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Controversy Over Cantor's Theory
In mathematical logic, the theory of infinite sets was first developed by Georg Cantor. Although this work has become a thoroughly standard fixture of classical set theory, it has been criticized in several areas by mathematicians and philosophers. Cantor's theorem implies that there are sets having cardinality greater than the infinite cardinality of the set of natural numbers. Cantor's argument for this theorem is presented with one small change. This argument can be improved by using a definition he gave later. The resulting argument uses only five axioms of set theory. Cantor's set theory was controversial at the start, but later became largely accepted. Most modern mathematics textbooks implicitly use Cantor's views on mathematical infinity. For example, a line (geometry), line is generally presented as the infinite set of its points, and it is commonly taught that there are more real numbers than rational numbers (see cardinality of the continuum). Cantor's argument Cant ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Logic
Mathematical logic is the study of logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal systems of logic such as their expressive or deductive power. However, it can also include uses of logic to characterize correct mathematical reasoning or to establish foundations of mathematics. Since its inception, mathematical logic has both contributed to and been motivated by the study of foundations of mathematics. This study began in the late 19th century with the development of axiomatic frameworks for geometry, arithmetic, and Mathematical analysis, analysis. In the early 20th century it was shaped by David Hilbert's Hilbert's program, program to prove the consistency of foundational theories. Results of Kurt Gödel, Gerhard Gentzen, and others provided partial resolution to the program, and clarified the issues involved in pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Leopold Kronecker
Leopold Kronecker (; 7 December 1823 – 29 December 1891) was a German mathematician who worked on number theory, algebra and logic. He criticized Georg Cantor's work on set theory, and was quoted by as having said, "'" ("God made the integers, all else is the work of man").The English translation is from Gray. In a footnote, Gray attributes the German quote to "Weber 1891/92, 19, quoting from a lecture of Kronecker's of 1886". Weber, Heinrich L. 1891–1892Kronecker 2:5-23. (The quote is on p. 19.) Kronecker was a student and lifelong friend of . Biography Leopold Kronecker was born ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Image
An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensional picture, that resembles a subject. In the context of signal processing, an image is a distributed amplitude of color(s). In optics, the term “image” may refer specifically to a 2D image. An image does not have to use the entire visual system to be a visual representation. A popular example of this is of a greyscale image, which uses the visual system's sensitivity to brightness across all wavelengths, without taking into account different colors. A black and white visual representation of something is still an image, even though it does not make full use of the visual system's capabilities. Images are typically still, but in some cases can be moving or animated. Characteristics Images may be two or three-dimensional, such as a ph ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Onto
In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of its domain. It is not required that be unique; the function may map one or more elements of to the same element of . The term ''surjective'' and the related terms ''injective'' and ''bijective'' were introduced by Nicolas Bourbaki, a group of mainly French 20th-century mathematicians who, under this pseudonym, wrote a series of books presenting an exposition of modern advanced mathematics, beginning in 1935. The French word '' sur'' means ''over'' or ''above'', and relates to the fact that the image of the domain of a surjective function completely covers the function's codomain. Any function induces a surjection by restricting its codomain to the image of its domain. Every surjective function has a right inverse assuming the axiom ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John Penn Mayberry
John Penn Mayberry (18 November 1939 – 19 August 2016) was an American mathematical philosopher and creator of a distinctive Aristotelian philosophy of mathematics to which he gave expression in his book ''The Foundations of Mathematics in the Theory of Sets''. Following completion of a Ph.D. at Illinois under the supervision of Gaisi Takeuti, he took up, in 1966, a position in the mathematics department of the University of Bristol. He remained there until his retirement in 2004 as a Reader in Mathematics. Philosophical work Mayberry's philosophy rejects the Platonic tradition, which holds mathematics to be a transcendental science concerned with discovering truths about immaterial, but intelligible, objective entities, as metaphysically conceited. This stance sets him apart from what probably is the “silent majority” view among practising mathematicians. Roger Penrose eloquently expresses a typical Platonic position. :“The natural numbers were there before there were hu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equinumerous
In mathematics, two sets or classes ''A'' and ''B'' are equinumerous if there exists a one-to-one correspondence (or bijection) between them, that is, if there exists a function from ''A'' to ''B'' such that for every element ''y'' of ''B'', there is exactly one element ''x'' of ''A'' with ''f''(''x'') = ''y''. Equinumerous sets are said to have the same cardinality (number of elements). The study of cardinality is often called equinumerosity (''equalness-of-number''). The terms equipollence (''equalness-of-strength'') and equipotence (''equalness-of-power'') are sometimes used instead. Equinumerosity has the characteristic properties of an equivalence relation. The statement that two sets ''A'' and ''B'' are equinumerous is usually denoted :A \approx B \, or A \sim B, or , A, =, B, . The definition of equinumerosity using bijections can be applied to both finite and infinite sets, and allows one to state whether two sets have the same size even if they are infinite. Georg Cantor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gottlob Frege
Friedrich Ludwig Gottlob Frege (; ; 8 November 1848 – 26 July 1925) was a German philosopher, logician, and mathematician. He was a mathematics professor at the University of Jena, and is understood by many to be the father of analytic philosophy, concentrating on the philosophy of language, logic, and mathematics. Though he was largely ignored during his lifetime, Giuseppe Peano (1858–1932), Bertrand Russell (1872–1970), and, to some extent, Ludwig Wittgenstein (1889–1951) introduced his work to later generations of philosophers. Frege is widely considered to be the greatest logician since Aristotle, and one of the most profound philosophers of mathematics ever. His contributions include the development of modern logic in the ''Begriffsschrift'' and work in the foundations of mathematics. His book the ''Foundations of Arithmetic'' is the seminal text of the logicist project, and is cited by Michael Dummett as where to pinpoint the linguistic turn. His philosophical ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hume's Principle
Hume's principle or HP says that the number of ''F''s is equal to the number of ''G''s if and only if there is a one-to-one correspondence (a bijection) between the ''F''s and the ''G''s. HP can be stated formally in systems of second-order logic. Hume's principle is named for the Scottish philosopher David Hume and was coined by George Boolos. HP plays a central role in Gottlob Frege's philosophy of mathematics. Frege shows that HP and suitable definitions of arithmetical notions entail all axioms of what we now call second-order arithmetic. This result is known as Frege's theorem, which is the foundation for a philosophy of mathematics known as neo-logicism. Origins Hume's principle appears in Frege's ''Foundations of Arithmetic'' (§63), which quotes from Part III of Book I of David Hume's '' A Treatise of Human Nature'' (1740). Hume there sets out seven fundamental relations between ideas. Concerning one of these, proportion in quantity or number, Hume argues that our reaso ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
One-to-one Correspondence
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other set, and each element of the other set is paired with exactly one element of the first set. There are no unpaired elements. In mathematical terms, a bijective function is a one-to-one (injective) and onto (surjective) mapping of a set ''X'' to a set ''Y''. The term ''one-to-one correspondence'' must not be confused with ''one-to-one function'' (an injective function; see figures). A bijection from the set ''X'' to the set ''Y'' has an inverse function from ''Y'' to ''X''. If ''X'' and ''Y'' are finite sets, then the existence of a bijection means they have the same number of elements. For infinite sets, the picture is more complicated, leading to the concept of cardinal number—a way to distinguish the various sizes of infinite sets. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Axiom Of Power Set
In mathematics, the axiom of power set is one of the Zermelo–Fraenkel axioms of axiomatic set theory. In the formal language of the Zermelo–Fraenkel axioms, the axiom reads: :\forall x \, \exists y \, \forall z \, \in y \iff \forall w \, (w \in z \Rightarrow w \in x)/math> where ''y'' is the Power set of ''x'', \mathcal(x). In English, this says: :Given any set ''x'', there is a set \mathcal(x) such that, given any set ''z'', this set ''z'' is a member of \mathcal(x) if and only if every element of ''z'' is also an element of ''x''. More succinctly: ''for every set x, there is a set \mathcal(x) consisting precisely of the subsets of x.'' Note the subset relation \subseteq is not used in the formal definition as subset is not a primitive relation in formal set theory; rather, subset is defined in terms of set membership, \in. By the axiom of extensionality, the set \mathcal(x) is unique. The axiom of power set appears in most axiomatizations of set theory. It is g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Axiom Of Infinity
In axiomatic set theory and the branches of mathematics and philosophy that use it, the axiom of infinity is one of the axioms of Zermelo–Fraenkel set theory. It guarantees the existence of at least one infinite set, namely a set containing the natural numbers. It was first published by Ernst Zermelo as part of his set theory in 1908.Zermelo: ''Untersuchungen über die Grundlagen der Mengenlehre'', 1907, in: Mathematische Annalen 65 (1908), 261-281; Axiom des Unendlichen p. 266f. Formal statement In the formal language of the Zermelo–Fraenkel axioms, the axiom reads: :\exists \mathbf \, ( \empty \in \mathbf \, \land \, \forall x \in \mathbf \, ( \, ( x \cup \ ) \in \mathbf ) ) . In words, there is a set I (the set which is postulated to be infinite), such that the empty set is in I, and such that whenever any ''x'' is a member of I, the set formed by taking the union of ''x'' with its singleton is also a member of I. Such a set is sometimes called an inductive set. Inter ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Axiomatic Set Theory
Set theory is the branch of mathematical logic that studies Set (mathematics), 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 of set theory, 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 co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
