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