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In the mathematical field of model theory, the elementary diagram of a structure is the set of all sentences with parameters from the structure that are true in the structure. It is also called the complete diagram .
In model theory, a branch of mathematical logic, the diagram of a structure is a simple but powerful concept for proving useful properties of a theory, for example the amalgamation property and the joint embedding property, among others.
A structure is said to be a model of a theory if the language of is the same as the language of and every sentence in is satisfied by . Thus, for example, a "ring" is a structure for the language of rings that satisfies each of the ring axioms, and a model of ZFC set theory is a structure in the language of set theory that satisfies each of the ...
A sentence diagram is a pictorial representation of the grammatical structure of a sentence. The term "sentence diagram" is used more when teaching written language, where sentences are diagrammed. The model shows the relations between words and the nature of sentence structure and can be used as a tool to help recognize which potential ...
The complete theory of all sentences satisfied by a structure is also called the theory of that structure. It's a consequence of Gödel's completeness theorem (not to be confused with his incompleteness theorems ) that a theory has a model if and only if it is consistent , i.e. no contradiction is proved by the theory.
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 ϕ ∈ T {\displaystyle \phi \in T} of a deductively closed theory T {\displaystyle T} is then called a theorem of the theory.
Sentences are then built up out of atomic sentences by applying connectives and quantifiers. A set of sentences is called a theory; thus, individual sentences may be called theorems. To properly evaluate the truth (or falsehood) of a sentence, one must make reference to an interpretation of the theory.
A process theory is a system of ideas that explains how an entity changes and develops. [1] Process theories are often contrasted with variance theories, that is, systems of ideas that explain the variance in a dependent variable based on one or more independent variables. While process theories focus on how something happens, variance theories ...