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.

Historical Note

Gödel's Incompleteness Theorems answered the second of Hilbert's $23$ (then) unsolved problems of mathematics.

Hence it ended attempts, like those of Alfred North Whitehead and Bertrand Russell, to develop the whole of mathematics from a finite set of logical axioms.

It also damages the idea of finding a finite set of basic axioms of physics to define all natural phenomena.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): incompleteness theorems
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): incompleteness theorems
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Gödel's Incompleteness Theorems