Gödel's incompleteness theorems

related topics
{math, number, function}
{theory, work, human}

Gödel's incompleteness theorems are two theorems of mathematical logic that establish inherent limitations of all but the most trivial axiomatic systems for mathematics. The theorems, proven by Kurt Gödel in 1931, are important both in mathematical logic and in the philosophy of mathematics. The two results are widely interpreted as showing that Hilbert's program to find a complete and consistent set of axioms for all of mathematics is impossible, thus giving a negative answer to Hilbert's second problem.

The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an "effective procedure" (essentially, a computer program) is capable of proving all facts about the natural numbers. For any such system, there will always be statements about the natural numbers that are true, but that are unprovable within the system. The second incompleteness theorem shows that if such a system is also capable of proving certain basic facts about the natural numbers, then one particular arithmetic truth the system cannot prove is the consistency of the system itself.

Contents

Full article ▸

related documents
First-order logic
Geometric algebra
Vector space
Perl
Integral
Exponentiation
Model theory
Fourier transform
Propositional calculus
Lisp (programming language)
Vienna Development Method
Generic programming
Emmy Noether
Singular value decomposition
C++
System of linear equations
Combinatory logic
Formal power series
Discrete cosine transform
Orthogonal matrix
Dedekind domain
Hilbert's tenth problem
Singleton pattern
Closure (computer science)
Fourier series
Recurrence relation
Markov chain
Zermelo–Fraenkel set theory
Matrix multiplication
Addition