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Admissible Set
In set theory, a discipline within mathematics, an admissible set is a transitive set A\, such that \langle A,\in \rangle is a model of Kripke–Platek set theory (Barwise 1975). The smallest example of an admissible set is the set of hereditarily finite sets. Another example is the set of hereditarily countable sets. See also * Admissible ordinal References * Barwise, Jon (1975). ''Admissible Sets and Structures: An Approach to Definability Theory'', Perspectives in Mathematical Logic, Volume 7, Springer-VerlagElectronic versionon Project Euclid Project Euclid is a collaborative partnership between Cornell University Library and Duke University Press which seeks to advance scholarly communication in theoretical and applied mathematics and statistics through partnerships with independent an .... Set theory {{settheory-stub ... [...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] |
Transitive Set
In set theory, a branch of mathematics, a set A is called transitive if either of the following equivalent conditions hold: * whenever x \in A, and y \in x, then y \in A. * whenever x \in A, and x is not an urelement, then x is a subset of A. Similarly, a class M is transitive if every element of M is a subset of M. Examples Using the definition of ordinal numbers suggested by John von Neumann, ordinal numbers are defined as hereditarily transitive sets: an ordinal number is a transitive set whose members are also transitive (and thus ordinals). The class of all ordinals is a transitive class. Any of the stages V_\alpha and L_\alpha leading to the construction of the von Neumann universe V and Gödel's constructible universe L are transitive sets. The universes V and L themselves are transitive classes. This is a complete list of all finite transitive sets with up to 20 brackets: * \, * \, * \, * \, * \, * \, * \, * \, * \, * \, * \, * \, * \, * \, * \, * \, * \, * \, * \, * \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Model Theory
In mathematical logic, model theory is the study of the relationship between theory (mathematical logic), formal theories (a collection of Sentence (mathematical logic), sentences in a formal language expressing statements about a Structure (mathematical logic), mathematical structure), and their models (those structures in which the statements of the theory hold). The aspects investigated include the number and size of models of a theory, the relationship of different models to each other, and their interaction with the formal language itself. In particular, model theorists also investigate the sets that can be definable set, defined in a model of a theory, and the relationship of such definable sets to each other. As a separate discipline, model theory goes back to Alfred Tarski, who first used the term "Theory of Models" in publication in 1954. Since the 1970s, the subject has been shaped decisively by Saharon Shelah's stable theory, stability theory. Compared to other areas of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kripke–Platek Set Theory
The Kripke–Platek set theory (KP), pronounced , is an axiomatic set theory developed by Saul Kripke and Richard Platek. The theory can be thought of as roughly the predicative part of ZFC and is considerably weaker than it. Axioms In its formulation, a Δ0 formula is one all of whose quantifiers are bounded. This means any quantification is the form \forall u \in v or \exist u \in v. (See the Lévy hierarchy.) * Axiom of extensionality: Two sets are the same if and only if they have the same elements. * Axiom of induction: φ(''a'') being a formula, if for all sets ''x'' the assumption that φ(''y'') holds for all elements ''y'' of ''x'' entails that φ(''x'') holds, then φ(''x'') holds for all sets ''x''. * Axiom of empty set: There exists a set with no members, called the empty set and denoted . * Axiom of pairing: If ''x'', ''y'' are sets, then so is , a set containing ''x'' and ''y'' as its only elements. * Axiom of union: For any set ''x'', there is a set ''y'' s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hereditarily Finite Set
In mathematics and set theory, hereditarily finite sets are defined as finite sets whose elements are all hereditarily finite sets. In other words, the set itself is finite, and all of its elements are finite sets, recursively all the way down to the empty set. Formal definition A recursive definition of well-founded hereditarily finite sets is as follows: : ''Base case'': The empty set is a hereditarily finite set. : ''Recursion rule'': If ''a''1,...,''a''''k'' are hereditarily finite, then so is . The set \ is an example for such a hereditarily finite set and so is the empty set \emptyset=\. On the other hand, the sets \ or \ are examples of finite sets that are not ''hereditarily'' finite. For example, the first cannot be hereditarily finite since it contains at least one infinite set as an element, when = \. Discussion A symbol for the class of hereditarily finite sets is H_, standing for the cardinality of each of its member being smaller than \aleph_0. Whether H_ is a s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hereditarily Countable Set
In set theory, a set is called hereditarily countable if it is a countable set of hereditarily countable sets. This inductive definition is well-founded and can be expressed in the language of first-order set theory. A set is hereditarily countable if and only if it is countable, and every element of its transitive closure is countable. If the axiom of countable choice holds, then a set is hereditarily countable if and only if its transitive closure is countable. The class of all hereditarily countable sets can be proven to be a set from the axioms of Zermelo–Fraenkel set theory (ZF) without any form of the axiom of choice, and this set is designated H_. The hereditarily countable sets form a model of Kripke–Platek set theory with the axiom of infinity (KPI), if the axiom of countable choice is assumed in the metatheory. If x \in H_, then L_(x) \subset H_. More generally, a set is hereditarily of cardinality less than κ if it is of cardinality less than κ, and all its ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Admissible Ordinal
In set theory, an ordinal number ''α'' is an admissible ordinal if L''α'' is an admissible set (that is, a transitive model of Kripke–Platek set theory); in other words, ''α'' is admissible when ''α'' is a limit ordinal and L''α'' ⊧ Σ0-collection.. See in particulap. 265. The term was coined by Richard Platek in 1966. The first two admissible ordinals are ω and \omega_1^ (the least non-recursive ordinal, also called the Church–Kleene ordinal). Any regular uncountable cardinal is an admissible ordinal. By a theorem of Sacks, the countable admissible ordinals are exactly those constructed in a manner similar to the Church–Kleene ordinal, but for Turing machines with oracles. One sometimes writes \omega_\alpha^ for the \alpha-th ordinal that is either admissible or a limit of admissibles; an ordinal that is both is called ''recursively inaccessible''. There exists a theory of large ordinals in this manner that is highly parallel to that of (small) large car ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jon Barwise
Kenneth Jon Barwise (; June 29, 1942 – March 5, 2000) was an American mathematician, philosopher and logician who proposed some fundamental revisions to the way that logic is understood and used. Education and career Born in Independence, Missouri to Kenneth T. and Evelyn Barwise, Jon was a precocious child. A pupil of Solomon Feferman at Stanford University, Barwise started his research in infinitary logic. After positions as assistant professor at Yale University and the University of Wisconsin, during which time his interests turned to natural language, he returned to Stanford in 1983 to direct the Center for the Study of Language and Information. He began teaching at Indiana University in 1990. He was elected a Fellow of the American Academy of Arts and Sciences in 1999. In his last year, Barwise was invited to give the 2000 Gödel Lecture; he died prior to the lecture. Philosophical and logical work Barwise contended that, by being explicit about the context in w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Project Euclid
Project Euclid is a collaborative partnership between Cornell University Library and Duke University Press which seeks to advance scholarly communication in theoretical and applied mathematics and statistics through partnerships with independent and society publishers. It was created to provide a platform for small publishers of scholarly journals to move from print to electronic in a cost-effective way. Through a combination of support by subscribing libraries and participating publishers, Project Euclid has made 70% of its journal articles available as open access. As of 2010, Project Euclid provided access to over one million pages of open-access content. Mission and goals Project Euclid's stated mission is to advance scholarly communication in the field of theoretical and applied mathematics and statistics. Through a "mixture of open access, subscription, and hosted subscription content it provides a way for small publishers (especially societies) to host their math or statisti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
