Mathematics for AIXI and Gödel machine

Classical logic up to Goedel's theorem.
Fairly standard Bayesian probability.
A little bit of coding (up to Kraft's inequality).
Computability and algorithmic complexity.
Solomonoff induction.

It's all pretty basic stuff, but taken from a variety of disciplines.

The things you directly need are algorithmic complexity theory (Classic textbook: Li and Vitanyi) and some way of understanding proofs (Probably start with regular mathematical logic / model theory, not sure of a standout textbook here, maybe Manin?, then look into modal logic, classic textbook by Boolos).

Prerequisites for those are mathematical logic, set theory, probability theory, and some amount of discrete math.

What (parts of) the areas mentioned in the other posts do you already know? That may significantly affect the specific recommendation and approach.

Elementary probability theory, elementary theoretical computer science, Solomonoff's theory of inductive inference, reinforcement learning.

I think you have to invent, not just to learn this.