Integer interval ($[a,b]\cap \mathbb{Z}$)

A finite integer interval is a set $\{a,a+1,\dots ,b\}$ with integers $a\le b$. Its cardinality is $b-a+1$, including both endpoints. For example, $\{3,4,5,6,7\}$ contains five integers, although the real interval from $3$ to $7$ has length four.

Infinite integer intervals allow one or both endpoints to be unbounded. Statements about bounded gaps in words quantify over finite integer intervals; their endpoints are indices, rather than arbitrary real positions.