more from Kurt Gödel

### Single Idea 10072

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

Full Idea

First Incompleteness Theorem: any properly axiomatised and consistent theory of basic arithmetic must remain incomplete, whatever our efforts to complete it by throwing further axioms into the mix.

Gist of Idea

First Incompleteness: arithmetic must always be incomplete

Source

report of Kurt Gödel (On Formally Undecidable Propositions [1931]) by Peter Smith - Intro to Gödel's Theorems 1.2

Book Reference

Smith,Peter: 'An Introduction to Gödel's Theorems' [CUP 2007], p.5

A Reaction

This is because it is always possible to formulate a well-formed sentence which is not provable within the theory.