more from Kurt Gödel

### Single Idea 10118

#### [catalogued under 6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / g. Incompleteness of Arithmetic]

Full Idea

First Incompleteness Theorem: If S is a sufficiently powerful formal system, then if S is consistent then S is syntactically incomplete.

Gist of Idea

First Incompleteness: a decent consistent system is syntactically incomplete

Source

report of Kurt Gödel (On Formally Undecidable Propositions [1931]) by A.George / D.J.Velleman - Philosophies of Mathematics Ch.6

Book Reference

George,A/Velleman D.J.: 'Philosophies of Mathematics' [Blackwell 2002], p.161

A Reaction

Gödel found a single sentence, effectively saying 'I am unprovable in S', which is neither provable nor refutable in S.