Модельдер аралық қағидасы немесе көшіру принципі

That all statements of some language that are true for some structure are true for another structure

In model theory, a transfer principle states that all statements of some language that are true for some structure are true for another structure. One of the first examples was the Lefschetz principle, which states that any sentence in the first order language of fields that is true for the complex numbers is also true for any algebraically closed field of characteristic 0.

An incipient form of a transfer principle was described by Leibniz under the name of "the Law of Continuity". Here infinitesimals are expected to have the "same" properties as appreciable numbers. The transfer principle can also be viewed as a rigorous formalization of the principle of permanence. Similar tendencies are found in Cauchy, who used infinitesimals to define both the continuity of functions (in Cours d'Analyse) and a form of the Dirac delta function. The theorem to the effect that each proposition valid over R, is also valid over *R, is called the transfer principle. There are several different versions of the transfer principle, depending on what model of nonstandard mathematics is being used. In terms of model theory, the transfer principle states that a map from a standard model to a nonstandard model is an elementary embedding (an embedding preserving the truth values of all statements in a language), or sometimes a bounded elementary embedding (similar, but only for statements with bounded quantifiers). The transfer principle appears to lead to contradictions if it is not handled correctly. For example, since the hyperreal numbers form a non Archimedean ordered field and the reals form an Archimedean ordered field, the property of being Archimedean ("every positive real is larger than for some positive integer ") seems at first sight not to satisfy the transfer principle. The statement "every positive hyperreal is larger than for some positive integer " is false; however the correct interpretation is "every positive hyperreal is larger than for some positive hyperinteger ". In other words, the hyperreals appear to be Archimedean to an internal observer living in the nonstandard universe, but appear
to be non Archimedean to an external observer outside the universe. A freshman level accessible formulation of the transfer principle is Keisler's book Elementary Calculus: An Infinitesimal Approach.

The transfer principle however doesn't mean that R and *R have identical behavior. For instance, in *R there exists an element ω such that

but there is no such number in R. This is possible because the nonexistence of this number cannot be expressed as a first order statement of the above type. A hyperreal number like ω is called infinitely large; the reciprocals of the infinitely large numbers are the infinitesimals. The hyperreals *R form an ordered field containing the reals R as a subfield. Unlike the reals, the hyperreals do not form a standard metric space, but by virtue of their order they carry an order topology.

The hyperreals can be developed either axiomatically or by more constructively oriented methods. The essence of the axiomatic approach is to assert (1) the existence of at least one infinitesimal number, and (2) the validity of the transfer principle. In the following subsection we give a detailed outline of a more constructive approach. This method allows one to construct the hyperreals if given a set theoretic object called an ultrafilter, but the ultrafilter itself cannot be explicitly constructed. Vladimir Kanovei and Shelah give a construction of a definable, countably saturated elementary extension of the structure consisting of the reals and all finitary relations on it. In its most general form, transfer is a bounded elementary embedding between structures.