Editing Nonstandard analysis (section)

== Internal sets ==
A set ''x'' is ''internal'' if and only if ''x'' is an element of *''A'' for some element ''A'' of {{math|''V''('''R''')}}. *''A'' itself is internal if ''A'' belongs to {{math|''V''('''R''')}}.

We now formulate the basic logical framework of nonstandard analysis:
* ''Extension principle'': The mapping * is the identity on {{math|'''R'''}}.
* ''Transfer principle'': For any formula {{math|''P''(''x''<sub>1</sub>, ..., ''x<sub>n</sub>'')}} with bounded quantification and with free variables {{math|''x''<sub>1</sub>, ..., ''x<sub>n</sub>''}}, and for any elements {{math|''A''<sub>1</sub>, ..., ''A<sub>n</sub>''}} of {{math|''V''('''R''')}}, the following equivalence holds:

::<math>P(A_1, \ldots, A_n) \iff P(*A_1, \ldots, *A_n) </math>

* ''Countable saturation'': If {''A''<sub>''k''</sub>}<sub>''k'' ∈ '''N'''</sub> is a decreasing sequence of nonempty internal sets, with ''k'' ranging over the natural numbers, then

::<math>\bigcap_k A_k \neq \emptyset </math>

One can show using ultraproducts that such a map * exists. Elements of {{math|''V''('''R''')}} are called ''standard''. Elements of {{math|*'''R'''}} are called [[hyperreal number]]s.