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Forcing (set Theory)
In the mathematical discipline of set theory, forcing is a technique for proving consistency and independence (mathematical logic), independence results. Intuitively, forcing can be thought of as a technique to expand the set theoretical universe (mathematics), universe V to a larger universe V[G] by introducing a new "generic" object G. Forcing was first used by Paul Cohen (mathematician), Paul Cohen in 1963, to prove the independence of the axiom of choice and the continuum hypothesis from Zermelo–Fraenkel set theory. It has been considerably reworked and simplified in the following years, and has since served as a powerful technique, both in set theory and in areas of mathematical logic such as recursion theory. Descriptive set theory uses the notions of forcing from both recursion theory and set theory. Forcing has also been used in model theory, but it is common in model theory to define generic filter, genericity directly without mention of forcing. Intuition Forcing is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Recursion Theory
Computability theory, also known as recursion theory, is a branch of mathematical logic, computer science, and the theory of computation that originated in the 1930s with the study of computable functions and Turing degrees. The field has since expanded to include the study of generalized computability and definable set, definability. In these areas, computability theory overlaps with proof theory and effective descriptive set theory. Basic questions addressed by computability theory include: * What does it mean for a function (mathematics), function on the natural numbers to be computable? * How can noncomputable functions be classified into a hierarchy based on their level of noncomputability? Although there is considerable overlap in terms of knowledge and methods, mathematical computability theorists study the theory of relative computability, reducibility notions, and degree structures; those in the computer science field focus on the theory of computational complexity theory ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ordinal Number
In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets. A finite set can be enumerated by successively labeling each element with the least natural number that has not been previously used. To extend this process to various infinite sets, ordinal numbers are defined more generally using linearly ordered greek letter variables that include the natural numbers and have the property that every set of ordinals has a least or "smallest" element (this is needed for giving a meaning to "the least unused element"). This more general definition allows us to define an ordinal number \omega (omega) to be the least element that is greater than every natural number, along with ordinal numbers , , etc., which are even greater than . A linear order such that every non-empty subset has a least element is called a well-order. The axiom of choice implies that every set can be well-orde ... [...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, with respect to any given property, 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 "prop ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Axiom Schema
In mathematical logic, an axiom schema (plural: axiom schemata or axiom schemas) generalizes the notion of axiom. Formal definition An axiom schema is a formula in the metalanguage of an axiomatic system, in which one or more schematic variables appear. These variables, which are metalinguistic constructs, stand for any term or subformula of the system, which may or may not be required to satisfy certain conditions. Often, such conditions require that certain variables be free, or that certain variables not appear in the subformula or term. Examples Two well known instances of axiom schemata are the: * induction schema that is part of Peano's axioms for the arithmetic of the natural numbers; * axiom schema of replacement that is part of the standard ZFC axiomatization of set theory. Czesław Ryll-Nardzewski proved that Peano arithmetic cannot be finitely axiomatized, and Richard Montague proved that ZFC cannot be finitely axiomatized. Hence, the axiom schemata cannot be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mostowski Collapse Lemma
In mathematical logic, the Mostowski collapse lemma, also known as the Shepherdson–Mostowski collapse, is a theorem of set theory introduced by and . Statement Suppose that ''R'' is a binary relation on a class ''X'' such that *''R'' is set-like: ''R''−1 'x''= is a set for every ''x'', *''R'' is well-founded: every nonempty subset ''S'' of ''X'' contains an ''R''-minimal element (i.e. an element ''x'' ∈ ''S'' such that ''R''−1 'x''∩ ''S'' is empty), *''R'' is extensional: ''R''−1 'x''≠ ''R''−1 'y''for every distinct elements ''x'' and ''y'' of ''X'' The Mostowski collapse lemma states that for every such ''R'' there exists a unique transitive class (possibly proper) whose structure under the membership relation is isomorphic to (''X'', ''R''), and the isomorphism is unique. The isomorphism maps each element ''x'' of ''X'' to the set of images of elements ''y'' of ''X'' such that ''y R x'' (Jech 2003:69). Generalizations Every well-founded set-like relati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Standard Model (set Theory)
In set theory, a standard model for a theory ''T'' is a model ''M'' for ''T'' where the membership relation ∈''M'' is the same as the membership relation ∈ of a set theoretical universe ''V'' (restricted to the domain of ''M''). In other words, ''M'' is a substructure of ''V.'' A standard model ''M'' that satisfies the additional transitivity condition that ''x'' ∈ ''y ∈'' ''M'' implies ''x'' ∈ ''M'' is a standard transitive model (or simply a transitive model). Usually, when one talks about a model ''M'' of set theory, it is assumed that ''M'' is a set model, i.e. the domain of ''M'' is a set in ''V.'' If the domain of ''M'' is a proper class, then ''M'' is a class model. An inner model In set theory, a branch of mathematical logic, an inner model for a theory ''T'' is a substructure of a model ''M'' of a set theory that is both a model for ''T'' and contains all the ordinals of ''M''. Definition Let ''L'' = ⟨∈� ... is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boolean-valued Model
In mathematical logic, a Boolean-valued model is a generalization of the ordinary Tarskian notion of structure from model theory. In a Boolean-valued model, the truth values of propositions are not limited to "true" and "false", but instead take values in some fixed complete Boolean algebra. Boolean-valued models were introduced by Dana Scott, Robert M. Solovay, and Petr Vopěnka in the 1960s in order to help understand Paul Cohen's method of forcing. They are also related to Heyting algebra semantics in intuitionistic logic. Definition Fix a complete Boolean algebra ''B''''B'' here is assumed to be ''nondegenerate''; that is, 0 and 1 must be distinct elements of ''B''. Authors writing on Boolean-valued models typically take this requirement to be part of the definition of "Boolean algebra", but authors writing on Boolean algebras in general often do not. and a first-order language ''L''; the signature of ''L'' will consist of a collection of constant symbols, function symbol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ramified Forcing
In the mathematical discipline of set theory, ramified forcing is the original form of forcing introduced by to prove the independence of the continuum hypothesis from Zermelo–Fraenkel set theory. Ramified forcing starts with a model of set theory in which the axiom of constructibility, , holds, and then builds up a larger model of Zermelo–Fraenkel set theory by adding a generic subset of a partially ordered set to , imitating Kurt Gödel's constructible hierarchy. Dana Scott and Robert Solovay realized that the use of constructible sets was an unnecessary complication, and could be replaced by a simpler construction similar to John von Neumann John von Neumann ( ; ; December 28, 1903 – February 8, 1957) was a Hungarian and American mathematician, physicist, computer scientist and engineer. Von Neumann had perhaps the widest coverage of any mathematician of his time, in ...'s construction of the universe as a union of sets for ordinals . Their simpli ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Axiom Of Replacement
In set theory, the axiom schema of replacement is a schema of axioms in Zermelo–Fraenkel set theory (ZF) that asserts that the image of any set under any definable mapping is also a set. It is necessary for the construction of certain infinite sets in ZF. The axiom schema is motivated by the idea that whether a class is a set depends only on the cardinality of the class, not on the rank of its elements. Thus, if one class is "small enough" to be a set, and there is a surjection from that class to a second class, the axiom states that the second class is also a set. However, because ZFC only speaks of sets, not proper classes, the schema is stated only for definable surjections, which are identified with their defining formulas. Statement Suppose P is a definable binary relation (which may be a proper class) such that for every set x there is a unique set y such that P(x,y) holds. There is a corresponding definable function F_P, where F_P(x)=y if and only if P(x,y). Consid ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Relation (mathematics)
In mathematics, a relation denotes some kind of ''relationship'' between two mathematical object, objects in a Set (mathematics), set, which may or may not hold. As an example, "''is less than''" is a relation on the set of natural numbers; it holds, for instance, between the values and (denoted as ), and likewise between and (denoted as ), but not between the values and nor between and , that is, and both evaluate to false. As another example, "''is sister of'' is a relation on the set of all people, it holds e.g. between Marie Curie and Bronisława Dłuska, and likewise vice versa. Set members may not be in relation "to a certain degree" – either they are in relation or they are not. Formally, a relation over a set can be seen as a set of ordered pairs of members of . The relation holds between and if is a member of . For example, the relation "''is less than''" on the natural numbers is an infinite set of pairs of natural numbers that contains both and , b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Well-ordering
In mathematics, a well-order (or well-ordering or well-order relation) on a set is a total ordering on with the property that every non-empty subset of has a least element in this ordering. The set together with the ordering is then called a well-ordered set (or woset). In some academic articles and textbooks these terms are instead written as wellorder, wellordered, and wellordering or well order, well ordered, and well ordering. Every non-empty well-ordered set has a least element. Every element of a well-ordered set, except a possible greatest element, has a unique successor (next element), namely the least element of the subset of all elements greater than . There may be elements, besides the least element, that have no predecessor (see below for an example). A well-ordered set contains for every subset with an upper bound a least upper bound, namely the least element of the subset of all upper bounds of in . If ≤ is a non-strict well ordering, then < is a stri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Order Type
In mathematics, especially in set theory, two ordered sets and are said to have the same order type if they are order isomorphic, that is, if there exists a bijection (each element pairs with exactly one in the other set) f\colon X \to Y such that both and its inverse are monotonic (preserving orders of elements). In the special case when is totally ordered, monotonicity of already implies monotonicity of its inverse. One and the same set may be equipped with different orders. Since order-equivalence is an equivalence relation, it partitions the class of all ordered sets into equivalence classes. Notation If a set X has order type denoted \sigma, the order type of the reversed order, the dual of X, is denoted \sigma^. The order type of a well-ordered set is sometimes expressed as . Examples The order type of the integers and rationals is usually denoted \pi and \eta, respectively. The set of integers and the set of even integers have the same order type, becaus ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
