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Alternative Set Theory
In a general sense, an alternative set theory is any of the alternative mathematical approaches to the concept of set and any alternative to the de facto standard set theory described in axiomatic set theory by the axioms of Zermelo–Fraenkel set theory. More specifically, Alternative Set Theory (or AST) may refer to a particular set theory developed in the 1970s and 1980s by Petr Vopěnka and his students. Vopěnka's Alternative Set Theory Vopěnka's Alternative Set Theory builds on some ideas of the theory of semisets, but also introduces more radical changes: for example, all sets are "formally" finite, which means that sets in AST satisfy the law of mathematical induction for set- formulas (more precisely: the part of AST that consists of axioms related to sets only is equivalent to the Zermelo–Fraenkel (or ZF) set theory, in which the axiom of infinity is replaced by its negation). However, some of these sets contain subclasses that are not sets, which makes them diffe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Set (mathematics)
A set is the mathematical model for a collection of different things; a set contains '' elements'' or ''members'', which can be mathematical objects of any kind: numbers, symbols, points in space, lines, other geometrical shapes, variables, or even other sets. The set with no element is the empty set; a set with a single element is a singleton. A set may have a finite number of elements or be an infinite set. Two sets are equal if they have precisely the same elements. Sets are ubiquitous in modern mathematics. Indeed, set theory, more specifically Zermelo–Fraenkel set theory, has been the standard way to provide rigorous foundations for all branches of mathematics since the first half of the 20th century. History The concept of a set emerged in mathematics at the end of the 19th century. The German word for set, ''Menge'', was coined by Bernard Bolzano in his work '' Paradoxes of the Infinite''. Georg Cantor, one of the founders of set theory, gave the followi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Type Theory
In mathematics, logic, and computer science, a type theory is the formal presentation of a specific type system, and in general type theory is the academic study of type systems. Some type theories serve as alternatives to set theory as a foundation of mathematics. Two influential type theories that were proposed as foundations are Alonzo Church's typed λ-calculus and Per Martin-Löf's intuitionistic type theory. Most computerized proof-writing systems use a type theory for their foundation. A common one is Thierry Coquand's Calculus of Inductive Constructions. History Type theory was created to avoid a paradox in a mathematical foundation based on naive set theory and formal logic. Russell's paradox, which was discovered by Bertrand Russell, existed because a set could be defined using "all possible sets", which included itself. Between 1902 and 1908, Bertrand Russell proposed various "theories of type" to fix the problem. By 1908 Russell arrived at a "ramifi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Non-well-founded Set Theory
Non-well-founded set theories are variants of axiomatic set theory that allow sets to be elements of themselves and otherwise violate the rule of well-foundedness. In non-well-founded set theories, the foundation axiom of ZFC is replaced by axioms implying its negation. The study of non-well-founded sets was initiated by Dmitry Mirimanoff in a series of papers between 1917 and 1920, in which he formulated the distinction between well-founded and non-well-founded sets; he did not regard well-foundedness as an axiom. Although a number of axiomatic systems of non-well-founded sets were proposed afterwards, they did not find much in the way of applications until Peter Aczel’s hyperset theory in 1988. The theory of non-well-founded sets has been applied in the logical modelling of non-terminating computational processes in computer science (process algebra and final semantics), linguistics and natural language semantics ( situation theory), philosophy (work on the Liar Parado ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
General Set Theory
General set theory (GST) is George Boolos's (1998) name for a fragment of the axiomatic set theory Z. GST is sufficient for all mathematics not requiring infinite sets, and is the weakest known set theory whose theorems include the Peano axioms. Ontology The ontology of GST is identical to that of ZFC, and hence is thoroughly canonical. GST features a single primitive ontological notion, that of set, and a single ontological assumption, namely that all individuals in the universe of discourse (hence all mathematical objects) are sets. There is a single primitive binary relation, set membership; that set ''a'' is a member of set ''b'' is written ''a ∈ b'' (usually read "''a'' is an element of ''b''"). Axioms The symbolic axioms below are from Boolos (1998: 196), and govern how sets behave and interact. As with Z, the background logic for GST is first order logic with identity. Indeed, GST is the fragment of Z obtained by omitting the axioms Union, Power Set, Elementary ... [...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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Constructive Set Theory
Constructive set theory is an approach to mathematical constructivism following the program of axiomatic set theory. The same first-order language with "=" and "\in" of classical set theory is usually used, so this is not to be confused with a constructive types approach. On the other hand, some constructive theories are indeed motivated by their interpretability in type theories. In addition to rejecting the principle of excluded middle (), constructive set theories often require some logical quantifiers in their axioms to be bounded, motivated by results tied to impredicativity. Introduction Constructive outlook Use of intuitionistic logic The logic of the set theories discussed here is constructive in that it rejects , i.e. that the disjunction \phi \lor \neg \phi automatically holds for all propositions. As a rule, to prove the excluded middle for a proposition P, i.e. to prove the particular disjunction P \lor \neg P, either P or \neg P needs to be explicitly prov ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Scott–Potter Set Theory
An approach to the foundations of mathematics that is of relatively recent origin, Scott–Potter set theory is a collection of nested axiomatic set theories set out by the philosopher Michael Potter, building on earlier work by the mathematician Dana Scott and the philosopher George Boolos. Potter (1990, 2004) clarified and simplified the approach of Scott (1974), and showed how the resulting axiomatic set theory can do what is expected of such theory, namely grounding the cardinal and ordinal numbers, Peano arithmetic and the other usual number systems, and the theory of relations. ZU etc. Preliminaries This section and the next follow Part I of Potter (2004) closely. The background logic is first-order logic with identity. The ontology includes urelements as well as sets, which makes it clear that there can be sets of entities defined by first-order theories not based on sets. The urelements are not essential in that other mathematical structures can be defined as sets, and i ... [...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] |
S (set Theory)
S is an axiomatic set theory set out by George Boolos in his 1989 article, "Iteration Again". S, a first-order theory, is two-sorted because its ontology includes “stages” as well as sets. Boolos designed S to embody his understanding of the “iterative conception of set“ and the associated iterative hierarchy. S has the important property that all axioms of Zermelo set theory ''Z'', except the axiom of extensionality and the axiom of choice, are theorems of S or a slight modification thereof. Ontology Any grouping together of mathematical, abstract, or concrete objects, however formed, is a ''collection'', a synonym for what other set theories refer to as a class. The things that make up a collection are called elements or members. A common instance of a collection is the domain of discourse of a first-order theory. All sets are collections, but there are collections that are not sets. A synonym for collections that are not sets is proper class. An essential task of a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Naive Set Theory
Naive set theory is any of several theories of sets used in the discussion of the foundations of mathematics. Unlike axiomatic set theories, which are defined using formal logic, naive set theory is defined informally, in natural language. It describes the aspects of mathematical sets familiar in discrete mathematics (for example Venn diagrams and symbolic reasoning about their Boolean algebra), and suffices for the everyday use of set theory concepts in contemporary mathematics. Sets are of great importance in mathematics; in modern formal treatments, most mathematical objects (numbers, relations, functions, etc.) are defined in terms of sets. Naive set theory suffices for many purposes, while also serving as a stepping-stone towards more formal treatments. Method A ''naive theory'' in the sense of "naive set theory" is a non-formalized theory, that is, a theory that uses natural language to describe sets and operations on sets. The words ''and'', ''or'', ''if ... then'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Internal Set Theory
Internal set theory (IST) is a mathematical theory of sets developed by Edward Nelson that provides an axiomatic basis for a portion of the nonstandard analysis introduced by Abraham Robinson. Instead of adding new elements to the real numbers, Nelson's approach modifies the axiomatic foundations through syntactic enrichment. Thus, the axioms introduce a new term, "standard", which can be used to make discriminations not possible under the conventional ZFC axioms for sets. Thus, IST is an enrichment of ZFC: all axioms of ZFC are satisfied for all classical predicates, while the new unary predicate "standard" satisfies three additional axioms I, S, and T. In particular, suitable nonstandard elements within the set of real numbers can be shown to have properties that correspond to the properties of infinitesimal and unlimited elements. Nelson's formulation is made more accessible for the lay-mathematician by leaving out many of the complexities of meta-mathematical logic tha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Positive Set Theory
In mathematical logic, positive set theory is the name for a class of alternative set theories in which the axiom of comprehension holds for at least the positive formulas \phi (the smallest class of formulas containing atomic membership and equality formulas and closed under conjunction, disjunction, existential and universal quantification). Typically, the motivation for these theories is topological: the sets are the classes which are closed under a certain topology. The closure conditions for the various constructions allowed in building positive formulas are readily motivated (and one can further justify the use of universal quantifiers bounded in sets to get generalized positive comprehension): the justification of the existential quantifier seems to require that the topology be compact. Axioms The set theory \mathrm^+_\infty of Olivier Esser consists of the following axioms: Extensionality \forall x \forall y (\forall z (z \in x \leftrightarrow z \in y) \to x = y) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
