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

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Cantor's original proof shows that the interval [0,1] is not countably infinite. The proof by contradiction proceeds as follows: Assume (for the sake of argument) that the interval [0,1] is countably infinite. We may then enumerate all numbers in this interval as a sequence, ( r1, r2, r3, ... ) We already know that each of these numbers may be represented as a decimal expansion. We arrange the numbers in a list (they do not need to be in orde ...

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Cantor's diagonal argument, 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 II - Cantor's diagonal argument - Real numbers

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

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The important idea of Cantor's, which got set theory going as a new field of study, was to define two sets A and B to have the same number of members (the same cardinality) when there is a way of pairing off members of A exhaustively with members of B. Then the set N of natural numbers has the same cardinality as the set Q of rational numbers (they are both said to be countably infinite), even though N is a proper subset of Q. On the other hand, the set R of real numbers d ...

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Axiomatic set theory, 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

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In linguistics, cardinal numbers is the name given to number words that are used for quantity (one, two, three), as opposed to ordinal numbers, words that are used for order (first, second, third). See names of numbers in English. In mathematics, cardinal numbers, or cardinals for short, are a generalized kind of number used to denote the size of a set. While for finite sets the size is given by a natural number, the number of elements, cardinal numbers (cardinality ...

Including:

• Cardinal number - History
• Cardinal number - Motivation
• Cardinal number - Formal definition
• Cardinal number - Cardinal arithmetic
• Cardinal number - The continuum hypothesis

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"Crank" (or kook, crackpot, or quack) is a pejorative term for a person who writes or speaks in an authoritative fashion about a particular subject, often of a scientific or pseudo-scientific nature, but is perceived as holding false or even ludicrous beliefs. Crank is also used as a noun to describe the opinions of such people (see American Heritage Dictionary 2000 - noun definition 3). Usage of the label is often subjective, with proponents of competing theories labeling each other cranks, but the term pr ...

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• Crank person - Crank tactics
• Crank person - Cranks on the Internet
• Crank person - Related terminology
• Crank person - Topics typically associated with the crank label
• Crank person - Physics computer science and mathematics
• Crank person - Medicine
• Crank person - Politics economics and law
• Crank person - Paranormal and spiritual

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In mathematics, the cardinality of a set is a measure of the "number of elements of the set". There are two approaches to cardinality – one which compares sets directly using bijections, injections, and surjections, and another which uses cardinal numbers. Cardinality - Comparing sets. We say that two sets A and B have the same cardinality if there exists a bijection, i.e. a injective and surjective function, from A to B. For example, the set E = {2, 4, 6, ...} of positi ...

Including:

• Cardinality - Comparing sets
• Cardinality - Countable and uncountable sets
• Cardinality - Cardinal numbers
• Cardinality - Examples and other properties

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In theology and the philosophy of religion, an ontological argument for the existence of God is an argument that God's existence can be proved a priori, that is, by intuition and reason alone. In the context of the Abrahamic religions, it was first proposed by the medieval philosopher Anselm of Canterbury in his Proslogion, and important variations have been developed by philosophers such as René Descartes, Gottfried Leibniz, Norman Malcolm, Charles Hartshorne, and Alvin Plantinga. A modal logic versi ...

Including:

• Ontological argument - Anselm's argument
• Ontological argument - Philosophical assumptions underlying the argument
• Ontological argument - A modern description of the argument
• Ontological argument - Criticisms and Objections
• Ontological argument - Gaunilo's island
• Ontological argument - Necessary nonexistence
• Ontological argument - Existence as a property
• Ontological argument - Miscellaneous
• Ontological argument - Revisionists
• Ontological argument - Descartes' ontological arguments
• Ontological argument - Plantinga's modal form and contemporary discussion
• Ontological argument - Bibliography

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In mathematics, an uncountable or nondenumerable set is a set which is not countable. Here, "countable" means countably infinite or finite, so by definition, all uncountable sets are infinite. Explicitly, a set X is uncountable if and only if there does not exist a surjective function from the natural numbers N to X. Not all uncountable sets have the same size; the sizes of infinite sets are analyzed with the theory of cardinal numbers. Formally, an uncountable set is defined as one whose cardinality is strictly greater than ...

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In mathematics, theoretical computer science and mathematical logic, the computable numbers, also known as the recursive numbers, are the subset of the real numbers consisting of the numbers which can be computed by a finite, terminating algorithm. They can be defined equivalently using the axioms of recursive functions, Turing machines or lambda-calculus. In contrast, the reals require the more powerful axioms of Zermelo-Fraenkel set theory. The computable numbers form a real closed field and can be used in the place of real numbe ...

Including:

• Computable number - Formal definition
• Computable number - Properties
• Computable number - Computing digit strings
• Computable number - Uncomputable numbers
• Computable number - Can computable numbers be used instead of the reals?

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In the philosophy of mathematics, constructivism asserts that it is necessary to find (or "construct") a mathematical object to prove that it exists. When one assumes that an object does not exist and derives a contradiction from that assumption, one still has not found the object and therefore not proved its existence, according to constructivists. See constructive proof. Constructivism is often confused with intuitionism, but in fact, intuitionism is only one kind of constructivism. Intuitionism maintains that the foundations ...

Including:

• Constructivism mathematics - Constructivist mathematics
• Constructivism mathematics - Example from real analysis
• Constructivism mathematics - Cardinality
• Constructivism mathematics - Attitude of mathematicians
• Constructivism mathematics - Mathematicians who have contributed to constructivism
• Constructivism mathematics - Branches

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Since its inception, there have been some mathematicians who have objected to using set theory as a foundation for mathematics, claiming that it is just a game which includes elements of fantasy. Notably, Henri Poincaré is supposed to have said "set theory is a disease from which mathematics will one day recover", (this quotation is part of the folklore of mathematics; the original source is unknown) and Errett Bishop dismissed set th ...

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Axiomatic set theory, 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

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In 1917, Dmitry Mirimanov (also spelled Mirimanoff) introduced the concept of well-foundedness: a set, x0, is well founded iff it has no infinite descending membership sequence: · · · In ZFC, there is no infinite descending ∈-sequence by the axiom of regularity (for a proof see Axiom of regularity). In fact, the axiom of regularity is often called the foundation axiom since it can be proved within ZFC- (that is, ZFC wit ...

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Axiomatic set theory, 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 II - Axiomatic set theory - Well-foundedness and hypersets

From these initial axioms for sets one can construct all other mathematical concepts and objects: number - discrete and continuous, order, relation, function , etc. For example, whilst the elements of a set have no intrinsic ordering it is possible to construct models of ordered lists. The essential step is to be able to model the ordered pair ( a, b ) which represents the pairing of two objects in this order. The defining property of an ordered pair is that ( a, b ) = ( c, d ) if and only if a = c and b = d. The approach is basically to specify th ...

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Axiomatic set theory, 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 II - Axiomatic set theory - Set theory ZFC foundations for mathematics

The axioms for set theory now most often studied and used, although put in their final form by Skolem, are called the Zermelo-Fraenkel set theory (ZF). Actually, this term usually excludes the axiom of choice, which was once more controversial than it is today. When this axiom is included, the resulting system is called ZFC. An important feature of ZFC is that every object that it deals with is a set. In particular, every element of a set is itself a set. Other familiar mathematical objects, s ...

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Axiomatic set theory, 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 II - Axiomatic set theory - Axioms for set theory

In mathematics the term countable is used to describe the size of a set, i.e. the number of elements it contains. The notion of an infinite set is not elementary; it requires a strong sense of abstraction and precision. A set is called countable if the number of elements is finite or if it has the same number of elements as the natural numbers. (Cantor defined a countable set as a set which can be put into one-to-one correspondence with a subset of the natural numbers). The term countable stems from the fac ...

Including:

• Countable set - Definition
• Countable set - Gentle introduction
• Countable set - A more formal introduction
• Countable set - Further theorems about uncountable sets

Read more here: » Countable set: Encyclopedia - Countable set

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

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Many important statements are independent of ZFC, see the list of statements undecidable in ZFC. The independence is usually proved by forcing, that is, it is shown that every countable transitive model of ZFC (plus, occasionally, large cardinal axioms) can be expanded to satisfy the statement in question, and (through a different expansion) its negation. An independence proof by forcing automatically proves independence from arithmetical statements, other concrete statements, and large cardinal axioms. Some statements independent of ZFC can ...

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Axiomatic set theory, 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 II - Axiomatic set theory - Independence in ZFC

Mathematics is often defined as the study of topics such as quantity, structure, space, and change. Another view, held by many mathematicians, is that mathematics is the body of knowledge justified by deductive reasoning, starting from axioms and definitions. Practical mathematics, in nearly every society, is used for such purposes as accounting, measuring land, or predicting astronomical events. Mathematical discovery or research often involves discovering and cataloging patterns, without regard for application. Today, the natural sciences, engineering, economics, and medici ...

Including:

• Mathematics - History
• Mathematics - Inspiration pure and applied mathematics and aesthetics
• Mathematics - Notation language and rigor
• Mathematics - Is mathematics a science?
• Mathematics - Overview of fields of mathematics
• Mathematics - Major themes in mathematics
• Mathematics - Quantity
• Mathematics - Structure
• Mathematics - Space
• Mathematics - Change
• Mathematics - Foundations and methods
• Mathematics - Discrete mathematics
• Mathematics - Applied mathematics
• Mathematics - Important theorems
• Mathematics - Important conjectures
• Mathematics - History and the world of mathematicians
• Mathematics - Mathematics and other fields
• Mathematics - Common misconceptions

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Crank person - Physics computer science and mathematics. Claims to have produced solutions to problems which have been proven to be unsolvable, such as the geometric construction problems of squaring the circle, doubling the cube and trisecting the angle. (It should be noted that all of these problems have solutions if one is permitted tools beyond a straightedge and compass). producing unified Theories of Everything, and particularly doing so with high school or undergraduate level physics knowled ...

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Crank person, Crank person - Crank tactics, Crank person - Cranks on the Internet, Crank person - Related terminology, Crank person - Topics typically associated with the crank label, Crank person - Physics computer science and mathematics, Crank person - Medicine, Crank person - Politics economics and law, Crank person - Paranormal and spiritual

Read more here: » Crank person: Encyclopedia II - Crank person - Topics typically associated with the crank label