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Full Idea
In order to be sure that new expression can be translated into expressions not containing them, it is necessary to have a survey of all possible expressions, and this can be furnished only by syntactical considerations.
Gist of Idea
A logical system needs a syntactical survey of all possible expressions
Source
Kurt Gödel (Russell's Mathematical Logic [1944], p.448)
Book Ref
'Philosophy of Mathematics: readings (2nd)', ed/tr. Benacerraf/Putnam [CUP 1983], p.448
A Reaction
[compressed]
| 10082 | There are infinite sets that are not enumerable [Cantor, by Smith,P] |
| 10038 | A logical system needs a syntactical survey of all possible expressions [Gödel] |
| 9997 | For a reasonable language, the set of valid wff's can always be enumerated [Enderton] |
| 10768 | Effective enumeration might be proved but not specified, so it won't guarantee knowledge [Tharp] |
| 10764 | A complete logic has an effective enumeration of the valid formulas [Tharp] |
| 10081 | A set is 'enumerable' is all of its elements can result from a natural number function [Smith,P] |
| 10083 | A set is 'effectively enumerable' if a computer could eventually list every member [Smith,P] |
| 10084 | A finite set of finitely specifiable objects is always effectively enumerable (e.g. primes) [Smith,P] |
| 10085 | The set of ordered pairs of natural numbers <i,j> is effectively enumerable [Smith,P] |
| 10601 | The thorems of a nice arithmetic can be enumerated, but not the truths (so they're diffferent) [Smith,P] |