Transfer principle for the hyperreals

Statement

The transfer principle concerns the logical relation between the properties of the real numbers R, and the properties of a larger field denoted *R called the hyperreal numbers. The field *R includes, in particular, infinitesimal ("infinitely small") numbers, providing a rigorous mathematical realisation of a project initiated by Leibniz.

The idea is to express analysis over R in a suitable language of mathematical logic, and then point out that this language applies equally well to *R. This turns out to be possible because at the set-theoretic level, the propositions in such a language are interpreted to apply only to internal sets rather than to all sets. As Robinson put it, the sentences of [the theory] are interpreted in *R in Henkin's sense.

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 $1/n$ for some positive integer $n$") seems at first sight not to satisfy the transfer principle. The statement "every positive hyperreal is larger than $1/n$ for some positive integer $n$" is false; however the correct interpretation is "every positive hyperreal is larger than $1/n$ for some positive hyperinteger $n$". 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.