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Omega-categorical Theory
In mathematical logic, an omega-categorical theory is a theory that has exactly one countably infinite model up to isomorphism. Omega-categoricity is the special case κ = \aleph_0 = ω of κ-categoricity, and omega-categorical theories are also referred to as ω-categorical. The notion is most important for countable first-order theories. Equivalent conditions for omega-categoricity Many conditions on a theory are equivalent to the property of omega-categoricity. In 1959 Erwin Engeler, Czesław Ryll-Nardzewski and Lars Svenonius, proved several independently.Rami Grossberg, José Iovino and Olivier Lessmann''A primer of simple theories''/ref> Despite this, the literature still widely refers to the Ryll-Nardzewski theorem as a name for these conditions. The conditions included with the theorem vary between authors.Hodges, Model Theory, p. 341.Rothmaler, p. 200. Given a countable complete first-order theory ''T'' with infinite models, the following are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Logic
Mathematical logic is the study of logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal systems of logic such as their expressive or deductive power. However, it can also include uses of logic to characterize correct mathematical reasoning or to establish foundations of mathematics. Since its inception, mathematical logic has both contributed to and been motivated by the study of foundations of mathematics. This study began in the late 19th century with the development of axiomatic frameworks for geometry, arithmetic, and Mathematical analysis, analysis. In the early 20th century it was shaped by David Hilbert's Hilbert's program, program to prove the consistency of foundational theories. Results of Kurt Gödel, Gerhard Gentzen, and others provided partial resolution to the program, and clarified the issues involved in pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Oligomorphic Group
In mathematics, a permutation group is a group ''G'' whose elements are permutations of a given set ''M'' and whose group operation is the composition of permutations in ''G'' (which are thought of as bijective functions from the set ''M'' to itself). The group of ''all'' permutations of a set ''M'' is the symmetric group of ''M'', often written as Sym(''M''). The term ''permutation group'' thus means a subgroup of the symmetric group. If then Sym(''M'') is usually denoted by S''n'', and may be called the ''symmetric group on n letters''. By Cayley's theorem, every group is isomorphic to some permutation group. The way in which the elements of a permutation group permute the elements of the set is called its group action. Group actions have applications in the study of symmetries, combinatorics and many other branches of mathematics, physics and chemistry. Basic properties and terminology Being a subgroup of a symmetric group, all that is necessary for a set of permutation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtraction and division are defined and satisfy certain basic rules. The most common examples of finite fields are given by the integers mod when is a prime number. The ''order'' of a finite field is its number of elements, which is either a prime number or a prime power. For every prime number and every positive integer there are fields of order p^k, all of which are isomorphic. Finite fields are fundamental in a number of areas of mathematics and computer science, including number theory, algebraic geometry, Galois theory, finite geometry, cryptography and coding theory. Properties A finite field is a finite set which is a field; this means that multiplication, addition, subtraction and division (excluding division by zero) are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rado Graph
In the mathematical field of graph theory, the Rado graph, Erdős–Rényi graph, or random graph is a countably infinite graph that can be constructed (with probability one) by choosing independently at random for each pair of its vertices whether to connect the vertices by an edge. The names of this graph honor Richard Rado, Paul Erdős, and Alfréd Rényi, mathematicians who studied it in the early 1960s; it appears even earlier in the work of . The Rado graph can also be constructed non-randomly, by symmetrizing the membership relation of the hereditarily finite sets, by applying the BIT predicate to the binary representations of the natural numbers, or as an infinite Paley graph that has edges connecting pairs of prime numbers congruent to 1 mod 4 that are quadratic residues modulo each other. Every finite or countably infinite graph is an induced subgraph of the Rado graph, and can be found as an induced subgraph by a greedy algorithm that builds up the subgraph one ve ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cantor's Isomorphism Theorem
In order theory and model theory, branches of mathematics, Cantor's isomorphism theorem states that every two countable dense unbounded linear orders are order-isomorphic. For instance, there is an isomorphism (a one-to-one order-preserving correspondence) between the numerical ordering of the rational numbers and the numerical ordering of the dyadic rationals, that can be described by Minkowski's question-mark function. The isomorphism theorem is named after Georg Cantor, who used it to characterize the (uncountable) ordering on the real numbers. It can be proved by a back-and-forth method that is also sometimes attributed to Cantor. The same back-and-forth method also proves that countable dense unbounded orders are highly symmetric, and can be applied to other kinds of structures. However, Cantor's original proof only used the "going forth" half of this method. In terms of model theory, the isomorphism theorem can be expressed by saying that the first-order theory of unbound ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime Model
In mathematics, and in particular model theory, a prime model is a model that is as simple as possible. Specifically, a model P is prime if it admits an elementary embedding into any model M to which it is elementarily equivalent (that is, into any model M satisfying the same complete theory as P). Cardinality In contrast with the notion of saturated model, prime models are restricted to very specific cardinalities by the Löwenheim–Skolem theorem. If L is a first-order language with cardinality \kappa and T is a complete theory over L, then this theorem guarantees a model for T of cardinality \max(\kappa,\aleph_0). Therefore no prime model of T can have larger cardinality since at the very least it must be elementarily embedded in such a model. This still leaves much ambiguity in the actual cardinality. In the case of countable languages, all prime models are at most countably infinite. Relationship with saturated models There is a duality between the definitions of prime a ... [...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] |
Atomic Model (mathematical Logic)
In model theory, a subfield of mathematical logic, an atomic model is a model such that the complete type of every tuple is axiomatized by a single formula. Such types are called principal types, and the formulas that axiomatize them are called complete formulas. Definitions Let ''T'' be a theory. A complete type ''p''(''x''1, ..., ''x''''n'') is called principal or atomic (relative to ''T'') if it is axiomatized relative to ''T'' by a single formula ''φ''(''x''1, ..., ''x''''n'') ∈ ''p''(''x''1, ..., ''x''''n''). A formula ''φ'' is called complete in ''T'' if for every formula ''ψ''(''x''1, ..., ''x''''n''), the theory ''T'' ∪ entails exactly one of ''ψ'' and ¬''ψ''.Some authors refer to complete formulas as "atomic formulas", but this is inconsistent with the purely syntactical notion of an atom or atomic formula as a formula that does not contain a proper subformula. It follows that a complete type is principal if and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lindenbaum–Tarski Algebra
In mathematical logic, the Lindenbaum–Tarski algebra (or Lindenbaum algebra) of a logical theory ''T'' consists of the equivalence classes of sentences of the theory (i.e., the quotient, under the equivalence relation ~ defined such that ''p'' ~ ''q'' exactly when ''p'' and ''q'' are provably equivalent in ''T''). That is, two sentences are equivalent if the theory ''T'' proves that each implies the other. The Lindenbaum–Tarski algebra is thus the quotient algebra obtained by factoring the algebra of formulas by this congruence relation. The algebra is named for logicians Adolf Lindenbaum and Alfred Tarski. It was first introduced by Tarski in 1935 as a device to establish correspondence between classical propositional calculus and Boolean algebras. The Lindenbaum–Tarski algebra is considered the origin of the modern algebraic logic.; here: pages 1-2 Operations The operations in a Lindenbaum–Tarski algebra ''A'' are inherited from those in the underlying theory ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Type (model Theory)
In model theory and related areas of mathematics, a type is an object that describes how a (real or possible) element or finite collection of elements in a mathematical structure might behave. More precisely, it is a set of first-order formulas in a language ''L'' with free variables ''x''1, ''x''2,…, ''x''''n'' that are true of a sequence of elements of an ''L''-structure \mathcal. Depending on the context, types can be complete or partial and they may use a fixed set of constants, ''A'', from the structure \mathcal. The question of which types represent actual elements of \mathcal leads to the ideas of saturated models and omitting types. Formal definition Consider a structure \mathcal for a language ''L''. Let ''M'' be the universe of the structure. For every ''A'' ⊆ ''M'', let ''L''(''A'') be the language obtained from ''L'' by adding a constant ''c''''a'' for every ''a'' ∈ ''A''. In other words, :L(A) = L \cup \. A 1-type (of \mathcal) over ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complete Theory
In mathematical logic, a theory is complete if it is consistent and for every closed formula in the theory's language, either that formula or its negation is provable. That is, for every sentence \varphi, the theory T contains the sentence or its negation but not both (that is, either T \vdash \varphi or T \vdash \neg \varphi). Recursively axiomatizable first-order theories that are consistent and rich enough to allow general mathematical reasoning to be formulated cannot be complete, as demonstrated by Gödel's first incompleteness theorem. This sense of ''complete'' is distinct from the notion of a complete ''logic'', which asserts that for every theory that can be formulated in the logic, all semantically valid statements are provable theorems (for an appropriate sense of "semantically valid"). Gödel's completeness theorem is about this latter kind of completeness. Complete theories are closed under a number of conditions internally modelling the T-schema: * For a set of for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theory (mathematical Logic)
In mathematical logic, a theory (also called a formal theory) is a set of sentences in a formal language. In most scenarios, a deductive system is first understood from context, after which an element \phi\in T of a deductively closed theory T is then called a theorem of the theory. In many deductive systems there is usually a subset \Sigma \subseteq T that is called "the set of axioms" of the theory T, in which case the deductive system is also called an "axiomatic system". By definition, every axiom is automatically a theorem. A first-order theory is a set of first-order sentences (theorems) recursively obtained by the inference rules of the system applied to the set of axioms. General theories (as expressed in formal language) When defining theories for foundational purposes, additional care must be taken, as normal set-theoretic language may not be appropriate. The construction of a theory begins by specifying a definite non-empty ''conceptual class'' \mathcal, the element ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
