 In SIA the set \(\bN \) of natural numbers can be defined to be the smallest subset of \(\bR\) which contains 0 and is closed under the operation of adding 1. In some models of SIA, \(\bR\) satisfies the Archimedean principle that every real number is majorized by a natural number. However, models of SIA have been constructed (Moerdijk &amp; Reyes 1991) in which \(\bR\) is not Archimedean in this sense. In these models it is more natural to consider, in place of \(\bN \), the set \(\bNs\) of smooth natural numbers defined by