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5. Theory of Logic / K. Features of Logics / 8. Enumerability

[whether all formulae in a system can be specified]

10 ideas
There are infinite sets that are not enumerable [Cantor, by Smith,P]
A logical system needs a syntactical survey of all possible expressions [Gödel]
For a reasonable language, the set of valid wff's can always be enumerated [Enderton]
A complete logic has an effective enumeration of the valid formulas [Tharp]
Effective enumeration might be proved but not specified, so it won't guarantee knowledge [Tharp]
A set is 'enumerable' is all of its elements can result from a natural number function [Smith,P]
A set is 'effectively enumerable' if a computer could eventually list every member [Smith,P]
A finite set of finitely specifiable objects is always effectively enumerable (e.g. primes) [Smith,P]
The set of ordered pairs of natural numbers <i,j> is effectively enumerable [Smith,P]
The thorems of a nice arithmetic can be enumerated, but not the truths (so they're diffferent) [Smith,P]