Gödel's Incompleteness Theorems/First
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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.
The result follows by Proof by Contradiction.
$\blacksquare$
Also see
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