Gödel's Incompleteness Theorems

Theorem

Gödel's First Incompleteness 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.

Gödel's Second Incompleteness 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.

Let $\map {\mathrm {Cons} } T$ be the propositional function which states that $T$ is consistent.

Then it is not possible to prove $\map {\mathrm {Cons} } T$ by means of formal statements within $T$ itself.

Also see

• Gödel's Completeness 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): Entry: incompleteness theorems
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Entry: Gödel's Incompleteness Theorems