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".
[1]: 903 In 1955, Jerzy Łoś proved the transfer principle for any hyperreal number system.
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.
As Robinson put it, the sentences of [the theory] are interpreted in *R in Henkin's sense.
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).
[clarification needed] 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
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.
, partly because the reals do not correspond to any physical reality (in the sense of measurement and computation) different from that represented by
Thus, the notion of an irrational number is meaningless to even the most powerful floating-point computer.
The necessity for such an extension stems not from physical observation but rather from the internal requirements of mathematical coherence.
As already mentioned above, the mathematical justification for this latest extension was delayed by three centuries.
Keisler wrote: The self-consistent development of the hyperreals turned out to be possible if every true first-order logic statement that uses basic arithmetic (the natural numbers, plus, times, comparison) and quantifies only over the real numbers was assumed to be true in a reinterpreted form if we presume that it quantifies over hyperreal numbers.
Note that, in many formulas in analysis, quantification is over higher-order objects such as functions and sets, which makes the transfer principle somewhat more subtle than the above examples suggest.
The transfer principle however doesn't mean that R and *R have identical behavior.
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.
Vladimir Kanovei and Shelah[3] 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.
Some other members of *R, the reciprocals y of the nonzero infinitesimals, are infinite, i.e., The underlying set of the field *R is the image of R under a mapping A ↦ *A from subsets A of R to subsets of *R. In every case with equality if and only if A is finite.
Thus, the well-ordering property of the natural numbers by transfer yields the fact that every internal subset of