1. This method is primarily used to construct recursively enumerable sets with particular properties. 2. It is complete but does not have a recursively enumerable set of axioms. 3. Yuri Matiyasevich utilized a method involving Fibonacci numbers, which computably enumerable set is Diophantine. 4. The Matiyasevich / MDRP theorem has shown that recursively enumerable sets are equivalent to Diophantine sets. 5. The domain of any universal computable function is a computably enumerable set but never a computable set. 6. The preimage of a recursively enumerable set under a partial recursive function is a recursively enumerable set. 7. The preimage of a recursively enumerable set under a partial recursive function is a recursively enumerable set . 8. Other texts use the definition in terms of enumerations, which is equivalent for recursively enumerable sets . 9. The set of all provable sentences in an effective axiomatic system is always a recursively enumerable set . 10. As intermediate results, Post defined natural types of recursively enumerable sets like the hypersimple and hyperhypersimple sets.
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