Editing Hyperreal number (section)

== Transfer principle ==
{{Main|Transfer principle}}
The idea of the hyperreal system is to extend the real numbers '''R''' to form a system *'''R''' that includes infinitesimal and infinite numbers, but without changing any of the elementary axioms of algebra. Any statement of the form "for any number x&nbsp;..." that is true for the reals is also true for the hyperreals. For example, the axiom that states "for any number ''x'', ''x''&nbsp;+&nbsp;0&nbsp;=&nbsp;''x''" still applies. The same is true for [[quantification (logic)|quantification]] over several numbers, e.g., "for any numbers ''x'' and ''y'', ''xy''&nbsp;=&nbsp;''yx''." This ability to carry over statements from the reals to the hyperreals is called the [[transfer principle]]. However, statements of the form "for any ''set'' of numbers ''S'' ..." may not carry over. The only properties that differ between the reals and the hyperreals are those that rely on quantification over [[Set (mathematics)|sets]], or other higher-level structures such as functions and relations, which are typically constructed out of sets. Each real set, function, and relation has its natural hyperreal extension, satisfying the same first-order properties.  The kinds of logical sentences that obey this restriction on quantification are referred to as statements in [[first-order logic]].

The transfer principle, however, does not mean that '''R''' and *'''R''' have identical behavior. For instance, in *'''R''' there exists an element ''&omega;'' such that
: <math> 1<\omega, \quad 1+1<\omega, \quad 1+1+1<\omega, \quad 1+1+1+1<\omega, \ldots. </math>
but there is no such number in '''R'''. (In other words, *'''R''' is not [[Archimedean property|Archimedean]].) This is possible because the nonexistence of ''&omega;'' cannot be expressed as a first-order statement.