Georg Cantor

Cantor was born in St Petersburg, Russia, the son of a Danish merchant, Georg Waldemar Cantor, and a Russian musician, Maria Anna Böhm. In 1856, the family moved to Germany and he continued his education in German schools, earning his doctorate from the University of Berlin in 1867. In 1890, he founded together with other mathematicians the Deutsche Mathematiker-Vereinigung and became the first president of the society. Cantor recognized that infinite sets can have different sizes, distinguished between countable and uncountab ...

See also:

Georg Cantor, Georg Cantor - Biography

Read more here: » Georg Cantor: Encyclopedia II - Georg Cantor - Biography

Cantor was born in St Petersburg, Russia, the son of a Danish merchant, Georg Waldemar Cantor, and a musician of German descent, Maria Anna Böhm. In 1856, the family moved to Germany and he continued his education in German schools, earning his doctorate from the University of Berlin in 1867. In 1890, he founded together with other mathematicians the Deutsche Mathematiker-Vereinigung and ...

See also:

Georg Cantor, Georg Cantor - Biography

Read more here: » Georg Cantor: Encyclopedia II - Georg Cantor - Biography

In the branch of mathematics known as set theory, the aleph numbers are a series of numbers used to represent the cardinality (or size) of infinite sets. They are named after the symbol used to denote them, the Hebrew letter aleph (). The cardinality of the natural numbers is aleph-null () (also aleph-naught, aleph-nought); the next larger cardinality is aleph-one , then and so on. Continuing in this manner, it is possible to define a cardinal number for every ordinal number α, as will be described below. The concept goes back to Georg Cantor, who defined the notion of cardinality and realized t ...

Including:

• Aleph number - Aleph-null
• Aleph number - Aleph-one
• Aleph number - The continuum hypothesis
• Aleph number - Aleph-ω
• Aleph number - Aleph-α for general α

Read more here: » Aleph number: Encyclopedia - Aleph number

The definition of h can be visualized with the following diagram. Displayed are parts of the (disjoint) sets A and B together with parts of the mappings f and g. If the set A ∪ B, together with the two maps, is interpreted as a directed graph, then this bipartite graph has several connected components. These can be divided into four types: paths extending infinitely to both directions, finite cycles of even length, infinite paths starting in the set A, and infini ...

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Cantor–Bernstein–Schroeder theorem, Cantor–Bernstein–Schroeder theorem - Visualization, Cantor–Bernstein–Schroeder theorem - Original proof

Read more here: » Cantor–Bernstein–Schroeder theorem: Encyclopedia II - Cantor–Bernstein–Schroeder theorem - Visualization

is the cardinality of the set of all countably infinite ordinal numbers, called ω1 or Ω. This definition implies (already in ZF, Zermelo-Fraenkel set theory without the axiom of choice) that no cardinal number is between and . If the axiom of choice (AC) is used, it can be further proved that the class of cardinal numbers is totally ordered, and thus is the second-smallest infinite cardinal number. Using AC we can show one of the most useful properties of the set Ω (the standard example of a set o ...

See also:

Aleph number, Aleph number - Aleph-null, Aleph number - Aleph-one, Aleph number - The continuum hypothesis, Aleph number - Aleph-ω, Aleph number - Aleph-α for general α, Aleph number - Fixed points of aleph, Aleph number - Popular culture

Read more here: » Aleph number: Encyclopedia II - Aleph number - Aleph-one

is the cardinality of the set of all countably infinite ordinal numbers, called ω1 or Ω. This definition implies (already in ZF, Zermelo-Fraenkel set theory without the axiom of choice) that no cardinal number is between and . If the axiom of choice (AC) is used, it can be further proved that the class of cardinal numbers is totally ordered, and thus is the second-smallest infinite cardinal number. Using AC we can show one of the most useful properties of the set Ω (the standard example of a set o ...

See also:

Aleph number, Aleph number - Aleph-null, Aleph number - Aleph-one, Aleph number - The continuum hypothesis, Aleph number - Aleph-ω, Aleph number - Aleph-α for general α

Read more here: » Aleph number: Encyclopedia II - Aleph number - Aleph-one

Charles Sanders Santiago Peirce (pronounced purse), (September 10, 1839, Cambridge, Massachusetts – April 19, 1914, Milford, Pennsylvania) was an American polymath. Although educated as a chemist and employed as a scientist for 30 years, he is now mostly seen as a philosopher. He is the greatest American builder of architectonic systems, and his admirers deem him the most important systemat ...

Including:

• Charles Peirce - Life
• Charles Peirce - Reception
• Charles Peirce - Works
• Charles Peirce - Peirce's philosophy
• Charles Peirce - Pragmatism
• Charles Peirce - Scholastic realism
• Charles Peirce - Formal perspective
• Charles Peirce - Dynamics of representation
• Charles Peirce - Normative sciences
• Charles Peirce - Parallels with Leibniz
• Charles Peirce - Bibliography
• Charles Peirce - Primary literature
• Charles Peirce - Secondary literature

Read more here: » Charles Peirce: Encyclopedia - Charles Peirce

1918 (MCMXVIII) was a common year starting on Tuesday of the Gregorian calendar (see link for calendar) or a common year starting on Wednesday of the Julian calendar. 1918 - Events. 1918 - January-February. January 8 - President Woodrow Wilson announces his "Fourteen Points" for the aftermath of World War I. January 22 - Manitoba, Canada film censor board bans comedies January 24 - a decree of the Council of People's Com ...

Including:

• 1918 - Events
• 1918 - January-February
• 1918 - March-April
• 1918 - May-July
• 1918 - August-October
• 1918 - November
• 1918 - December
• 1918 - Unknown dates
• 1918 - Births
• 1918 - January-February
• 1918 - March-April
• 1918 - May-August
• 1918 - September-December
• 1918 - Deaths
• 1918 - Nobel Prizes

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The Right Honourable Bertrand Arthur William Russell, 3rd Earl Russell, OM, FRS (18 May 1872 – 2 February 1970), was an influential British logician, philosopher, and mathematician, working mostly in the 20th century. A prolific writer, Bertrand Russell was also a populariser of philosophy and a commentator on a large variety of topics, ranging from very serious issues to the mundane. Continuing a family tradition in political affairs, he was a prominent liberal as well as a socialist and anti-war activist for most of his long life. ...

Including:

• Bertrand Russell - Biography
• Bertrand Russell - Russell's philosophical work
• Bertrand Russell - Analytic philosophy
• Bertrand Russell - Epistemology
• Bertrand Russell - Ethics
• Bertrand Russell - Logical atomism
• Bertrand Russell - Logic and mathematics
• Bertrand Russell - Philosophy of language
• Bertrand Russell - Philosophy of science
• Bertrand Russell - Religion and theology
• Bertrand Russell - Influence on philosophy
• Bertrand Russell - Russell's activism
• Bertrand Russell - Pacifism war and nuclear weapons
• Bertrand Russell - Communism and socialism
• Bertrand Russell - Women's suffrage
• Bertrand Russell - Sexuality
• Bertrand Russell - Eugenics and race
• Bertrand Russell - Russell summing up his life
• Bertrand Russell - Comments about Russell
• Bertrand Russell - As a man
• Bertrand Russell - As a philosopher
• Bertrand Russell - As a writer and his place in history
• Bertrand Russell - As a mathematician and logician
• Bertrand Russell - As an activist
• Bertrand Russell - As a recipient of the Nobel Prize for Literature
• Bertrand Russell - From a daughter
• Bertrand Russell - Quotes
• Bertrand Russell - Asides
• Bertrand Russell - Succession

Read more here: » Bertrand Russell: Encyclopedia - Bertrand Russell

Set theory is a branch of mathematics created principally by the German mathematician Georg Cantor at the end of the 19th century. Initially controversial, set theory has come to play the role of a foundational theory in modern mathematics, in the sense of a theory invoked to justify assumptions made in mathematics concerning the existence of mathematical objects (such as numbers or functions) and their properties. Formal versions of set theory also have a foundational role to play as specifying a theoretical ideal of mathematical rig ...

Including:

• Axiomatic set theory - The origins of rigorous set theory
• Axiomatic set theory - Axioms for set theory
• Axiomatic set theory - Independence in ZFC
• Axiomatic set theory - Set theory ZFC foundations for mathematics
• Axiomatic set theory - Well-foundedness and hypersets
• Axiomatic set theory - Objections to set theory

Read more here: » Axiomatic set theory: Encyclopedia - Axiomatic set theory

Canada - Mexico - South Africa - U.S. Rail Transport - Science - Sports Births - Deaths 1845 was a common year starting on Wednesday (see link for calendar). 1845 - Events. January 29 - The Raven by Edgar Allan Poe is published for the first time (New York Evening Mirror). February 7 - In the British Museum, drunken visitor smashes Portland Vase - it takes months to repair March 1 - President John Tyler signs a bill authorizing the United Sta ...

Including:

• 1845 - Events
• 1845 - Month/day unknown
• 1845 - Ongoing Events
• 1845 - Births
• 1845 - Deaths

Read more here: » 1845: Encyclopedia - 1845

Logic, from Classical Greek λόγος (logos), originally meaning the word, or what is spoken, (but coming to mean thought or reason) is most often said to be the study of arguments, although the exact definition of logic is a matter of controversy among philosophers. However the subject is grounded, the task of the logician is the same: to advance an account of valid and fallacious inference to allow ...

Including:

• Logic - Nature of logic
• Logic - Informal formal and symbolic logic
• Logic - Rival conceptions of logic
• Logic - History of logic
• Logic - Relation to other sciences
• Logic - Deductive and inductive reasoning
• Logic - Topics in logic
• Logic - Syllogistic logic
• Logic - Predicate logic
• Logic - Modal logic
• Logic - Deduction and reasoning
• Logic - Mathematical logic
• Logic - Philosophical logic
• Logic - Logic and computation
• Logic - Controversies in logic
• Logic - Bivalence and the law of the excluded middle
• Logic - Implication: strict or material?
• Logic - Tolerating the impossible
• Logic - Is logic empirical?

Read more here: » Logic: Encyclopedia - Logic

Infinity is a term with very distinct, separate meanings which arise in theology, philosophy, mathematics and everyday life. Popular or colloquial usage of the term often does not accord with its more technical meanings. The word infinity comes from Latin : "Infinito", unending. In theology, for example in the work of theologians such as Duns Scotus, the infinite nature of God invokes a sense of being without constraint, rather than a sense of being unlimited in quantity. In philosophy, infinity can be attrib ...

Including:

• Infinity - History
• Infinity - Ancient view of infinity
• Infinity - Views from the Renaissance to modern times
• Infinity - Modern philosophical views
• Infinity - Infinity symbol
• Infinity - Mathematical infinity
• Infinity - Infinity in real analysis
• Infinity - Infinity in complex analysis
• Infinity - Arithmetic properties of infinity
• Infinity - Infinity in set theory
• Infinity - Mathematics without infinity
• Infinity - Use of infinity in common speech
• Infinity - Physical infinity
• Infinity - Infinity in cosmology
• Infinity - Three types of infinities
• Infinity - Infinity in science fiction
• Infinity - Note

Read more here: » Infinity: Encyclopedia - Infinity

In mathematics, the continuum hypothesis is a hypothesis about the possible sizes of infinite sets. Georg Cantor introduced the concept of cardinality to compare the sizes of infinite sets, and he showed that the set of integers is strictly smaller than the set of real numbers. The continuum hypothesis states the following: There is no set whose size is strictly between that of the integers and that of the real numbers. Or mathematically speaking, noting that the cardinality for the integers is ("aleph-null") and the cardinality of the real numbers is , the continuum hypothesis says: ...

Including:

• Continuum hypothesis - The size of a set
• Continuum hypothesis - Impossibility of proof and disproof
• Continuum hypothesis - Arguments pro and con
• Continuum hypothesis - The generalized continuum hypothesis

Read more here: » Continuum hypothesis: Encyclopedia - Continuum hypothesis

In mathematics, and particularly in applications to set theory and the foundations of mathematics, a universe or universal class (or if a set, universal set) is, roughly speaking, a class that is large enough to contain (in some sense) all of the sets that one may wish to use. Universe mathematics - In a specific context. There are several precise versions of this general idea. Perhaps the simplest is that any set can be a universe, so long as you are studying that particular set. So if ...

Including:

• Universe mathematics - In a specific context
• Universe mathematics - In ordinary mathematics
• Universe mathematics - In set theory
• Universe mathematics - In category theory

Read more here: » Universe mathematics: Encyclopedia - Universe mathematics

White Light is a work of science fiction by Rudy Rucker published in 1980 by Ace Books. The novel deals mainly with the concept of infinity and relies heavily on principles of Set theory. White Light Rudy Rucker novel - Synopsis. The book is the story of a mathematics teacher at SUCAS (a state college in New York) named Felix Rayman, who starts experimenting with lucid dreaming. On one of these out of body experiences Felix loses his physical body and is forced to travel into another dimensio ...

Including:

• White Light Rudy Rucker novel - Synopsis
• White Light Rudy Rucker novel - White Light and Transrealism

Read more here: » White Light Rudy Rucker novel: Encyclopedia - White Light Rudy Rucker novel

John von Neumann (Neumann János) (December 28, 1903 – February 8, 1957) was a Jewish Hungarian-born mathematician and polymath who made important contributions in quantum physics, functional analysis, set theory, computer science, economics and many other mathematical fields. Most notably, von Neumann was a pioneer of the modern digital computer and the application of operator theory to quantum mechanics (see Von Neumann algebra), member of the Manhattan Project Team, creator of game theory and the concept of cellular automata. ...

Including:

• John von Neumann - Biography
• John von Neumann - Logic
• John von Neumann - Quantum Mechanics
• John von Neumann - Economics
• John von Neumann - Armaments
• John von Neumann - Computer Science
• John von Neumann - Politics and Social Affairs
• John von Neumann - Honors
• John von Neumann - Students

Read more here: » John von Neumann: Encyclopedia - John von Neumann

Cantor's diagonal argument is a proof devised by Georg Cantor to demonstrate that the real numbers are not countably infinite. (It is also called the diagonalization argument or the diagonal slash argument or the diagonal method.) The diagonal argument was not Cantor's first proof of the uncountability of the real numbers, but was published three years after his first proof. His original argument did not mention decimal expansions, nor any other numeral system. Since this technique was first used, si ...

Including:

• Cantor's diagonal argument - Real numbers
• Cantor's diagonal argument - Why this does not work on integers
• Cantor's diagonal argument - General sets

Read more here: » Cantor's diagonal argument: Encyclopedia - Cantor's diagonal argument

To define aleph-α for arbitrary ordinal number α, we need the successor cardinal operation, which assigns to any cardinal number ρ the next bigger cardinal ρ + . We can then define the aleph numbers as follows and for λ, an infinite limit ordinal, ...

See also:

Aleph number, Aleph number - Aleph-null, Aleph number - Aleph-one, Aleph number - The continuum hypothesis, Aleph number - Aleph-ω, Aleph number - Aleph-α for general α, Aleph number - Fixed points of aleph, Aleph number - Popular culture

Read more here: » Aleph number: Encyclopedia II - Aleph number - Aleph-α for general α

Conventionally the smallest infinite ordinal is denoted ω, and the cardinal number is the smallest upper bound of Aleph-ω is the first uncountable cardinal number that can be demonstrated within Zermelo-Fraenkel set theory not to be equal to the cardinality of the set of all real numbers; for any positive integer n we can consistently assume that , and moreover it is possible to assume is as large as we like. We are only forced to avoid setting it to certain special cardinals with cofinality , meaning th ...

See also:

Aleph number, Aleph number - Aleph-null, Aleph number - Aleph-one, Aleph number - The continuum hypothesis, Aleph number - Aleph-ω, Aleph number - Aleph-α for general α, Aleph number - Fixed points of aleph, Aleph number - Popular culture

Read more here: » Aleph number: Encyclopedia II - Aleph number - Aleph-ω