# The arithmetic hierarchy of real functions

> Source: <https://www.lesswrong.com/posts/kLiJnEfavGAxXfiNr/the-arithmetic-hierarchy-of-real-functions>
> Published: 2026-06-28 19:11:19+00:00

I wrote a fairly accessible [introduction to real hypercomputation](https://www.sciencedirect.com/science/article/pii/S0890540126000738) with Marcus Hutter. The focus is on enabling applications to algorithmic information theory. This project was intended to build my technical foundations for studying AIXI, but took me a bit further afield and down some rabbit holes. In the future I will prefer to focus more tightly on AI safety.

Feedback would be appreciated. In particular, I needed to introduce an extra extensionality assumption for the real domain case, which I am still not sure is necessary.

**Errata:** The diagram of results currently has theorems misnumbered due to a typographical error.

Thanks to the LTFF for supporting my work over most of the research process.