Theory (mathematical logic)

theorytheoriesformal theoriesformal theoryalgebraic theoryassociatedfirst-order theoryformal systemset of beliefssubtheories
In mathematical logic, a theory (also called a formal theory) is a set of sentences in a formal language.wikipedia
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Sentence (mathematical logic)

sentencesentencesclosed formula
In mathematical logic, a theory (also called a formal theory) is a set of sentences in a formal language. The complete theory of a structure A is the set of all first-order sentences over the signature of A which are satisfied by A.
A set of sentences is called a theory; thus, individual sentences may be called theorems.

Mathematical logic

formal logicsymbolic logiclogic
In mathematical logic, a theory (also called a formal theory) is a set of sentences in a formal language.
Here a theory is a set of formulas in a particular formal logic and signature, while a model is a structure that gives a concrete interpretation of the theory.

Satisfiability

satisfiablesatisfiability problemsatisfies
This means there is a structure M that satisfies every sentence in the theory.
The four concepts can be raised to apply to whole theories: a theory is satisfiable (valid) if one (all) of the interpretations make(s) each of the axioms of the theory true, and a theory is unsatisfiable (invalid) if all (one) of the interpretations make(s) each of the axioms of the theory false.

Complete theory

completemaximal consistent setcompleteness
A complete consistent theory (or just a complete theory) is a consistent theory T such that for every sentence φ in its language, either φ is provable from T or T \cup {φ} is inconsistent.
In mathematical logic, a theory is complete if, for every formula in the theory's language, that formula or its negation is demonstrable.

First-order logic

predicate logicfirst-orderpredicate calculus
A first-order theory is a set of first-order sentences. For first-order logic, the most important case, it follows from the completeness theorem that the two meanings coincide. The complete theory of a structure A is the set of all first-order sentences over the signature of A which are satisfied by A.
For more information on this subject see List of first-order theories and Theory (mathematical logic)

Ω-consistent theory

ω-consistencyω-consistentω-consistency
In other logics, such as second-order logic, there are syntactically consistent theories that are not satisfiable, such as ω-inconsistent theories.
In mathematical logic, an ω-consistent (or omega-consistent, also called numerically segregative) theory is a theory (collection of sentences) that is not only (syntactically) consistent (that is, does not prove a contradiction), but also avoids proving certain infinite combinations of sentences that are intuitively contradictory.

Consistency

consistentinconsistentlogically consistent
A complete consistent theory (or just a complete theory) is a consistent theory T such that for every sentence φ in its language, either φ is provable from T or T \cup {φ} is inconsistent.
In classical deductive logic, a consistent theory is one that does not contain a contradiction.

Theorem

theoremspropositionconverse
An element \phi\in T of a theory T is then called an axiom of the theory, and any sentence that follows from the axioms (T\vdash\phi) is called a theorem of the theory.
A set of formal theorems may be referred to as a formal theory. A theorem whose interpretation is a true statement about a formal system is called a metatheorem.

Formal system

logical systemdeductive systemsystem of logic
Usually a deductive system is understood from context.
Theory (mathematical logic)

Löwenheim–Skolem theorem

(downward) Löwenheim–Skolem propertydownward Löwenheim–Skolem theoremLöwenheim-Skolem theorem
Löwenheim–Skolem theorem
In mathematical logic, the Löwenheim–Skolem theorem, named for Leopold Löwenheim and Thoralf Skolem, states that if a countable first-order theory has an infinite model, then for every infinite cardinal number κ it has a model of size κ.

True arithmetic

The first of these, called the theory of true arithmetic, cannot be written as the set of logical consequences of any enumerable set of axioms.
This is the theory associated with the standard model of the Peano axioms in the language of the first-order Peano axioms.

Decidability (logic)

decidabledecidabilityundecidable
The theory of (R, +, ×, 0, 1, =) was shown by Tarski to be decidable; it is the theory of real closed fields.
A theory (set of sentences closed under logical consequence) in a fixed logical system is decidable if there is an effective method for determining whether arbitrary formulas are included in the theory.

Interpretability

interpretableinterpretationinterpret
Interpretability
Assume T and S are formal theories.

Structure (mathematical logic)

structuremodelstructures
A satisfiable theory is a theory that has a model.
A structure \mathcal{M} is said to be a model of a theory T if the language of \mathcal{M} is the same as the language of T and every sentence in T is satisfied by \mathcal{M}.

Deduction theorem

Reiterationvirtual rule of inference
Deduction theorem
*If T is a theory and F, G are formulas with F closed, and, then.

Zermelo–Fraenkel set theory

ZFZFCZermelo–Fraenkel
Theories obtained this way include ZFC and Peano arithmetic.
studied a subtheory of ZFC consisting of the axioms of extensionality, union, powerset, replacement, and choice.

Formal language

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

Axiom

axiomspostulateaxiomatic
An element \phi\in T of a theory T is then called an axiom of the theory, and any sentence that follows from the axioms (T\vdash\phi) is called a theorem of the theory.

Logical consequence

entailsentailmentfollows from
In a deductive theory, any sentence which is a logical consequence of one or more of the axioms is also a sentence of that theory.

Principle of explosion

ex falso quodlibetexplosionabsurdity constant
In a deductive system (such as first-order logic) that satisfies the principle of explosion, this is equivalent to requiring that there is no sentence φ such that both φ and its negation can be proven from the theory.

Gödel's completeness theorem

completeness theoremcompletenesscomplete
For first-order logic, the most important case, it follows from the completeness theorem that the two meanings coincide.

Second-order logic

second-ordersecond order logicexistential second-order logic
In other logics, such as second-order logic, there are syntactically consistent theories that are not satisfiable, such as ω-inconsistent theories.

One-to-many (data model)

one-to-manymany-to-oneone-to-many relationships
An interpretation of a theory is the relationship between a theory and some contensive subject matter when there is a many-to-one correspondence between certain elementary statements of the theory, and certain contensive statements related to the subject matter.

Signature (logic)

signaturesignatureslanguage
The complete theory of a structure A is the set of all first-order sentences over the signature of A which are satisfied by A.

Reduct

expandedexpansion
The elementary diagram of A is the set eldiag A of all first-order σ'-sentences that are satisfied by A or, equivalently, the complete (first-order) theory of the natural expansion of A to the signature σ'.