# Gödel's First Incompleteness Theorem

## Theorem

Let $T$ be the set of theorems of some recursive set of sentences in the language of arithmetic such that $T$ contains minimal arithmetic.

$T$ cannot be both consistent and complete.

### Corollary

If $T$ is both consistent and complete, it does not contain minimal arithmetic.

## Proof

Aiming for a contradiction, suppose that such a $T$ is consistent and complete.

By the Undecidability Theorem, since $T$ is consistent and contains $Q$, it is not recursive.

But, by Complete Recursively Axiomatized Theories are Recursive, since $T$ is complete and is the set of theorems of a recursive set, it is recursive.

Thus, we have a contradiction.

$\blacksquare$

## Source of Name

This entry was named for Kurt Friedrich Gödel.

## Sources

- 1974: George S. Boolos and Richard C. Jeffrey:
*Computability and Logic*: $\S 15$: Theorem $6$