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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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Logic
Mathematical logic is the study of Logic#Formal logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory (also known as computability 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 th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tarski's Axioms
Tarski's axioms are an axiom system for Euclidean geometry, specifically for that portion of Euclidean geometry that is formulable in first-order logic with identity (i.e. is formulable as an elementary theory). As such, it does not require an underlying set theory. The only primitive objects of the system are "points" and the only primitive predicates are "betweenness" (expressing the fact that a point lies on a line segment between two other points) and "congruence" (expressing the fact that the distance between two points equals the distance between two other points). The system contains infinitely many axioms. The axiom system is due to Alfred Tarski who first presented it in 1926. Other modern axiomizations of Euclidean geometry are Hilbert's axioms (1899) and Birkhoff's axioms (1932). Using his axiom system, Tarski was able to show that the first-order theory of Euclidean geometry is consistent, complete and decidable: every sentence in its language is either provable o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lindenbaum's Lemma
In mathematical logic, Lindenbaum's lemma, named after Adolf Lindenbaum, states that any consistent theory of predicate logic can be extended to a complete consistent theory. The lemma is a special case of the ultrafilter lemma for Boolean algebras, applied to the Lindenbaum algebra of a theory. Uses It is used in the proof of Gödel's completeness theorem, among other places. Extensions The effective version of the lemma's statement, "every consistent computably enumerable theory can be extended to a complete consistent computably enumerable theory," fails (provided Peano arithmetic is consistent) by Gödel's incompleteness theorem. History The lemma was not published by Adolf Lindenbaum; it is originally attributed to him by Alfred Tarski Alfred Tarski (; ; born Alfred Teitelbaum;School of Mathematics and Statistics, University of St Andrews ''School of Mathematics and Statistics, University of St Andrews''. January 14, 1901 – October 26, 1983) was a Polish-American ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elementary Diagram
In the mathematical field of model theory, the elementary diagram of a structure is the set of all sentences The ''Sentences'' (. ) is a compendium of Christian theology written by Peter Lombard around 1150. It was the most important religious textbook of the Middle Ages. Background The sentence genre emerged from works like Prosper of Aquitaine's ... with parameters from the structure that are true in the structure. It is also called the complete diagram. Definition Let ''M'' be a structure in a first-order language ''L''. An extended language ''L''(''M'') is obtained by adding to ''L'' a constant symbol ''c''''a'' for every element ''a'' of ''M''. The structure ''M'' can be viewed as an ''L''(''M'') structure in which the symbols in ''L'' are interpreted as before, and each new constant ''c''''a'' is interpreted as the element ''a''. The elementary diagram of ''M'' is the set of all ''L''(''M'') sentences that are true in ''M'' (Marker 2002:44). See also * Diagram ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
True Arithmetic
In mathematical logic, true arithmetic is the set of all true first-order statements about the arithmetic of natural numbers. This is the theory associated with the standard model of the Peano axioms in the language of the first-order Peano axioms. True arithmetic is occasionally called Skolem arithmetic, though this term usually refers to the different theory of natural numbers with multiplication. Definition The signature of Peano arithmetic includes the addition, multiplication, and successor function symbols, the equality and less-than relation symbols, and a constant symbol for 0. The (well-formed) formulas of the language of first-order arithmetic are built up from these symbols together with the logical symbols in the usual manner of first-order logic. The structure \mathcal is defined to be a model of Peano arithmetic as follows. * The domain of discourse is the set \mathbb of natural numbers, * The symbol 0 is interpreted as the number 0, * The function symbols are i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 equ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Countable Set
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers; this means that each element in the set may be associated to a unique natural number, or that the elements of the set can be counted one at a time, although the counting may never finish due to an infinite number of elements. In more technical terms, assuming the axiom of countable choice, a set is ''countable'' if its cardinality (the number of elements of the set) is not greater than that of the natural numbers. A countable set that is not finite is said to be countably infinite. The concept is attributed to Georg Cantor, who proved the existence of uncountable sets, that is, sets that are not countable; for example the set of the real numbers. A note on terminology Although the terms "countable" and "countably infinite" as def ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Morley's Categoricity Theorem
In mathematical logic, a theory is categorical if it has exactly one model ( up to isomorphism). Such a theory can be viewed as ''defining'' its model, uniquely characterizing the model's structure. In first-order logic, only theories with a finite model can be categorical. Higher-order logic contains categorical theories with an infinite model. For example, the second-order Peano axioms are categorical, having a unique model whose domain is the set of natural numbers \mathbb. In model theory, the notion of a categorical theory is refined with respect to cardinality. A theory is -categorical (or categorical in ) if it has exactly one model of cardinality up to isomorphism. Morley's categoricity theorem is a theorem of stating that if a first-order theory in a countable language is categorical in some uncountable cardinality, then it is categorical in all uncountable cardinalities. extended Morley's theorem to uncountable languages: if the language has cardinality and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Closed Field
In mathematics, a real closed field is a field F that has the same first-order properties as the field of real numbers. Some examples are the field of real numbers, the field of real algebraic numbers, and the field of hyperreal numbers. Definition A real closed field is a field ''F'' in which any of the following equivalent conditions is true: #''F'' is elementarily equivalent to the real numbers. In other words, it has the same first-order properties as the reals: any sentence in the first-order language of fields is true in ''F'' if and only if it is true in the reals. #There is a total order on ''F'' making it an ordered field such that, in this ordering, every positive element of ''F'' has a square root in ''F'' and any polynomial of odd degree with coefficients in ''F'' has at least one root in ''F''. #''F'' is a formally real field such that every polynomial of odd degree with coefficients in ''F'' has at least one root in ''F'', and for every element ''a'' o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Characteristic (algebra)
In mathematics, the characteristic of a ring , often denoted , is defined to be the smallest positive number of copies of the ring's multiplicative identity () that will sum to the additive identity (). If no such number exists, the ring is said to have characteristic zero. That is, is the smallest positive number such that: : \underbrace_ = 0 if such a number exists, and otherwise. Motivation The special definition of the characteristic zero is motivated by the equivalent definitions characterized in the next section, where the characteristic zero is not required to be considered separately. The characteristic may also be taken to be the exponent of the ring's additive group, that is, the smallest positive integer such that: : \underbrace_ = 0 for every element of the ring (again, if exists; otherwise zero). This definition applies in the more general class of rngs (see '); for (unital) rings the two definitions are equivalent due to their distributive law. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraically Closed Field
In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in . In other words, a field is algebraically closed if the fundamental theorem of algebra holds for it. Every field K is contained in an algebraically closed field C, and the roots in C of the polynomials with coefficients in K form an algebraically closed field called an algebraic closure of K. Given two algebraic closures of K there are isomorphisms between them that fix the elements of K. Algebraically closed fields appear in the following chain of class inclusions: Examples As an example, the field of real numbers is not algebraically closed, because the polynomial equation x^2+1=0 has no solution in real numbers, even though all its coefficients (1 and 0) are real. The same argument proves that no subfield of the real field is algebraically closed; in particular, the field of rational numbers is not algebraically cl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
