A more general class of sets consists of the recursively enumerable sets, also called "'semidecidable "'sets.
22.
This can be used to give a rigorous proof of G�del's first incompleteness theorem, because no recursively enumerable set is productive.
23.
The proof above shows that for each recursively enumerable set of axioms there is a recursive set of axioms with the same deductive closure.
24.
Matiyasevich showed this problem to be unsolvable by mapping a Diophantine equation to a recursively enumerable set and invoking G�del's Incompleteness Theorem.
25.
The theory of algebraically closed fields of a given characteristic is complete, consistent, and has an infinite but recursively enumerable set of axioms.
26.
Very soon after this, Friedberg and Muchnik independently solved Post's problem by establishing the existence of recursively enumerable sets of intermediate degree.
27.
Any proof ( in some traditional sense ) is just a finite string of characters taken from an enumerable set, and is therefore enumerable.
28.
For example, it may be used to represent sparse bitsets, i . e ., subsets of a much larger, fixed enumerable set.
29.
The fundamental results establish a robust, canonical class of computable functions with numerous independent, equivalent characterizations using Turing machines, lattice of recursively enumerable sets.
30.
Also, since all functions in these languages are total, algorithms for recursively enumerable sets cannot be written in these languages, in contrast with Turing machines.
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