Gödel's First Incompleteness Theorem

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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