Is there an element of the hyperreals minus the reals that isn't a hyperirrational?

Question

I was looking to see if there exist elements in *R-R that aren't in *Q? The ideal answer would offer some different ways to understand and see this as I still lack an intuition for these sorts of questions beyond trying to crudely apply the transfer theorem.

Answer 1

Removing $\mathbb{R}$ from $^*\mathbb{R}$ doesn't change much:

• Suppose $N$ is an "infinite integer" (= $N\in{}^*\mathbb{N}\setminus\mathbb{N}$). Then $N\in {}^*\mathbb{R}\setminus\mathbb{R}$ and $N\in {}^*\mathbb{Q}$ (we have $\mathbb{N}\subseteq\mathbb{Q}$, so ${}^*\mathbb{N}\subseteq{}^*\mathbb{Q}$).

• Conversely, if $N$ is an infinite integer, then $N\cdot \pi\in{}^*\mathbb{R}\setminus\mathbb{R}$ and $N\cdot\pi\not\in{}^*\mathbb{Q}$ (as above - note that by transfer, $\pi\not\in\mathbb{Q}\implies\pi\not\in{}^*\mathbb{Q}$).

In general, in "big" situations you'll see all the "standard" phenomena reappearing at the "nonstandard" levels.