Cantor's diagonal argument

Gödel believed strongly that CH is false. To him, his independence proof only showed that the prevalent set of axioms was defective. Gödel was a platonist and therefore had no problems with asserting truth and falsehood of statements independent of their provability. Cohen, however, was a formalist, but even he tended towards rejecting CH. Historically, mathematicians who favor a "rich" and "large" universe of sets were against CH, while those favoring a "neat" and "controllable" universe favored CH. More recently, some experts (e.g ...

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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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Here's a short, and very general description of the ontological argument: 1) God is the greatest possible being and thus possesses all perfections. 2) Existence is a perfection. 3) God exists. This is a shorter modern version of the argument. Anselm framed the argument as a reductio ad absurdum wherein he tried to show that the assumption that God does not exist leads to a logical contradiction. The following steps more closely follow Anselm's line of reasoning: 1) God is the entity t ...

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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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Most of the mathematical notation we use today was not invented until the 16th Century. Before that, mathematics was written out in words, a painstaking process that limited mathematical discovery. Modern notation makes mathematics much easier for the professional, but beginners often find it daunting. It is extremely compressed: a few symbols contain a great deal of information. Like musical notation, modern mathematical notation has a strict grammar (under the influence of ...

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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

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Among some people, there is a misconception that reductio ad absurdum just means "a silly argument". In general practice, a reductio ad absurdum is a tactic in which the logic of an argument is challenged by reducing the concept to its most absurd extreme. It is thus often similar in nature to the slippery slope argument. For example: A — I don't think the police should arrest teenagers for soft drug possession. B — So, you are basically arguing the police should not enforce the law ...

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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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The computable complex numbers form an algebraically closed field, and for many purposes is large enough already without requiring the noncomputable construction of the real and complex numbers. It contains all algebraic numbers as well as many known transcendental mathematical constants. There are however many real numbers which are not computable: the set of all computable numbers is countable (because the set of algorithms is) while the set of real ...

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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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A real number a is said to be computable if it can be approximated by some algorithm (or Turing machine), in the following sense: given any integer , the algorithm produces an integer k such that: Or, equivalently, there exists an algorithm which, given any real error bound ε > 0, produces a rational number r such that: A complex number is called com ...

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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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Note that, up until this point, we have only defined the term "cardinality" in a strictly functional role: we have not actually defined the "cardinality" of a set as a specified object itself. We now outline such an approach. The relation of having the same cardinality is called equinumerosity, and this is an equivalence relation on the class of all sets. The equivalence class of a set A under this relation then consists of all those sets which have the same cardinality as A. There are then two main approaches to the def ...

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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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Such a property allows for the comparison of how many elements are contained in two or more sets without resorting to an intermediate set (viz. the natural numbers). Within the realm of uncountable sets, there exists a class of sets Y such that | Y | = c (cardinality of set of real numbers). Such sets are said to have "cardinality of the continuum." It can be proven that there exists no set X such that for any set Y, | Y | â ...

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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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Traditionally, mathematicians have been suspicious, if not downright antagonistic, towards mathematical constructivism, largely because of the limitations that it poses for constructive analysis. These views were forcefully expressed by David Hilbert in 1928, when he wrote in Die Grundlagen der Mathematik, "Taking the principle of excluded middle from the mathematician would be the same, say, as proscribing the telescope to the astronomer or to the boxer the use of his fists" [1]. (The law of excluded middle is not valid in cons ...

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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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Mathematics is not a closed intellectual system, in which everything has already been worked out. There is no shortage of open problems. Pseudomathematics is a form of mathematics-like activity undertaken outside academia, and occasionally by mathematicians themselves. It often consists of determined attacks on famous questions, consisting of proof-attempts made in an isolated way (that is, long papers not supported by previously published theory). The relationship to generally-accepted mathematics is similar to that between pseudosci ...

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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

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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

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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 - Common misconceptions

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Carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. If one considers science to be strictly about the physical world, then mathematics, or at least pure mathematics, is not a science. An alternative view is that certain scientific fields (such as theoretical physics) are mathematics with axioms that are intended to correspond to reality. In fact, the theoretical physicist, J. M. Ziman, proposed that science is < ...

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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

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As noted above, the major disciplines within mathematics first arose out of the need to do calculations in commerce, to understand the relationships between numbers, to measure land, and to predict astronomical events. These four needs can be roughly related to the broad subdivision of mathematics into the study of quantity, structure, space, and change (i.e. arithmetic, algebra, geometry and analysis). In addition to these main concerns, there are also subdivisions dedicated to exploring links from the heart of mathematics to other fields: to logic, to set theory (foundations) and to the emp ...

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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

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It is interesting to note that Gödel believed strongly that CH is false. To him, his independence proof only showed that the prevalent set of axioms was defective. Gödel was a platonist and therefore had no problems with asserting truth and falsehood of statements independent of their provability. Cohen, however, was a formalist, but even he tended towards rejecting CH. Historically, mathematicians who favor a "rich" and "large" universe of sets were against CH, while those favoring a "neat" and "controllable" universe favored CH. M ...

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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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We can define arithmetic operations on cardinal numbers that generalize the ordinary operations for natural numbers. If X and Y are disjoint, addition is given by the union of X and Y |X| + |Y| = |X ∪ Y| The product of cardinals by the cartesian product |X| |Y| = |X × Y| Exponentiation is given by |X| ...

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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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The famous continuum hypothesis asserts that c is also the first aleph number ℵ1. In other words, the continuum hypothesis states that there is no set A whose cardinality lies strictly between ℵ0 and c However, this statement is now known to be independent of the axioms of Zermelo-Fraenkel set theory (ZFC). That is, both the hypothesis and its negation are consistent with these axioms. In fact, for every nonzero natural number n, the equality c = ℵ_{See also:}

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 generalized continuum hypothesis (GCH) states that if an infinite set's cardinality lies between that of an infinite set S and that of the power set of S, then it either has the same cardinality as the set S or the same cardinality as the power set of S. That is, for any infinite cardinal λ there is no cardinal κ such that λ < κ < 2λ. An equivalent condition is that for every ordinal α. Another equ ...

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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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Since the Cantor set is defined as the set of points not excluded, the proportion of the unit interval remaining can be found by total length removed. This total is the geometric series So that the proportion left is 1 – 1 = 0. Alternatively, it can be observed that each step leaves 2/3 of the length in the previous stage, so that the amount remaining is 2/3 × 2/3 × 2/3 ...

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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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Cantor believed the continuum hypothesis to be true and tried for many years to prove it, in vain. It became the first on David Hilbert's list of important open questions that was presented at the International Mathematical Congress in the year 1900 in Paris. Kurt Gödel showed in 1940 that the continuum hypothesis (CH for short) cannot be disproved from the standard Zermelo-Fraenkel set theory, even if the axiom of choice is adopted. Paul Cohen showed in 1963 that CH cannot be proven from those same axioms either. Hence, CH is ind ...

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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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Instead of repeatedly removing the middle third of every piece as in the Cantor set, we could also keep removing any other fixed percentage (other than 0% and 100%) from the middle. The resulting sets are all homeomorphic to the Cantor set and also have Lebesgue measure 0. In the case where the middle 8/10 of the interval is removed, we get a remarkably accessible case — the set consists of all numbers in [0,1] that can be written as a decimal consisting entirely of 0's and 9's. By removing progressively smaller percentages of the r ...

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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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Obviously Anselm thought this argument was valid and persuasive, and it still has occasional defenders, but many, perhaps most, contemporary philosophers believe that the ontological argument, at least as Anselm articulated it, does not stand up to strict logical scrutiny. Some of those who have argued that the ontological argument fails are content to leave it at that, either because they do not believe that God exists, or because they believe the existence of God is demonstrated on other grounds. Others, like Gottfried Leibniz, Norman Malcolm, Charles Hartshorne, Kurt Gödel and Alvin Plantinga have ...

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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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