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

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.

## 1 Answer

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.