Definition:Initial Segment of Natural Numbers

Also denoted as
The usual notation for these are $\N_n$ and $\N^*_n$, but the notations $\N_{< n}$ and $\N^*_{\le n}$ are less ambiguous.

Also defined as
Some sources consider $n$ as an integer and use the symbology:
 * $\Z \left({n}\right) = \left\{{1, 2, \ldots, n}\right\} = \left\{{z \in \Z: 1 \le z \le n}\right\}$

but this is rare.

Some sources use $\mathbf P_n$ or similar, for either $\N_{< n}$ or $\N^*_{\le n}$, where $\mathbf P$ may stand for positive.

There is considerable inconsistency in the literature. , for example, has $S_n = \left\{{1, 2, \ldots, n - 1}\right\}$.