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Absolutely Infinite
The Absolute Infinite (''symbol'': Ω) is an extension of the idea of infinity proposed by mathematician Georg Cantor. It can be thought of as a number that is bigger than any other conceivable or inconceivable quantity, either finite or transfinite. Cantor linked the Absolute Infinite with God, Cited as ''Cantor 1883b'' by Jané; with biography by Adolf Fraenkel; reprinted Hildesheim: Georg Olms, 1962, and Berlin: Springer-Verlag, 1980, . Original article. and believed that it had various mathematical properties, including the reflection principle: every property of the Absolute Infinite is also held by some smaller object.''Infinity: New Research and Frontiers'' by Michael Heller and W. Hugh Woodin (2011)p. 11 Cantor's view Cantor said: Cantor also mentioned the idea in his letters to Richard Dedekind (text in square brackets not present in original): The Burali-Forti paradox The idea that the collection of all ordinal numbers cannot logically exist seems paradoxic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Infinity
Infinity is that which is boundless, endless, or larger than any natural number. It is often denoted by the infinity symbol . Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions among philosophers. In the 17th century, with the introduction of the infinity symbol and the infinitesimal calculus, mathematicians began to work with infinite series and what some mathematicians (including l'Hôpital and Bernoulli) regarded as infinitely small quantities, but infinity continued to be associated with endless processes. As mathematicians struggled with the foundation of calculus, it remained unclear whether infinity could be considered as a number or magnitude and, if so, how this could be done. At the end of the 19th century, Georg Cantor enlarged the mathematical study of infinity by studying infinite sets and infinite numbers, showing that they can be of various sizes. For example, if a line is viewed as the set of all o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zermelo Set Theory
Zermelo set theory (sometimes denoted by Z-), as set out in a seminal paper in 1908 by Ernst Zermelo, is the ancestor of modern Zermelo–Fraenkel set theory (ZF) and its extensions, such as von Neumann–Bernays–Gödel set theory (NBG). It bears certain differences from its descendants, which are not always understood, and are frequently misquoted. This article sets out the original axioms, with the original text (translated into English) and original numbering. The axioms of Zermelo set theory The axioms of Zermelo set theory are stated for objects, some of which (but not necessarily all) are sets, and the remaining objects are urelements and not sets. Zermelo's language implicitly includes a membership relation ∈, an equality relation = (if it is not included in the underlying logic), and a unary predicate saying whether an object is a set. Later versions of set theory often assume that all objects are sets so there are no urelements and there is no need for the unary p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
The Ultimate (philosophy)
Georg Wilhelm Friedrich Hegel (; ; 27 August 1770 – 14 November 1831) was a German philosopher. He is one of the most important figures in German idealism and one of the founding figures of modern Western philosophy. His influence extends across the entire range of contemporary philosophical topics, from metaphysical issues in epistemology and ontology, to political philosophy, the philosophy of history, philosophy of art, philosophy of religion, and the history of philosophy. Born in 1770 in Stuttgart during the transitional period between the Enlightenment and the Romantic movement in the Germanic regions of Europe, Hegel lived through and was influenced by the French Revolution and the Napoleonic wars. His fame rests chiefly upon ''The Phenomenology of Spirit'', ''The Science of Logic'', and his lectures at the University of Berlin on topics from his ''Encyclopedia of the Philosophical Sciences''. Throughout his work, Hegel strove to address and correct the problemat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reflection Principle
In set theory, a branch of mathematics, a reflection principle says that it is possible to find sets that resemble the class of all sets. There are several different forms of the reflection principle depending on exactly what is meant by "resemble". Weak forms of the reflection principle are theorems of ZF set theory due to , while stronger forms can be new and very powerful axioms for set theory. The name "reflection principle" comes from the fact that properties of the universe of all sets are "reflected" down to a smaller set. Motivation A naive version of the reflection principle states that "for any property of the universe of all sets we can find a set with the same property". This leads to an immediate contradiction: the universe of all sets contains all sets, but there is no set with the property that it contains all sets. To get useful (and non-contradictory) reflection principles we need to be more careful about what we mean by "property" and what properties we allow. T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monadology
The ''Monadology'' (french: La Monadologie, 1714) is one of Gottfried Leibniz's best known works of his later philosophy. It is a short text which presents, in some 90 paragraphs, a metaphysics of simple substances, or '' monads''. Text During his last stay in Vienna from 1712 to September 1714, Leibniz wrote two short texts in French which were meant as concise expositions of his philosophy. After his death, ''Principes de la nature et de la grâce fondés en raison'', which was intended for prince Eugene of Savoy, appeared in French in the Netherlands. Christian Wolff and collaborators published translations in German and Latin of the second text which came to be known as ''The Monadology''. Without having seen the Dutch publication of the ''Principes'' they had assumed that it was the French original of the ''Monadology,'' which in fact remained unpublished until 1840. The German translation appeared in 1720 as ''Lehrsätze über die Monadologie'' and the following year ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Limitation Of Size
In the philosophy of mathematics, specifically the philosophical foundations of set theory, limitation of size is a concept developed by Philip Jourdain and/or Georg Cantor to avoid Cantor's paradox. It identifies certain "inconsistent multiplicities", in Cantor's terminology, that cannot be sets because they are "too large". In modern terminology these are called proper classes. Use The axiom of limitation of size is an axiom in some versions of von Neumann–Bernays–Gödel set theory In the foundations of mathematics, von Neumann–Bernays–Gödel set theory (NBG) is an axiomatic set theory that is a conservative extension of Zermelo–Fraenkel–choice set theory (ZFC). NBG introduces the notion of class, which is a colle ... or Morse–Kelley set theory. This axiom says that any class that is not "too large" is a set, and a set cannot be "too large". "Too large" is defined as being large enough that the class of all sets can be mapped one-to-one into it. References * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Actual Infinity
In the philosophy of mathematics, the abstraction of actual infinity involves the acceptance (if the axiom of infinity is included) of infinite entities as given, actual and completed objects. These might include the set of natural numbers, extended real numbers, transfinite numbers, or even an infinite sequence of rational numbers. Actual infinity is to be contrasted with potential infinity, in which a non-terminating process (such as "add 1 to the previous number") produces a sequence with no last element, and where each individual result is finite and is achieved in a finite number of steps. As a result, potential infinity is often formalized using the concept of Limit (mathematics), limit. Anaximander The ancient Greek term for the potential or improper infinite was ''Apeiron (cosmology), apeiron'' (unlimited or indefinite), in contrast to the actual or proper infinite ''aphorismenon''. ''Apeiron'' stands opposed to that which has a ''peras'' (limit). These notions are tod ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Willard Van Orman Quine
Willard Van Orman Quine (; known to his friends as "Van"; June 25, 1908 – December 25, 2000) was an American philosopher and logician in the analytic tradition, recognized as "one of the most influential philosophers of the twentieth century". From 1930 until his death 70 years later, Quine was continually affiliated with Harvard University in one way or another, first as a student, then as a professor. He filled the Edgar Pierce Chair of Philosophy at Harvard from 1956 to 1978. Quine was a teacher of logic and set theory. Quine was famous for his position that first order logic is the only kind worthy of the name, and developed his own system of mathematics and set theory, known as New Foundations. In philosophy of mathematics, he and his Harvard colleague Hilary Putnam developed the Quine–Putnam indispensability argument, an argument for the reality of mathematical entities.Colyvan, Mark"Indispensability Arguments in the Philosophy of Mathematics" The Stanford Encyclopedi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
New Foundations
In mathematical logic, New Foundations (NF) is an axiomatic set theory, conceived by Willard Van Orman Quine as a simplification of the theory of types of ''Principia Mathematica''. Quine first proposed NF in a 1937 article titled "New Foundations for Mathematical Logic"; hence the name. Much of this entry discusses NFU, an important variant of NF due to Jensen (1969) and clarified by Holmes (1998). In 1940 and in a revision in 1951, Quine introduced an extension of NF sometimes called "Mathematical Logic" or "ML", that included proper classes as well as sets. New Foundations has a universal set, so it is a non-well-founded set theory. That is to say, it is an axiomatic set theory that allows infinite descending chains of membership, such as … xn ∈ xn-1 ∈ … ∈ x2 ∈ x1. It avoids Russell's paradox by permitting only stratifiable formulas to be defined using the axiom schema of comprehension. For instance, x ∈ y is a stratifiable formula, but x ∈ x is not. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Set Theory
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concerned with those that are relevant to mathematics as a whole. The modern study of set theory was initiated by the German mathematicians Richard Dedekind and Georg Cantor in the 1870s. In particular, Georg Cantor is commonly considered the founder of set theory. The non-formalized systems investigated during this early stage go under the name of '' naive set theory''. After the discovery of paradoxes within naive set theory (such as Russell's paradox, Cantor's paradox and the Burali-Forti paradox) various axiomatic systems were proposed in the early twentieth century, of which Zermelo–Fraenkel set theory (with or without the axiom of choice) is still the best-known and most studied. Set theory is commonly employed as a foundational ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proper Class
Proper may refer to: Mathematics * Proper map, in topology, a property of continuous function between topological spaces, if inverse images of compact subsets are compact * Proper morphism, in algebraic geometry, an analogue of a proper map for algebraic varieties * Proper transfer function, a transfer function in control theory in which the degree of the numerator does not exceed the degree of the denominator * Proper equilibrium, in game theory, a refinement of the Nash equilibrium * Proper subset * Proper space * Proper complex random variable Other uses * Proper (liturgy), the part of a Christian liturgy that is specific to the date within the Liturgical Year * Proper frame, such system of reference in which object is stationary (non moving), sometimes also called a co-moving frame * Proper (heraldry), in heraldry, means depicted in natural colors * Proper Records, a UK record label * Proper (album), an album by Into It. Over It. released in 2011 * Proper (often capitaliz ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Meta-language
In logic and linguistics, a metalanguage is a language used to describe another language, often called the ''object language''. Expressions in a metalanguage are often distinguished from those in the object language by the use of italics, quotation marks, or writing on a separate line. The structure of sentences and phrases in a metalanguage can be described by a metasyntax. Types There are a variety of recognized metalanguages, including ''embedded'', ''ordered'', and ''nested'' (or ''hierarchical'') metalanguages. Embedded An embedded metalanguage is a language formally, naturally and firmly fixed in an object language. This idea is found in Douglas Hofstadter's book, '' Gödel, Escher, Bach'', in a discussion of the relationship between formal languages and number theory: "... it is in the nature of any formalization of number theory that its metalanguage is embedded within it." It occurs in natural, or informal, languages, as well—such as in English, where words ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
