# Skolem, only ×

**A system based only on primes: Skolem arithmetic**

This is the theory of whole numbers with multiplication only, no addition. Every number is a unique bundle of primes (12 = 2·2·3), so multiplication means “combine the bundles”. In that world you can talk about primes, divisibility and “infinitely many primes” with no trouble.

It is complete: every statement it can express is provable or disprovable, and a computer can even decide which. Gödel does not touch it. Addition alone (Presburger arithmetic) is also complete.

Connects to:
- [primes: provable](https://mapideas.org/i/s_primes)
- [why it is safe](https://mapideas.org/i/s_safe)
- [primes in the trick](https://mapideas.org/i/th_primes)

Sources:
- [Wikipedia: Presburger arithmetic (the addition-only sibling)](https://en.wikipedia.org/wiki/Presburger_arithmetic)

Source: https://mapideas.org/i/s_skolem · open on the map: https://mapideas.org/#s_skolem