A finding closely related to the completeness theorem is the Löwenheim-Skolem theorem (1915, 1920), named after Leopold Löwenheim, a German schoolteacher, and Skolem, which says that if a sentence (or a formal system) has any model, it has a countable or enumerable model (i.e., a model whose members can be matched with the positive integers).
Thus Gödel was able to assert that the set of theorems of mathematics is recursively enumerable, and, more recently, the American linguist Noam Chomsky (born 1928) could say that the set of grammatical sentences of a natural language, such as English, is recursively enumerable.
We have seen that, although the set V¹ of valid formulas of first-order logic is computably enumerable, the corresponding set V² for second-order logic (with the standard semantics) is vastly more complex.
Then A2 is a complete computably enumerable subset of true second-order arithmetic.
set, i.e., the complement of a computably enumerable set.
Making evidence available to the learner in some fixed order can certainly alter the picture quite radically (Gold proved that if some primitive-recursive generator controls the text it can in effect encode the identity of the target language so that all computably enumerable languages become identifiable from text).
The set of all finite grammars formulable in any given metalanguage is computably enumerable, so grammars can be systematically numbered.
This is because the set of valid FPO sentences is not recursively enumerable.
Suppose for the sake of contradiction that the set of valid IFL sentences is recursively enumerable.
While intuitively plausible, μKF suffers the same expressive incompleteness as KF: Since the minimal Kripkean fixed point forms a complete Π11 set and the internal theory of μKF remains recursively enumerable, there are standard models of the theory in which the interpretation of the truth predicate is not actually the minimal fixed point.
Among the French mathematicians, the semi-intuitionist Borel (1908) introduced the distinction between effectively enumerable sets and denumerable ones.
