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Full Idea
The Enumerability Theorem says that for a reasonable language, the set of valid wff's can be effectively enumerated.
Clarification
a 'wff' is a well-formed formula
Gist of Idea
For a reasonable language, the set of valid wff's can always be enumerated
Source
Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], 2.5)
Book Ref
Enderton,Herbert B.: 'A Mathematical Introduction to Logic' [Academic Press 2001], p.142
A Reaction
There are criteria for what makes a 'reasonable' language (probably specified to ensure enumerability!). Predicates and functions must be decidable, and the language must be finite.
| 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] |