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
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