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Full Idea
It has turned out that the solution of certain arithmetical problems requires the use of assumptions essentially transcending arithmetic.
Gist of Idea
Some arithmetical problems require assumptions which transcend arithmetic
Source
Kurt Gödel (Russell's Mathematical Logic [1944], p.449)
Book Ref
'Philosophy of Mathematics: readings (2nd)', ed/tr. Benacerraf/Putnam [CUP 1983], p.449
A Reaction
A nice statement of the famous result, from the great man himself, in the plainest possible English.
| 10041 | Impredicative Definitions refer to the totality to which the object itself belongs [Gödel] |
| 21716 | In simple type theory the axiom of Separation is better than Reducibility [Gödel, by Linsky,B] |
| 10035 | Mathematical Logic is a non-numerical branch of mathematics, and the supreme science [Gödel] |
| 10038 | A logical system needs a syntactical survey of all possible expressions [Gödel] |
| 10039 | Some arithmetical problems require assumptions which transcend arithmetic [Gödel] |
| 10042 | Reference to a totality need not refer to a conjunction of all its elements [Gödel] |
| 10043 | Mathematical objects are as essential as physical objects are for perception [Gödel] |
| 10045 | Impredicative definitions are admitted into ordinary mathematics [Gödel] |
| 10046 | The generalized Continuum Hypothesis asserts a discontinuity in cardinal numbers [Gödel] |