The MIU system

The MIU-system in GEB of Hofstadter :

1. Alphabet

The system uses only three symbols:

• M
• I
• U

2. Formation Rules (Syntax)

A string is a well-formed formula (WFF) if it consists only of the symbols M, I, and U, starting with the letter M.

• Examples of WFFs: MI, MUIU, MU
• Examples of non-WFFs: IMU (does not start with M), ABC (invalid alphabet)

3. Axioms

There is exactly one starting point:

• MI

4. Rules of Inference

Let $x$ and $y$ represent any string of symbols. The system has four rules for deriving new theorems:

1. Rule I: If a string ends in I, you can add a U at the end.

• From MI, you get MIU.

1. Rule II: If you have M$x$, you can add M$xx$.

• From MIU, you get MIUIU.

3. Rule III: If III appears in a string, you can replace it with U.

• From MIII, you get MU.

1. Rule IV: If UU appears in a string, you can drop it entirely.

• From MUU, you get M.

Connection to Gödel numbering

Despite its simplicity, the MIU-system illustrates a foundational idea in metamathematics. By assigning numbers to its symbols (e.g., M↦3, I↦1, U↦0), every string becomes a number and every typographical rule becomes an arithmetic operation. This arithmetization — the core of Gödel numbering — allows a richer system like TNT to express statements about MIU-theorems using only arithmetic formulas.

The formula MUMON in TNT, for instance, asserts that the number 30 (the Gödel number of MU) is reachable from the axiom 31 via the arithmetic versions of Rules I–IV.