It is known (see [1, 3.1.5]) that the order type of the nonstandard natural number system $^\ast N$ has the form $\omega + (\omega^\ast + \omega) \theta$, where $\theta$ is a dense order type without first or last element and $\omega$ is the order type of $N$. Concerning this, Zakon [2] examined $^\ast N$ more closely and investigated the nonstandard real number system $^\ast R$, as an ordered set, as an additive group and as a uniform space. He raised five questions which remained unsolved. These questions are concerned with the cofinality and coinitiality of $\theta$ (which depend on the underlying nonstandard universe $^\ast U$). In this paper, we shall treat nonstandard models where the cofinality and coinitiality of $\theta$ coincide with some appropriated cardinals. Using these nonstandard models, we shall give answers to three of these questions and partial answers to the other to questions in [2].