Hyperinteger

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 non-standard analysis, there exists a natural extension:

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

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

Internal sets

The set ∗ Z of all hyperintegers is an internal subset of the hyperreal line ∗ R . The set of all finite hyperintegers (i.e. 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 N and ∗ N . Note that the latter gives a non-standard model of arithmetic in the sense of Skolem.