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Full Idea
The theorems of any properly axiomatized theory can be effectively enumerated. However, the truths of any sufficiently expressive arithmetic can't be effectively enumerated. Hence the theorems and truths of arithmetic cannot be the same.
Clarification
In an expressive arithmetic we can enumerate theorems, but not truths
Gist of Idea
The thorems of a nice arithmetic can be enumerated, but not the truths (so they're diffferent)
Source
Peter Smith (Intro to Gödel's Theorems [2007], 05 Intro)
Book Ref
Smith,Peter: 'An Introduction to Gödel's Theorems' [CUP 2007], p.37
| 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] |
| 10764 | A complete logic has an effective enumeration of the valid formulas [Tharp] |
| 10768 | Effective enumeration might be proved but not specified, so it won't guarantee knowledge [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] |