I know that there are an infinite number of hyperreals. But is it true that there are only two hyperreals with standard part equal to $0$ (the "finite" infinitesimal one and the "infinite" hyperreal)?
Put differently, is it wrong to view the hyperreals as a field "generated" by $\mathbb{R} \cup \{\infty, 1/\infty\}$ whereby every real number $r \in \mathbb{R}$ is associated with its hyperreal shadow $s = r + 1/\infty$ with $s \approx r$ uniquely?
On
The wording of your question suggests that you are assuming that the standard part of an infinite hyperreal is equal to $0$. This is not correct. The standard part function is only defined on the subring of ${}^\ast\mathbb{R}$ given by limited (finite) hyperreals. Thus, the standard part of an infinite (unlimited) hyperreal is undefined.
Yes, it’s completely wrong. For example, if $\epsilon$ is any positive infinitesimal (i.e., a positive hyperreal whose standard part is $0$), then so is $\epsilon^2$, and of course $0<\epsilon^2<\epsilon$, so $\epsilon^2\ne\epsilon$, and $\epsilon^2$ is therefore another hyperreal whose standard part is $0$. For that matter, $\epsilon x$ is a positive infinitesimal for each positive standard real number $x$, and no two of these infinitesimals are equal.