Diagram (mathematical logic)

Short description: Concept in model theory

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.

## Definition

Let ℒ be a first-order language and T be a theory over ℒ. For a model 𝔄 of T one expands ℒ to a new language

ℒA:=ℒ∪{ca:a∈A}

by adding a new constant symbol ca for each element a in A, where A is a subset of the domain of 𝔄. Now one may expand 𝔄 to the model

𝔄A:=(𝔄,a)a∈A.

The positive diagram of 𝔄, sometimes denoted D+(𝔄), is the set of all those atomic sentences which hold in 𝔄 while the negative diagram, denoted D−(𝔄), thereof is the set of all those atomic sentences which do not hold in 𝔄.

The diagram D(𝔄) of 𝔄 is the set of all atomic sentences and negations of atomic sentences of ℒA that hold in 𝔄A.[1][2] Symbolically, D(𝔄)=D+(𝔄)∪¬D−(𝔄).

## See also

* Elementary diagram

## References

1. ↑ Hodges, Wilfrid (1993). Model theory. Cambridge University Press. ISBN 9780521304429. https://archive.org/details/modeltheory0000hodg.
2. ↑ Chang, C. C.; Keisler, H. Jerome (2012). Model Theory (Third ed.). Dover Publications. pp. 672 pages.

* v
* t
* e

Mathematical logic

General|

* Formal language
* Formation rule
* Formal proof
* Formal semantics
* Well-formed formula
* Set
* Element
* Class
* Classical logic
* Axiom
* Rule of inference
* Relation
* Theorem
* Logical consequence
* Type theory
* Symbol
* Syntax
* Theory

Systems|

* Formal system
* Deductive system
* Axiomatic system
* Hilbert style systems
* Natural deduction
* Sequent calculus

Traditional logic|

* Proposition
* Inference
* Argument
* Validity
* Cogency
* Syllogism
* Square of opposition
* Venn diagram

Propositional calculus
and Boolean logic

* Boolean functions
* Propositional calculus
* Propositional formula
* Logical connectives
* Truth tables
* Many-valued logic

Predicate logic|

* First-order
* Quantifiers
* Predicate
* Second-order
* Monadic predicate calculus

Naive set theory|

* Set
* Empty set
* Element
* Enumeration
* Extensionality
* Finite set
* Infinite set
* Subset
* Power set
* Countable set
* Uncountable set
* Recursive set
* Domain
* Codomain
* Image
* Map
* Function
* Relation
* Ordered pair

Set theory|

* Foundations of mathematics
* Zermelo–Fraenkel set theory
* Axiom of choice
* General set theory
* Kripke–Platek set theory
* Von Neumann–Bernays–Gödel set theory
* Morse–Kelley set theory
* Tarski–Grothendieck set theory

Model theory|

* Model
* Interpretation
* Non-standard model
* Finite model theory
* Truth value
* Validity

Proof theory|

* Formal proof
* Deductive system
* Formal system
* Theorem
* Logical consequence
* Rule of inference
* Syntax

Computability
theory|

* Recursion
* Recursive set
* Recursively enumerable set
* Decision problem
* Church–Turing thesis
* Computable function
* Primitive recursive function

0.00

(0 votes)

Original source: https://en.wikipedia.org/wiki/Diagram (mathematical logic). Read more

|

Retrieved from "https://handwiki.org/wiki/index.php?title=Diagram_(mathematical_logic)&oldid=175552"
*[v]: View this template
*[t]: Discuss this template
*[e]: Edit this template