# primes: provable

**Primes are provable, so where is the unprovable part?**

The system proves that there are infinitely many primes, since Euclid’s argument can be written inside it. Gödel does not say everything is unprovable. Peano arithmetic covers addition, multiplication, ordering, divisibility and primes, and primes are only one thing it can talk about.

The unprovable statements are of two kinds. One is the Gödel sentence, which looks like an ordinary claim about numbers but is built so that its meaning points back at itself. The other is natural examples, like Goodstein’s theorem: you rewrite a number in a special way, subtract 1, bump the base, and repeat. It is true that the sequence always ends at 0, and ordinary arithmetic cannot prove it. So incompleteness lives at the edges of everyday math.

Connects to:
- [what a system is](https://mapideas.org/i/s_sys)
- [Skolem, only ×](https://mapideas.org/i/s_skolem)
- [primes in the trick](https://mapideas.org/i/th_primes)
- [complete, incomplete](https://mapideas.org/i/w_comp)
- [Goldbach](https://mapideas.org/i/s_gold)

Sources:
- [Wikipedia: Euclid’s theorem](https://en.wikipedia.org/wiki/Euclid%27s_theorem)

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