Gödel's Incompleteness Theorems/First/Corollary
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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.
If $T$ is both consistent and complete, it does not contain minimal arithmetic.
Proof
This is simply the contrapositive of Gödel's First Incompleteness Theorem.
$\blacksquare$
Source of Name
This entry was named for Kurt Friedrich Gödel.
Sources
- 2007: George S. Boolos, John P. Burgess and Richard C. Jeffrey: Computability and Logic (5th ed.): $\S 15$: Theorem $6$: Corollary