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A mathematical model is an abstract description of a concrete system using mathematical concepts and language.
This page focuses on finitary first order model theory of infinite structures.. The relative emphasis placed on the class of models of a theory as opposed to the class of definable sets within a model fluctuated in the history of the subject, and the two directions are summarised by the pithy characterisations from 1973 and 1997 respectively:
A model describes how units of computations, memories, and communications are organized. [1] The computational complexity of an algorithm can be measured given a model of computation. Using a model allows studying the performance of algorithms independently of the variations that are specific to particular implementations and specific technology.
In model theory, the case of being algebraically closed and its prime field is especially important. While vector spaces are modular and affine spaces are "almost" modular (i.e. everywhere locally modular), algebraically closed fields are examples of the other extremity, not being even locally modular (i.e. none of the localizations is modular).
Since any model of P* is a model of P (since a model of a set of axioms is obviously also a model of any subset of that set of axioms), we have that our extended model is also a model of the Peano axioms. The element of this model corresponding to x cannot be a standard number, because as indicated it is larger than any standard number.
Model theory is the study of the relationship between formal theories (a collection of sentences in a formal language expressing statements about a mathematical structure), and their models (those structures in which the statements of the theory hold). The aspects investigated include the number and size of models of a theory, the relationship ...
In model theory and related areas of mathematics, a type is an object that describes how a (real or possible) element or finite collection of elements in a mathematical structure might behave. More precisely, it is a set of first-order formulas in a language L with free variables x 1 , x 2 ,..., x n that are true of a set of n -tuples of an L ...
The model is a parametric model if Θ ⊆ ℝ k for some positive integer k. When the model consists of absolutely continuous distributions, it is often specified in terms of corresponding probability density functions :