An alphabetical and subclassified list of mathematics articles is available. The following list of themes and links gives just one possible view. For a fuller treatment, see areas of mathematics or the list of mathematics lists. Mathematics - Quantity. This starts from explicit measurements of sizes of numbers or sets, or ways to find such measurements. See also:

Mathematics, 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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Cantor set - The Cantor set is uncountable. It can be shown that there are as many points left behind in this process as there were that were removed. To see this, we show that there is a function f from the Cantor set C to the closed interval [0,1] that is surjective (i.e. f maps from C onto [0,1]) so that the cardinality of C is no less than that of [0,1]. Since C is a subset of [0,1], its cardinality is also ...

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Cantor set, Cantor set - What's in the Cantor set?, Cantor set - Non-endpoints in the Cantor set, Cantor set - Properties, Cantor set - The Cantor set is uncountable, Cantor set - The Cantor set is a fractal, Cantor set - Topological and analytical properties, Cantor set - Variants of the Cantor set, Cantor set - Historical remarks, Cantor set - Historical references, Cantor set - Modern references

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The definable numbers form a field containing all the familiar real numbers such as 0, 1, π, e, et cetera. In particular, it contains all the numbers named in the mathematical constants article, and all algebraic numbers (and therefore all rational numbers). However, most real numbers are not definable: the set of all definable numbers is countably infinite (because the set of all logical formulas is) while the set of real numbers is uncountably infinite (see Cantor's diagonal argument). As a result, most real numbers have no description (in the same sense o ...

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Definable number, Definable number - General facts, Definable number - Notion does not exhaust unambiguously described numbers, Definable number - Other notions of definability, Definable number - Definability in other languages or structures, Definable number - Definability with ordinal parameters

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To state the hypothesis formally, we need a definition: we say that two sets S and T have the same cardinality or cardinal number if there exists a bijection . Intuitively, this means that it is possible to "pair off" elements of S with elements of T in such a fashion that every element of S is paired off with exactly one element of T and vice versa. Hence, the set {banana, apple, pear} has the same cardinality as {yellow, red, green}. With infinite sets such as the set of intege ...

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Continuum hypothesis, 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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The evolution of mathematics might be seen to be an ever-increasing series of abstractions, or alternatively an expansion of subject matter. The first abstraction was probably that of numbers. The realization that two apples and two oranges do have something in common, namely that they fill the hands of exactly one person, was a breakthrough in human thought. In addition to recognizing how to count concrete objects, prehistoric peoples also recognized how to count abstract quantities, like time -- days, seasons, years. Arithmetic (e.g., addition, subtraction, mul ...

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Mathematics, 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 - Mathematical tools, Mathematics - Common misconceptions

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Primitive recursive functions take natural numbers or tuples of natural numbers as arguments and produce a natural number. A function which takes n arguments is called n-ary. The basic primitive recursive functions are given by these axioms: The constant function 0 is primitive recursive. The successor function S, which takes one argument and returns the succeeding number as given by the Peano postulates, is primitive recursive. The projection functions Pin, which take n arguments and return ...

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Primitive recursive function, Primitive recursive function - Definition, Primitive recursive function - Examples, Primitive recursive function - Addition, Primitive recursive function - Subtraction, Primitive recursive function - Limitations, Primitive recursive function - Bibliography

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The elements of a finite set can be listed, say { a1, a2, ..., an }. However, insofar as a set is a logical description of the properties of its members, it need not be finite. To understand this, imagine that I ask you: how many words can you make out of Scrabble pieces if you are allowed to ask me for more pieces no matter how many you used up? The answer? As many as you like; you can go forever. But that doesn't mean they won't each of them be a word made out of scrabble ...

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Countable set, Countable set - Definition, Countable set - Gentle introduction, Countable set - A more formal introduction, Countable set - Further theorems about uncountable sets, Countable set - Mentionable and interesting numbers

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As argued above, many naïve objections depend on implicitly denying Hume's principle, and are therefore question-begging. Wittgenstein explicitly denies the principle, arguing that our concept of number depends essentially on counting. "Where the nonsense starts is with our habit of thinking of a large number as closer to infinity than a small one" The expressions "divisible into two parts" and "divisible without limit" have completely different forms. This is, of course, the same case as the one in which someone operat ...

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Controversy over Cantor's theory, Controversy over Cantor's theory - Preface, Controversy over Cantor's theory - Introduction, Controversy over Cantor's theory - Cantor's argument, Controversy over Cantor's theory - Reception of the argument, Controversy over Cantor's theory - Naïve objections, Controversy over Cantor's theory - Objections to Cantor's theorem, Controversy over Cantor's theory - Objections to Hume's principle, Controversy over Cantor's theory - Objection to the axiom of infinity, Controversy over Cantor's theory - Objections to the power set axiom, Controversy over Cantor's theory - Footnote

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Real number - Completeness. The main reason for introducing the reals is that the reals contain all limits. More technically, the reals are complete (in the sense of metric spaces or uniform spaces, which is a different sense than the Dedekind completeness of the order in the previous section). This means the following: A sequence (xn) of real numbers is called a Cauchy sequence if for any ε > 0 there exists an integer N (possibly depending on ε) such t ...

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Real number, Real number - History, Real number - Definition, Real number - Construction from the rational numbers, Real number - Axiomatic approach, Real number - Properties, Real number - Completeness, Real number - The complete ordered field, Real number - Advanced properties, Real number - Generalizations and extensions

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Georg Cantor introduced the concept of cardinality to compare the sizes of infinite sets. He famously showed that the set of real numbers is uncountably infinite; i.e. c is strictly greater than the cardinality of the natural numbers, ℵ0 (aleph-null) In other words, there are strictly more real numbers than there are integers. Cantor proved this statement in a couple of different ways. See Cantor's ...

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Cardinality of the continuum, Cardinality of the continuum - Properties, Cardinality of the continuum - The continuum hypothesis, Cardinality of the continuum - Sets with cardinality c

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The elements of a finite set can be listed, say { a1, a2, ..., an}. However, insofar as a set is a logical description of the properties of its members, it need not be finite. To understand this, imagine that I ask you: how many words can you make out of Scrabble pieces if you are allowed to ask me for more pieces no matter how many you used up? The answer? As many as you like; you can go forever. But that doesn't mean they won't each of them be a word made out of scrabble b ...

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

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Examples of impossible problems include the following constructions in Euclidean geometry using only a ruler and compass: Squaring the circle: Drawing a square having the same area as a given circle. Doubling the cube: Drawing a cube with twice the volume of a given cube. Trisecting the angle: Dividing a given angle into three smaller angles all of the same size. For 2,000 years people have tried and failed to find such constructions; the reasons were discovered in the 19th century, when it was pr ...

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Pseudomathematics, Pseudomathematics - Impossible problems, Pseudomathematics - Current trends in pseudomathematics

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The following dialogue is an example of reductio ad absurdum: A — You should respect C's belief, for all beliefs are of equal validity and cannot be denied. B — Isn't it right to deny D's belief? (where D believes something that is considered to be wrong by most people, such the earth's being flat) A — I agree it is right to deny D's belief. B — If it is right to deny D's belief, it is not true that no belief can be denied. Therefore, I can deny C's belief if I can giv ...

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Reductio ad absurdum, Reductio ad absurdum - In philosophy, Reductio ad absurdum - As a figure of speech, Reductio ad absurdum - In mathematics

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Constructivist mathematics use constructivist logic, which is essentially a removal of the law of the excluded middle from classical logic. This is not to say that the law of the excluded middle is denied entirely; special cases of the law will be provable as theorems. It is just that the law is not assumed as an axiom. (The law of non-contradiction, on the other hand, is still valid.) For instance, in Heyting arithmetic, one can prove that for any proposition p which does not contain quantifiers, is a theorem (where See also:

Constructivism mathematics, 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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Turing's original paper defined computable numbers as follows: A real number is computable if its digit sequence can be produced by some algorithm or Turing machine. The algorithm takes an integer as input and produces the n-th digit of the real number's decimal expansion as output. Turing was already aware of the fact that this definition is equivalent to the ε-approximation definition given above. The argument proceeds as follows: if a number is c ...

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Computable number, 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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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 positive even numbers has the same cardinality as the set N = {1, 2, 3, ...} of natural numbers, since the function f(n) = 2n is a bijection from N to E. We say that a set A has cardinality greater than or equal to the cardinality of B (and B has cardinality l ...

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

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In informal use, a cardinal number is what is normally referred to as a counting number. They may be identified with the natural numbers beginning with 0 (i.e. 0, 1, 2, ...). The counting numbers are exactly what can be defined formally as the finite cardinal numbers. Infinite cardinals only occur in higher-level mathematics and logic. More formally, a non-zero number can be used for two purposes: to describe the size of a set, or to describe the position of an element in a sequence. For finite sets and sequences it is e ...

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

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To state the hypothesis formally, we need a definition: we say that two sets S and T have the same cardinality or cardinal number if there exists a bijection . Intuitively, this means that it is possible to "pair off" elements of S with elements of T in such a fashion that every element of S is paired off with exactly one element of T and vice versa. Hence, the set {banana, appl ...

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Continuum hypothesis, 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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Ontological argument - Gaunilo's island. One of the earliest recorded objections to Anselm's argument was raised by one of Anselm's contemporaries, Gaunilo. Gaunilo invited his readers to think of the greatest, or most perfect, conceivable island. As a matter of fact, it is likely that no such island actually exists. However, his argument would then say that we aren't thinking of the greatest conceivable island, because the greatest conceivable island would exist, as well as having all ...

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Ontological argument, 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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"Kook" is a somewhat similar pejorative term that is usually used to describe a person whose areas of interest are perceived to be eccentric, fantastic, or insane. A person may be said to be a "kook" if they are seen to hold socially unacceptable beliefs, or perceptions that outrageously conflict with known scientific results, and appear to base their entire world views upon them. The term was coined in 1960 and originates from the word cuckoo, which is also the name of a bird, but which ...

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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

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Mathematics arises wherever there are difficult problems that involve quantity, structure, space, or change. At first these were found in commerce, land measurement and later astronomy; nowadays, all sciences suggest problems studied by mathematicians, and many problems arise within mathematics itself. Newton invented infinitesimal calculus and Feynman his Feynman path integral using a combination of reasoning and physical insight, and today's string theory also inspires new mathematics. Some mathematics is only relevant in the area that ins ...

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Mathematics, 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

Read more here: » Mathematics: Encyclopedia II - Mathematics - Inspiration, pure and applied mathematics, and aesthetics