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Complete theory

Index Complete theory

In mathematical logic, a theory is complete if, for every formula in the theory's language, that formula or its negation is demonstrable. [1]

20 relations: Algebraically closed field, Łoś–Vaught test, Classical logic, Contradiction, Dense order, Euclidean geometry, Gödel's completeness theorem, Gödel's incompleteness theorems, Kripke semantics, Mathematical logic, Modal logic, Model theory, Morley's categoricity theorem, Omega-categorical theory, Presburger arithmetic, Real closed field, T-schema, Tarski's axioms, Theory (mathematical logic), Zorn's lemma.

Algebraically closed field

In abstract algebra, an algebraically closed field F contains a root for every non-constant polynomial in F, the ring of polynomials in the variable x with coefficients in F.

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Łoś–Vaught test

In model theory, a branch of mathematical logic, the Łoś–Vaught test is a criterion for a theory to be complete, unable to be augmented without becoming inconsistent.

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Classical logic

Classical logic (or standard logic) is an intensively studied and widely used class of formal logics.

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Contradiction

In classical logic, a contradiction consists of a logical incompatibility between two or more propositions.

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Dense order

In mathematics, a partial order or total order.

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Euclidean geometry

Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements.

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Gödel's completeness theorem

Gödel's completeness theorem is a fundamental theorem in mathematical logic that establishes a correspondence between semantic truth and syntactic provability in first-order logic.

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Gödel's incompleteness theorems

Gödel's incompleteness theorems are two theorems of mathematical logic that demonstrate the inherent limitations of every formal axiomatic system containing basic arithmetic.

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Kripke semantics

Kripke semantics (also known as relational semantics or frame semantics, and often confused with possible world semantics) is a formal semantics for non-classical logic systems created in the late 1950s and early 1960s by Saul Kripke and André Joyal.

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Mathematical logic

Mathematical logic is a subfield of mathematics exploring the applications of formal logic to mathematics.

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Modal logic

Modal logic is a type of formal logic primarily developed in the 1960s that extends classical propositional and predicate logic to include operators expressing modality.

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Model theory

In mathematics, model theory is the study of classes of mathematical structures (e.g. groups, fields, graphs, universes of set theory) from the perspective of mathematical logic.

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Morley's categoricity theorem

In model theory, a branch of mathematical logic, a theory is κ-categorical (or categorical in κ) if it has exactly one model of cardinality κ up to isomorphism.

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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.

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Presburger arithmetic

Presburger arithmetic is the first-order theory of the natural numbers with addition, named in honor of Mojżesz Presburger, who introduced it in 1929.

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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.

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T-schema

The T-schema or truth schema (not to be confused with 'Convention T') is used to give an inductive definition of truth which lies at the heart of any realisation of Alfred Tarski's semantic theory of truth.

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Tarski's axioms

Tarski's axioms, due to Alfred Tarski, are an axiom set for the substantial fragment of Euclidean geometry, called "elementary," that is formulable in first-order logic with identity, and requiring no set theory.

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Theory (mathematical logic)

In mathematical logic, a theory (also called a formal theory) is a set of sentences in a formal language.

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Zorn's lemma

Zorn's lemma, also known as the Kuratowski–Zorn lemma, after mathematicians Max Zorn and Kazimierz Kuratowski, is a proposition of set theory that states that a partially ordered set containing upper bounds for every chain (that is, every totally ordered subset) necessarily contains at least one maximal element.

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Redirects here:

Maximal consistent set.

References

[1] https://en.wikipedia.org/wiki/Complete_theory

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