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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 Zermelo–Fraenkel set theory (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 ... [...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 the ... [...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. Formal statement Using first-order logic primitive symbols, the axiom can be expressed as follows: \exist \mathrm \ (\exist o \ (o \in \mathrm \ \land \lnot \exist n \ (n \in o)) \ \land \ \forall x \ (x \in \mathrm \Rightarrow \exist y \ (y \in \mathrm \ \land \ \forall a \ (a \in y \Leftrightarrow (a \in x \ \lor \ a = x))))). If the notations of both set-builder and empty set are allowed: \exists \mathrm \, ( \varnothing \in \mathrm \, \land \, \forall x \, (x \in \mathrm \Rightarrow \, ( x \cup \ ) \in \mathrm ) ). Some mathematicians may call a set built this way an inductive set. Hint: In English, it reads: " There exists a set ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ordered Pair
In mathematics, an ordered pair, denoted (''a'', ''b''), is a pair of objects in which their order is significant. The ordered pair (''a'', ''b'') is different from the ordered pair (''b'', ''a''), unless ''a'' = ''b''. In contrast, the '' unordered pair'', denoted , always equals the unordered pair . Ordered pairs are also called 2-tuples, or sequences (sometimes, lists in a computer science context) of length 2. Ordered pairs of scalars are sometimes called 2-dimensional vectors. (Technically, this is an abuse of terminology since an ordered pair need not be an element of a vector space.) The entries of an ordered pair can be other ordered pairs, enabling the recursive definition of ordered ''n''-tuples (ordered lists of ''n'' objects). For example, the ordered triple (''a'',''b'',''c'') can be defined as (''a'', (''b'',''c'')), i.e., as one pair nested in another. In the ordered pair (''a'', ''b''), the object ''a'' is called the ''first entry'', and the object ''b'' the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Limit Ordinal
In set theory, a limit ordinal is an ordinal number that is neither zero nor a successor ordinal. Alternatively, an ordinal λ is a limit ordinal if there is an ordinal less than λ, and whenever β is an ordinal less than λ, then there exists an ordinal γ such that β 0, are limits of limits, etc. Properties The classes of successor ordinals and limit ordinals (of various cofinalities) as well as zero exhaust the entire class of ordinals, so these cases are often used in proofs by transfinite induction or definitions by transfinite recursion. Limit ordinals represent a sort of "turning point" in such procedures, in which one must use limiting operations such as taking the union over all preceding ordinals. In principle, one could do anything at limit ordinals, but taking the union is continuous in the order topology and this is usually desirable. If we use the von Neumann cardinal assignment, every infinite cardinal number In mathematics, a cardinal number, or ... [...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] |
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 Structure (mathematical logic), models (those Structure (mathematical logic), 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 Shel ... [...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 holds: * 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] |
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 a .... Set theory {{settheory-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Axiom Of Regularity
In mathematics, the axiom of regularity (also known as the axiom of foundation) is an axiom of Zermelo–Fraenkel set theory that states that every Empty set, non-empty Set (mathematics), set ''A'' contains an element that is Disjoint sets, disjoint from ''A''. In first-order logic, the axiom reads: \forall x\,(x \neq \varnothing \rightarrow (\exists y \in x) (y \cap x = \varnothing)). The axiom of regularity together with the axiom of pairing implies that Russell paradox, no set is an element of itself, and that there is no infinite sequence (a_n) such that a_ is an element of a_i for all i. With the axiom of dependent choice (which is a weakened form of the axiom of choice), this result can be reversed: if there are no such infinite sequences, then the axiom of regularity is true. Hence, in this context the axiom of regularity is equivalent to the sentence that there are no downward infinite membership chains. The axiom was originally formulated by von Neumann; it was adopted in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Power Set Axiom
In mathematics, the axiom of power set is one of the Zermelo–Fraenkel axioms of axiomatic set theory. It guarantees for every set x the existence of a set \mathcal(x), the power set of x, consisting precisely of the subsets of x. 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 generally considered uncontroversial, although constructive set theory prefers a weaker version to resolve concerns about predicativity. Formal statement The subset relation \subseteq is not a primitive notion in formal set theory and is not used in the formal language of the Zermelo–Fraenkel axioms. Rather, the subset relation \subseteq is defined in terms of set membership, \in. Given this, in the formal language of the Zermelo–Fraenkel axioms, the axiom of power set 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 o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
First-order Logic
First-order logic, also called predicate logic, predicate calculus, or quantificational logic, is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantified variables over non-logical objects, and allows the use of sentences that contain variables. Rather than propositions such as "all humans are mortal", in first-order logic one can have expressions in the form "for all ''x'', if ''x'' is a human, then ''x'' is mortal", where "for all ''x"'' is a quantifier, ''x'' is a variable, and "... ''is a human''" and "... ''is mortal''" are predicates. This distinguishes it from propositional logic, which does not use quantifiers or relations; in this sense, propositional logic is the foundation of first-order logic. A theory about a topic, such as set theory, a theory for groups,A. Tarski, ''Undecidable Theories'' (1953), p. 77. Studies in Logic and the Foundation of Mathematics, North-Holland or a formal theory o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Law Of Excluded Middle
In logic, the law of excluded middle or the principle of excluded middle states that for every proposition, either this proposition or its negation is true. It is one of the three laws of thought, along with the law of noncontradiction and the law of identity; however, no system of logic is built on just these laws, and none of these laws provides inference rules, such as modus ponens or De Morgan's laws. The law is also known as the law/principle of the excluded third, in Latin ''principium tertii exclusi''. Another Latin designation for this law is ''tertium non datur'' or "no third ossibilityis given". In classical logic, the law is a tautology. In contemporary logic the principle is distinguished from the semantical principle of bivalence, which states that every proposition is either true or false. The principle of bivalence always implies the law of excluded middle, while the converse is not always true. A commonly cited counterexample uses statements unprovable n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
