Editing Transfer principle (section)

== Constructions of the hyperreals ==

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<ref name=kanovei2003>{{Citation | last1=Kanovei|first1=Vladimir| last2=Shelah|first2=Saharon| title=A definable nonstandard model of the reals| url=http://shelah.logic.at/files/825.pdf| journal=Journal of Symbolic Logic|volume=69|year=2004| pages=159–164 | doi=10.2178/jsl/1080938834|arxiv=math/0311165|s2cid=15104702 }}</ref> 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 equivalence|elementary embedding]] between structures.