# Hyperinteger

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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

The standard integer part function:

$[x]$

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:

$^*[\,\cdot\,]$

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

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

$^*\mathbb{Z}\setminus\mathbb{Z}$

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.