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In non-standard analysis, a hyperinteger N is a hyperreal number 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 (1,2,3,...) in the ultrapower construction of the hyperreals.

Discussion [edit]

The standard integer part function:


is defined for all real x and equals the greatest integer not exceeding x. By the transfer principle of non-standard analysis, there exists a natural extension:


defined for all hyperreal x, and we say that x is a hyperinteger if:

 x = {}^*\![x].

Thus the hyperintegers are the image of the integer part function on the hyperreals.

Internal sets [edit]

The set ^*\mathbb{Z} of all hyperintegers is an internal subset of the hyperreal line ^*\mathbb{R}. The set of all finite hyperintegers (i.e. \mathbb{Z} itself) is not an internal subset. Elements of the complement


are called, depending on the author, non-standard, unlimited, or infinite hyperintegers. The reciprocal of an infinite hyperinteger is an infinitesimal.

Positive hyperintegers are sometimes called hypernatural numbers. Similar remarks apply to the sets \mathbb{N} and ^*\mathbb{N}. Note that the latter gives a non-standard model of arithmetic in the sense of Skolem.

References [edit]