Hyperinteger

In nonstandard analysis, a hyperinteger "n" is a hyperreal number that is equal to its own integer part. A hyperinteger may be either finite or infinite. A finite hyperinteger is an ordinary integer. An example of an infinite hyperinteger is given by the class of the sequence in the ultrapower construction of the hyperreals.
Discussion.
The standard integer part function: 
is defined for all real "x" and equals the greatest integer not exceeding "x". By the transfer principle of nonstandard analysis, there exists a natural extension: 
defined for all hyperreal "x", and we say that "x" is a hyperinteger if formula_3 Thus the hyperintegers are the image of the integer part function on the hyperreals.
Internal sets.
The set formula_4 of all hyperintegers is an internal subset of the hyperreal line formula_5. The set of all finite hyperintegers (i.e. formula_6 itself) is not an internal subset. Elements of the complement formula_7 are called, depending on the author, "nonstandard", "unlimited", or "infinite" hyperintegers. The reciprocal of an infinite hyperinteger is always an infinitesimal.
Nonnegative hyperintegers are sometimes called "hypernatural" numbers. Similar remarks apply to the sets formula_8 and formula_9. Note that the latter gives a non-standard model of arithmetic in the sense of Skolem.