document: Friedberg numberings of families of n-computably enumerable sets We establish a number of results on numberings , in particular , on Friedberg numberings , of families of d.c.e. sets . First , it is proved that there exists a Friedberg numbering of the family of all d.c.e. sets . We also show that this result , patterned on Friedberg 's famous theorem for the family of all c.e. sets , holds for the family of all n-c.e. sets for any n > 2 . Second , it is stated that there exists an infinite family of d.c.e. sets without a Friedberg numbering . Third , it is shown that there exists an infinite family of c.e. sets -LRB- treated as a family of d.c.e. sets -RRB- with a numbering which is unique up to equivalence . Fourth , it is proved that there exists a family of d.c.e. sets with a least numbering -LRB- under reducibility -RRB- which is Friedberg but is not the only numbering -LRB- modulo reducibility -RRB-
doc_bio_tags: B I O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O
extractive_keyphrases: friedberg numberings infinite family
abstractive_keyphrases: computability theory families of n-computably enumerable sets