 \(\bNs\) is the set of points of intersection of the smooth curve \(y = \sin \pi x\) with the positive \(x\)-axis. In these models \(\bR\) can be shown to possess the Archimedean property provided that in the definition \(\bN \) is replaced by \(\bNs\). In these models, then, \(\bN \) is a proper subset of \(\bNs\): the members of \(\bNs - \bN \) may be considered nonstandard integers. Multiplicative inverses of nonstandard integers are infinitesimals, but, being themselves invertible, they are of a different type from the ones we have considered so far. It is quite easy to show that they, as well as the infinitesimals in \(J\) (and so also those in \(\Delta\) and \(I)\) are all contained in the set—a further infinitesimal neighborhood of 0—