Non-standard model
Encyclopedia
In model theory
, a discipline within mathematical logic
, a non-standard model is a model of a theory that is not isomorphic to the intended model
(or standard model). If the intended model is infinite and the language is first-order
, then the Löwenheim-Skolem theorems guarantee the existence of non-standard models. The non-standard models can be chosen as elementary extensions or elementary substructure
s of the intended model.
Non-standard models are studied in set theory
, non-standard analysis
, and non-standard model of arithmetic.
Model theory
In mathematics, model theory is the study of mathematical structures using tools from mathematical logic....
, a discipline within mathematical logic
Mathematical logic
Mathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...
, a non-standard model is a model of a theory that is not isomorphic to the intended model
Intended interpretation
One who constructs a syntactical system usually has in mind from the outset some interpretation of this system. While this intended interpretation can have no explicit indication in the syntactical rules - since these rules must be strictly formal - the author's intention respecting...
(or standard model). If the intended model is infinite and the language is first-order
First-order logic
First-order logic is a formal logical system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic...
, then the Löwenheim-Skolem theorems guarantee the existence of non-standard models. The non-standard models can be chosen as elementary extensions or elementary substructure
Elementary substructure
In model theory, a field within mathematical logic, two structures M and N of the same signature σ are called elementarily equivalent if they satisfy the same first-order σ-sentences....
s of the intended model.
Non-standard models are studied in set theory
Set theory
Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
, non-standard analysis
Non-standard analysis
Non-standard analysis is a branch of mathematics that formulates analysis using a rigorous notion of an infinitesimal number.Non-standard analysis was introduced in the early 1960s by the mathematician Abraham Robinson. He wrote:...
, and non-standard model of arithmetic.