Gödel's First Incompleteness Theorem
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.
Source of Name
This entry was named for Kurt Friedrich Gödel.
- 1974: George S. Boolos and Richard C. Jeffrey: Computability and Logic: $\S 15$: Theorem $6$
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Entry: Gödel's proof
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Entry: Gödel's Incompleteness Theorems