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Cantor's diagonal argument | A Wisdom Archive on Cantor's diagonal argument | | Cantor's diagonal argument A selection of articles related to Cantor's diagonal argument | |
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Cantor's diagonal argument
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| ARTICLES RELATED TO Cantor's diagonal argument | | | |  Cantor's diagonal argument: Encyclopedia II - Real number - Definition
Real number - Construction from the rational numbers.
The real numbers can be constructed as a completion of the rational numbers. For details and other construction of real numbers, see construction of real numbers.
Real number - Axiomatic approach.
Let R denote the set of all real numbers. Then:
The set R is a field, meaning that addition and multiplication are defined and have the usual properties.
The field R is ordered, meaning th ...
See also: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 Read more here: » Real number: Encyclopedia II - Real number - Definition |
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| | | | | Cantor's diagonal argument: Encyclopedia II - Cardinal number - History The cardinal numbers were invented by Georg Cantor, when he was developing the set theory now called naive set theory in 1874–1884.
He first established cardinality as an instrument to compare finite sets; e.g. the sets {1,2,3} and {2,3,4} are not equal, but have the same cardinality, namely three.
Cantor invented the one-to-one correspondence, which easily showed that two finite sets had the same cardinality if there was a one-to-one correspondence between the members of the set. Using this one-to-one correspon ...
See also:Cardinal number, Cardinal number - History, Cardinal number - Motivation, Cardinal number - Formal definition, Cardinal number - Cardinal arithmetic, Cardinal number - The continuum hypothesis Read more here: » Cardinal number: Encyclopedia II - Cardinal number - History |
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| | | | | | Cantor's diagonal argument: Encyclopedia II - Ontological argument - Plantinga's modal form and contemporary discussion Alvin Plantinga has given us a another version of the argument, one where the conclusion follows from the premises, assuming axiom S5 of modal logic. A version of his argument is as follows:
By definition a maximally great being is one that exists necessarily and necessarily is omniscient, omnipotent and perfectly good. (Premise)
Possibly a maximally great being exists. (Premise)
Therefore, possibly it is necessarily true that an omniscient, omnipotent and perfectly good being exists (By 1 and 2)
There ...
See also: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 Read more here: » Ontological argument: Encyclopedia II - Ontological argument - Plantinga's modal form and contemporary discussion |
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| | | | | Cantor's diagonal argument: Encyclopedia II - Ontological argument - Descartes' ontological arguments Descartes composed a number of ontological arguments which differed from Anselm's formulation in important ways. Generally speaking, it is less a formal argument than a natural intuition.
Descartes wrote in the Fifth Meditation:
But if the mere fact that I can produce from my thought the idea of something entails that everything which I clearly and distinctly perceive to belong to that thing really does belong to it, is not this a possible basis for another argument to prove the existence of God? Certainly, the idea of Go ...
See also: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 Read more here: » Ontological argument: Encyclopedia II - Ontological argument - Descartes' ontological arguments |
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| | | | | | Cantor's diagonal argument: Encyclopedia II - Controversy over Cantor's theory - Preface The pure mathematicians and applied mathematicians who object to Cantor's theory of sets claim that Cantor introduced into mathematics an element of fantasy that should be expunged. The basic "anti-Cantorian" argument was stated most elegantly and concisely by Hermann Weyl when he wrote:
...classical logic was abstracted from the mathematics of finite sets and their subsets...Forgetful of this limited origin, one afterwards mistook that logic for something above and prior to all mathematics, and finally applied it, without justification, to the mathematics of infinite sets. This is the Fall ...
See also: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 Read more here: » Controversy over Cantor's theory: Encyclopedia II - Controversy over Cantor's theory - Preface |
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