Gödel's Incompleteness Theorems/First

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.

The result follows by Proof by Contradiction.

$\blacksquare$

Also see

• Gödel's Second Incompleteness Theorem

Source of Name

This entry was named for Kurt Friedrich Gödel.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Gödel's proof
• 2007: George S. Boolos, John P. Burgess and Richard C. Jeffrey: Computability and Logic (5th ed.): $\S 15$: Theorem $6$
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Gödel's proof
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Gödel's Incompleteness Theorems