# Gödel's Incompleteness Theorems

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## 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

## 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**