, the principle of non-contradiction and the law of excluded middle. Modern work on Aristotle's logic builds on the tradition started in 1951 with the establishment by Łukasiewicz of a revolutionary paradigm; the Łukasiewicz approach was reinvigorated in the early 1970s in a series of papers by John Corcoran and Timothy Smiley—which inform modern translations of Prior Analytics by Robin Smith in 1989 and Gisela Striker in 2009. Łukasiewicz is regarded as one of the most important historians of logic. He grew up in Lwów and was the only child of Paweł Łukasiewicz, a captain in the Austrian army, Leopoldina, née Holtzer, the daughter of a civil servant, his family was Roman Catholic. He finished his gymnasium studies in philology and in 1897 went on to Lwów University, where he studied philosophy and mathematics.
He was a pupil of philosopher Kazimierz Twardowski. In 1902 he received a Doctor of Philosophy degree under the patronage of Emperor Franz Joseph I of Austria, who gave him a special doctoral ring with diamonds, he spent three years as a private teacher, in 1905 he received a scholarship to complete his philosophy studies at the University of Berlin and the University of Louvain in Belgium. Łukasiewicz continued studying for his habilitation qualification and in 1906 submitted his thesis to the University of Lwów. In 1906 he was appointed a lecturer at the University of Lwów where he was appointed Extraordinary Professor by Emperor Franz Joseph I, he taught there until the First World War. In 1915 he was invited to lecture as a full professor at the University of Warsaw which had re-opened after being closed down by the Tsarist government in the 19th century. In 1919 Łukasiewicz left the university to serve as Polish Minister of Religious Denominations and Public Education in the Paderewski government until 1920.
Łukasiewicz led the development of a Polish curriculum replacing the Russian and Austrian curricula used in partitioned Poland. The Łukasiewicz curriculum emphasized the early acquisition of mathematical concepts. In 1928 he married Regina Barwińska, he remained a professor at the University of Warsaw from 1920 until 1939 when the family house was destroyed by German bombs and the university was closed under German occupation. He had been a rector of the university twice. In this period Łukasiewicz and Stanisław Leśniewski founded the Lwów–Warsaw school of logic, made internationally famous by Alfred Tarski, Leśniewski's student. At the beginning of World War II he worked at the Warsaw Underground University as part of the secret system of education in Poland during World War II, he and his wife wanted to move to Switzerland but couldn't get permission from the German authorities. Instead, in the summer of 1944, they left Poland with the help of Heinrich Scholz and spent the last few months of the war in Münster, Germany hoping to somehow go on further to Switzerland.
Following the war, he worked at University College Dublin until his death. Jan Łukasiewicz's papers are held by the University of Manchester Library. A number of axiomatizations of classical propositional logic are due to Łukasiewicz. A elegant axiomatization features a mere three axioms and is still invoked to the present day, he was a pioneer investigator of multi-valued logics. He wrote on the philosophy of science, his approach to the making of scientific theories was similar to the thinking of Karl Popper. Łukasiewicz invented the Polish notation for the logical connectives around 1920. There is a quotation from his paper, Remarks on Nicod's Axiom and on "Generalizing Deduction", page 180. I used that notation for the first time in my article Łukasiewicz, p. 610, footnote." The reference cited by Łukasiewicz above is a lithographed report in Polish. The referring paper by Łukasiewicz Remarks on Nicod's Axiom and on "Generalizing Deduction" published in Polish in 1931, was reviewed by H. A. Pogorzelski in the Journal of Symbolic Logic in 1965.
In Łukasiewicz 1951 book, Aristotle's Syllogistic from the Standpoint of Modern Formal Logic, he mentions that the principle of his notation was to write the functors before the arguments to avoid brackets and that he had employed his notation in his logical papers since 1929. He goes on to cite, as an example, a 1930 paper he wrote with Alfred Tarski on the sentential calculus; this notation is the root of the idea of the recursive stack, a last-in, first-out computer memory store proposed by several researchers including Turing and Hamblin, first implemented in 1957. In 1960, Łukasiewicz notation concepts and stacks were used as the basis of the Burroughs B5000 computer designed by Robert S. Barton and his team at Burroughs Corporation in California; the concepts led to the design of the English Electric multi-programmed KDF9 computer system of 1963, which had two such hardware register stacks. A similar concept underlies the reverse Polish notation of the Friden EC-130 calculator and its successors, many Hewlett Packard calculators, the Forth programming language, the PostScript page description langu
In philosophy and mathematics, a logical form of a syntactic expression is a precisely-specified semantic version of that expression in a formal system. Informally, the logical form attempts to formalize a ambiguous statement into a statement with a precise, unambiguous logical interpretation with respect to a formal system. In an ideal formal language, the meaning of a logical form can be determined unambiguously from syntax alone. Logical forms are semantic, not syntactic constructs; the logical form of an argument is called the argument test form of the argument. The importance of the concept of form to logic was recognized in ancient times. Aristotle, in the Prior Analytics, was the first to employ variable letters to represent valid inferences. Therefore, Łukasiewicz claims that the introduction of variables was'one of Aristotle's greatest inventions'. According to the followers of Aristotle like Ammonius, only the logical principles stated in schematic terms belong to logic, not those given in concrete terms.
The concrete terms man, etc. are analogous to the substitution values of the schematic placeholders'A','B','C', which were called the'matter' of the argument. The term "logical form" itself was introduced by Bertrand Russell in 1914, in the context of his program to formalize natural language and reasoning, which he called philosophical logic. Russell wrote: "Some kind of knowledge of logical forms, though with most people it is not explicit, is involved in all understanding of discourse, it is the business of philosophical logic to extract this knowledge from its concrete integuments, to render it explicit and pure." To demonstrate the important notion of the form of an argument, substitute letters for similar items throughout the sentences in the original argument. Original argument All humans are mortal. Socrates is human. Therefore, Socrates is mortal. Argument form All H are M. S is H. Therefore, S is M. All we have done in the Argument form is to put'H' for'human' and'humans','M' for'mortal', and'S' for'Socrates'.
Moreover, each individual sentence of the Argument form is the sentence form of its respective sentence in the original argument. Attention is given to argument and sentence form, because form is what makes an argument valid or cogent. All logical form arguments are either deductive. Inductive logical forms include inductive generalization, statistical arguments, causal argument, arguments from analogy. Common deductive argument forms are hypothetical syllogism, categorical syllogism, argument by definition, argument based on mathematics, argument from definition; the most reliable forms of logic are modus ponens, modus tollens, chain arguments because if the premises of the argument are true the conclusion follows. Two invalid argument forms are denying the antecedent. Affirming the consequent All dogs are animals. Coco is an animal. Therefore, Coco is a dog. Denying the antecedent All cats are animals. Missy is not a cat. Therefore, Missy is not an animal. A logical argument, seen as an ordered set of sentences, has a logical form that derives from the form of its constituent sentences.
Some authors only define logical form with respect to whole arguments, as the schemata or inferential structure of the argument. In argumentation theory or informal logic, an argument form is sometimes seen as a broader notion than the logical form, it consists of stripping out all spurious grammatical features from the sentence, replacing all the expressions specific to the subject matter of the argument by schematic variables. Thus, for example, the expression'all A's are B's' shows the logical form, common to the sentences'all men are mortals','all cats are carnivores','all Greeks are philosophers' and so on; the fundamental difference between modern formal logic and traditional, or Aristotelian logic, lies in their differing analysis of the logical form of the sentences they treat: On the traditional view, the form of the sentence consists of a subject plus a sign of quantity. Thus:'all men are mortal'; the logical constants such as "all", "no" and so on, plus sentential connectives such as "and" and "or" were called syncategorematic terms.
This is a fixed scheme, where each judgment has a specific quantity and copula, determining the logical form of the sentence. The modern view is more complex, since a single judgement of Aristotle's system involves two or more logical connectives. For example, the sentence "All men are mortal" involves, in term logic, two non-logical terms "is a man" and "is mortal": the sentence is given by the judgement A. In predicate logic, the sentence involves the same two non-logical concepts, here analyzed as m and d, the sentence is given by ∀ x, involving the logical connectives for universal quantification and implication; the more complex modern view comes with more power. On the modern view, the fundamental form of a simple sentence is given by a recursive schema, like natural langu
Infimum and supremum
In mathematics, the infimum of a subset S of a ordered set T is the greatest element in T, less than or equal to all elements of S, if such an element exists. The term greatest lower bound is commonly used; the supremum of a subset S of a ordered set T is the least element in T, greater than or equal to all elements of S, if such an element exists. The supremum is referred to as the least upper bound; the infimum is in a precise sense dual to the concept of a supremum. Infima and suprema of real numbers are common special cases that are important in analysis, in Lebesgue integration. However, the general definitions remain valid in the more abstract setting of order theory where arbitrary ordered sets are considered; the concepts of infimum and supremum are similar to minimum and maximum, but are more useful in analysis because they better characterize special sets which may have no minimum or maximum. For instance, the positive real numbers ℝ+ does not have a minimum, because any given element of ℝ+ could be divided in half resulting in a smaller number, still in ℝ+.
There is, however one infimum of the positive real numbers: 0, smaller than all the positive real numbers and greater than any other real number which could be used as a lower bound. A lower bound of a subset S of a ordered set is an element a of P such that a ≤ x for all x in S. A lower bound a of S is called an infimum of S. An upper bound of a subset S of a ordered set is an element b of P such that b ≥ x for all x in S. An upper bound b of S is called a supremum of S if for all upper bounds z of S in P, z ≥ b. Infima and suprema do not exist. Existence of an infimum of a subset S of P can fail if S has no lower bound at all, or if the set of lower bounds does not contain a greatest element. However, if an infimum or supremum does exist, it is unique. Ordered sets for which certain infima are known to exist become interesting. For instance, a lattice is a ordered set in which all nonempty finite subsets have both a supremum and an infimum, a complete lattice is a ordered set in which all subsets have both a supremum and an infimum.
More information on the various classes of ordered sets that arise from such considerations are found in the article on completeness properties. If the supremum of a subset S exists, it is unique. If S contains a greatest element that element is the supremum. If S contains a least element that element is the infimum; the infimum of a subset S of a ordered set P, assuming it exists, does not belong to S. If it does, it is a minimum or least element of S. Similarly, if the supremum of S belongs to S, it is a maximum or greatest element of S. For example, consider the set of negative real numbers; this set has no greatest element, since for every element of the set, there is another, element. For instance, for any negative real number x, there is another negative real number x 2, greater. On the other hand, every real number greater than or equal to zero is an upper bound on this set. Hence, 0 is the least upper bound of the negative reals, so the supremum is 0; this set has a supremum but no greatest element.
However, the definition of maximal and minimal elements is more general. In particular, a set can have many maximal and minimal elements, whereas infima and suprema are unique. Whereas maxima and minima must be members of the subset, under consideration, the infimum and supremum of a subset need not be members of that subset themselves. A ordered set may have many minimal upper bounds without having a least upper bound. Minimal upper bounds are those upper bounds for which there is no smaller element, an upper bound; this does not say that each minimal upper bound is smaller than all other upper bounds, it is not greater. The distinction between "minimal" and "least" is only possible when the given order is not a total one. In a ordered set, like the real numbers, the concepts are the same; as an example, let S be the set of all finite subsets of natural numbers and consider the ordered set obtained by taking all sets from S together with the set of integers ℤ and the set of positive real numbers ℝ+, ordered by subset inclusion as above.
Both ℤ and ℝ+ are greater than all finite sets of natural numbers. Yet, neither is ℝ+ smaller than ℤ nor is the converse true: both sets are minimal upper bounds but none is a supremum; the least-upper-bound property is an example of the aforementioned completeness properties, typical for the set of real numbers. This property is sometimes called Dedekind completeness. If an ordered set S has the property that every nonempty subset of S having an upper bound has a least upper bound S is said to have the least-upper-bound property; as noted above, the set ℝ of all real numbers has the least-upper-bound property. The set ℤ of integers has the least-upper-bound property.
Mathematics includes the study of such topics as quantity, structure and change. Mathematicians use patterns to formulate new conjectures; when mathematical structures are good models of real phenomena mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity from as far back; the research required to solve mathematical problems can take years or centuries of sustained inquiry. Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Since the pioneering work of Giuseppe Peano, David Hilbert, others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. Mathematics developed at a slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.
Mathematics is essential in many fields, including natural science, medicine and the social sciences. Applied mathematics has led to new mathematical disciplines, such as statistics and game theory. Mathematicians engage in pure mathematics without having any application in mind, but practical applications for what began as pure mathematics are discovered later; the history of mathematics can be seen as an ever-increasing series of abstractions. The first abstraction, shared by many animals, was that of numbers: the realization that a collection of two apples and a collection of two oranges have something in common, namely quantity of their members; as evidenced by tallies found on bone, in addition to recognizing how to count physical objects, prehistoric peoples may have recognized how to count abstract quantities, like time – days, years. Evidence for more complex mathematics does not appear until around 3000 BC, when the Babylonians and Egyptians began using arithmetic and geometry for taxation and other financial calculations, for building and construction, for astronomy.
The most ancient mathematical texts from Mesopotamia and Egypt are from 2000–1800 BC. Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry, it is in Babylonian mathematics that elementary arithmetic first appear in the archaeological record. The Babylonians possessed a place-value system, used a sexagesimal numeral system, still in use today for measuring angles and time. Beginning in the 6th century BC with the Pythagoreans, the Ancient Greeks began a systematic study of mathematics as a subject in its own right with Greek mathematics. Around 300 BC, Euclid introduced the axiomatic method still used in mathematics today, consisting of definition, axiom and proof, his textbook Elements is considered the most successful and influential textbook of all time. The greatest mathematician of antiquity is held to be Archimedes of Syracuse, he developed formulas for calculating the surface area and volume of solids of revolution and used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, in a manner not too dissimilar from modern calculus.
Other notable achievements of Greek mathematics are conic sections, trigonometry (Hipparchus of Nicaea, the beginnings of algebra. The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today, evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics. Other notable developments of Indian mathematics include the modern definition of sine and cosine, an early form of infinite series. During the Golden Age of Islam during the 9th and 10th centuries, mathematics saw many important innovations building on Greek mathematics; the most notable achievement of Islamic mathematics was the development of algebra. Other notable achievements of the Islamic period are advances in spherical trigonometry and the addition of the decimal point to the Arabic numeral system. Many notable mathematicians from this period were Persian, such as Al-Khwarismi, Omar Khayyam and Sharaf al-Dīn al-Ṭūsī. During the early modern period, mathematics began to develop at an accelerating pace in Western Europe.
The development of calculus by Newton and Leibniz in the 17th century revolutionized mathematics. Leonhard Euler was the most notable mathematician of the 18th century, contributing numerous theorems and discoveries; the foremost mathematician of the 19th century was the German mathematician Carl Friedrich Gauss, who made numerous contributions to fields such as algebra, differential geometry, matrix theory, number theory, statistics. In the early 20th century, Kurt Gödel transformed mathematics by publishing his incompleteness theorems, which show that any axiomatic system, consistent will contain unprovable propositions. Mathematics has since been extended, there has been a fruitful interaction between mathematics and science, to