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Vaught Conjecture
The Vaught conjecture is a conjecture in the mathematical field of model theory originally proposed by Robert Lawson Vaught in 1961. It states that the number of countable models of a first-order complete theory in a countable language is finite or ℵ0 or 2. Morley showed that the number of countable models is finite or ℵ0 or ℵ1 or 2, which solves the conjecture except for the case of ℵ1 models when the continuum hypothesis fails. For this remaining case, has announced a counterexample to the Vaught conjecture and the topological Vaught conjecture. As of 2021, the counterexample has not been verified. Statement of the conjecture Let T be a first-order, countable, complete theory with infinite models. Let I(T, \alpha) denote the number of models of ''T'' of cardinality \alpha up to isomorphism, the spectrum of the theory T. Morley proved that if ''I''(''T'', ℵ0) is infinite then it must be ℵ0 or ℵ1 or the cardinality of the continuum. The Vaught conjecture is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conjecture
In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in 1995 by Andrew Wiles), have shaped much of mathematical history as new areas of mathematics are developed in order to prove them. Important examples Fermat's Last Theorem In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers a, ''b'', and ''c'' can satisfy the equation ''a^n + b^n = c^n'' for any integer value of ''n'' greater than two. This theorem was first conjectured by Pierre de Fermat in 1637 in the margin of a copy of '' Arithmetica'', where he claimed that he had a proof that was too large to fit in the margin. The first successful proof was released in 1994 by Andrew Wiles, and formally published in 1995, after 358 years of effort by mathe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Saturated Model
In mathematical logic, and particularly in its subfield model theory, a saturated model ''M'' is one that realizes as many complete types as may be "reasonably expected" given its size. For example, an ultrapower model of the hyperreals is \aleph_1-saturated, meaning that every descending nested sequence of internal sets has a nonempty intersection. Definition Let ''κ'' be a finite or infinite cardinal number and ''M'' a model in some first-order language. Then ''M'' is called ''κ''-saturated if for all subsets ''A'' ⊆ ''M'' of cardinality less than ''κ'', the model ''M'' realizes all complete types over ''A''. The model ''M'' is called saturated if it is , ''M'', -saturated where , ''M'', denotes the cardinality of ''M''. That is, it realizes all complete types over sets of parameters of size less than , ''M'', . According to some authors, a model ''M'' is called countably saturated if it is \aleph_1-saturated; that is, it realizes all complete types over count ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Israel Journal Of Mathematics
'' Israel Journal of Mathematics'' is a peer-reviewed mathematics journal published by the Hebrew University of Jerusalem (Magnes Press). Founded in 1963, as a continuation of the ''Bulletin of the Research Council of Israel'' (Section F), the journal publishes articles on all areas of mathematics. The journal is indexed by ''Mathematical Reviews'' and Zentralblatt MATH. Its 2009 MCQ was 0.70, and its 2009 impact factor The impact factor (IF) or journal impact factor (JIF) of an academic journal is a scientometric index calculated by Clarivate that reflects the yearly mean number of citations of articles published in the last two years in a given journal, as i ... was 0.754. External links * Mathematics journals Publications established in 1963 English-language journals Bimonthly journals Hebrew University of Jerusalem {{math-journal-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Morley's Categoricity Theorem
In mathematical logic, a theory is categorical if it has exactly one model (up to isomorphism). Such a theory can be viewed as ''defining'' its model, uniquely characterizing the model's structure. In first-order logic, only theories with a finite model can be categorical. Higher-order logic contains categorical theories with an infinite model. For example, the second-order Peano axioms are categorical, having a unique model whose domain is the set of natural numbers \mathbb. In model theory, the notion of a categorical theory is refined with respect to cardinality. A theory is -categorical (or categorical in ) if it has exactly one model of cardinality up to isomorphism. Morley's categoricity theorem is a theorem of stating that if a first-order theory in a countable language is categorical in some uncountable cardinality, then it is categorical in all uncountable cardinalities. extended Morley's theorem to uncountable languages: if the language has cardinality and a the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spectrum Of A Theory
In model theory, a branch of mathematical logic, the spectrum of a theory is given by the number of isomorphism classes of models in various cardinalities. More precisely, for any complete theory ''T'' in a language we write ''I''(''T'', ''κ'') for the number of models of ''T'' (up to isomorphism) of cardinality ''κ''. The spectrum problem is to describe the possible behaviors of ''I''(''T'', ''κ'') as a function of ''κ''. It has been almost completely solved for the case of a countable theory ''T''. Early results In this section ''T'' is a countable complete theory and ''κ'' is a cardinal. The Löwenheim–Skolem theorem shows that if ''I''(''T'',''κ'') is nonzero for one infinite cardinal then it is nonzero for all of them. Morley's categoricity theorem was the first main step in solving the spectrum problem: it states that if ''I''(''T'',''κ'') is 1 for some uncountable ''κ'' then it is 1 for all uncountable ''κ''. Robert Vaught showed that ''I''(''T'',ℵ0) canno ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equivalence Relation
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relation provides a partition of the underlying set into disjoint equivalence classes. Two elements of the given set are equivalent to each other if and only if they belong to the same equivalence class. Notation Various notations are used in the literature to denote that two elements a and b of a set are equivalent with respect to an equivalence relation R; the most common are "a \sim b" and "", which are used when R is implicit, and variations of "a \sim_R b", "", or "" to specify R explicitly. Non-equivalence may be written "" or "a \not\equiv b". Definition A binary relation \,\sim\, on a set X is said to be an equivalence relation, if and only if it is reflexive, symmetric and transitive. That is, for all a, b, and c in X: * a \sim a ( ref ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symmetric Group
In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group \mathrm_n defined over a finite set of n symbols consists of the permutations that can be performed on the n symbols. Since there are n! (n factorial) such permutation operations, the order (number of elements) of the symmetric group \mathrm_n is n!. Although symmetric groups can be defined on infinite sets, this article focuses on the finite symmetric groups: their applications, their elements, their conjugacy classes, a finite presentation, their subgroups, their automorphism groups, and their representation theory. For the remainder of this article, "symmetric group" will mean a symmetric group on a finite set. The symmetric group is important to diverse areas of mathematics such as Galois theory, invariant theory, the representatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Czesław Ryll-Nardzewski
Czesław Ryll-Nardzewski (; 7 October 1926 – 18 September 2015) was a Polish mathematician. Born in Wilno, Second Polish Republic (now Vilnius, Lithuania), he was a student of Hugo Steinhaus. At the age of 26 he became professor at Warsaw University. In 1959, he became a professor at the Wrocław University of Technology. He was the advisor of 18 PhD theses. His main research areas are measure theory, functional analysis, foundations of mathematics and probability theory. Several theorems bear his name: the Ryll-Nardzewski fixed point theorem, “9. Theorem of Ryll-Nardzewski” (p. 171), “(9.6) Theorem (Ryll-Nardzewski)” (p. 174) the Ryll-Nardzewski theorem See Theorem 7.3.1 Cf. (2.10) in model theory, and the Kuratowski and Ryll-Nardzewski measurable selection theorem. See Theorem 6.9.3 on p. 36 and the historical comment on p. 441 He became a member of the Polish Academy of Sciences The Polish Academy of Sciences ( pl, Polska Akademia Nauk, PAN) is a Polish state-spo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Andrzej Grzegorczyk
Andrzej Grzegorczyk (; 22 August 1922 – 20 March 2014) was a Polish logician, mathematician, philosopher, and ethicist noted for his work in computability, mathematical logic, and the foundations of mathematics. Historical family background Andrzej Grzegorczyk's foundational family background has its origins in the Polish intellectual, religious, patriotic, and nationalist traditions. He was the only child to the Galician family of well-educated and wealthy parents, his father Piotr Jan Grzegorczyk (1894–1968) was a Polish philologist and historian of Polish literature involved in literary criticism, bibliographic studies, and chronicles of the Polish cultural life. Andrzej's mother Zofia Jadwiga née Zdziarska was a Medical Doctor from a purely Polish landed gentry family. Rich historical family background was the most fundamental element in the shaping of Andrzej Grzegorczyk's intellectual formation and professional academic career. In particular, this heritage laid ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topological Space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points, along with an additional structure called a topology, which can be defined as a set of neighbourhoods for each point that satisfy some axioms formalizing the concept of closeness. There are several equivalent definitions of a topology, the most commonly used of which is the definition through open sets, which is easier than the others to manipulate. A topological space is the most general type of a mathematical space that allows for the definition of limits, continuity, and connectedness. Common types of topological spaces include Euclidean spaces, metric spaces and manifolds. Although very general, the concept of topological spaces is fundamental, and used in virtually every branch of modern mathematics. The study of topological spac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Natural Number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''Cardinal number, cardinal numbers'', and numbers used for ordering are called ''Ordinal number, ordinal numbers''. Natural numbers are sometimes used as labels, known as ''nominal numbers'', having none of the properties of numbers in a mathematical sense (e.g. sports Number (sports), jersey numbers). Some definitions, including the standard ISO/IEC 80000, ISO 80000-2, begin the natural numbers with , corresponding to the non-negative integers , whereas others start with , corresponding to the positive integers Texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the whole numbers, while in other writings, that term is used instead for the integers (including negative integers). The natural ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group Action
In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism group of the structure. It is said that the group ''acts'' on the space or structure. If a group acts on a structure, it will usually also act on objects built from that structure. For example, the group of Euclidean isometries acts on Euclidean space and also on the figures drawn in it. For example, it acts on the set of all triangles. Similarly, the group of symmetries of a polyhedron acts on the vertices, the edges, and the faces of the polyhedron. A group action on a vector space is called a representation of the group. In the case of a finite-dimensional vector space, it allows one to identify many groups with subgroups of , the group of the invertible matrices of dimension over a field . The symmetric group acts on any set wit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
