Algebra and Logic

, Volume 41, Issue 2, pp 81–86

# Friedberg Numberings of Families of n-Computably Enumerable Sets

• S. S. Goncharov
• S. Lempp
• D. R. Solomon
Article

## Abstract

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 (treated as a family of d.c.e. sets) 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 (under reducibility) which is Friedberg but is not the only numbering (modulo reducibility).

Friedberg numbering, computably enumerable set.

## REFERENCES

1. 1.
R. F. Friedberg, “Three theorems on recursive enumeration. I. Decomposition. II. Maximal set. III. Enumeration without duplication,” J. Symb. Log., 23, No. 3, 309-316 (1958).Google Scholar
2. 2.
Yu. L. Ershov, “Theorie der Numerierungen I,” Z. Math. Log. Grund. Math., 19, No. 4, 289-388 (1973); “Theorie der Numerierungen II,” Z. Math. Log. Grund. Math., 21, No. 6, 473-584 (1975); “Theorie der Numerierungen III,” Z. Math. Log. Grund. Math., 23, No. 4, 289-371 (1977).Google Scholar
3. 3.
Yu. L. Ershov, Theory of Numerations [in Russian], Nauka, Moscow (1977).Google Scholar
4. 4.
Yu. L. Ershov, “Theory of numberings,” in Handbook of Computability Theory, North-Holland, Amsterdam (1999) pp. 473-503.Google Scholar
5. 5.
A. H. Lachlan, “On recursive numeration without repetition,” Z. Math. Log. Grund. Math., 11, No. 3, 209-220 (1965); “A correction to: 'On recursive numeration without repetition',” Z. Math. Log. Grund. Math., 13, No. 2, 99-100 (1967).Google Scholar
6. 6.
M. B. Pour-El and W. A. Howard, “A structural criterion for recursive numeration without repetition,” Z. Math. Log. Grund. Math., 10, No. 2, 105-114 (1964).Google Scholar
7. 7.
M. B. Pour-El and H. Putnam, “Recursively enumerable classes and their applications to sequences of formal theories,” Arch. Math. Log. Gr., 8, 104-121 (1965).Google Scholar
8. 8.
H. Putnam, “Trial and error predicates and the solution to a problem of Mostowski,” J. Symb. Log., 30, No. 1, 49-57 (1965).Google Scholar
9. 9.
S. S. Goncharov and A. Sorbi, “Generalized computable numerations and nontrivial Rogers semilattices,” Algebra Logika, 36, No. 6, 621-641 (1997).Google Scholar
10. 10.
S. S. Goncharov, “One-to-one computable numerations,” Algebra Logika, 19, No. 5, 507-551 (1980).Google Scholar
11. 11.
A. I. Mal'tsev, Algorithms and Recursive Functions [in Russian], Nauka, Moscow (1965).Google Scholar
12. 12.
M. Kummer, “Recursive numerations without repetition revisited,” Lect. Notes Comp. Sc., Vol. 1432, Springer, Berlin (1990), pp. 255-275.Google Scholar
13. 13.
S. S. Goncharov, “Limit-equivalent constructivizations,” Trudy Inst. Mat. SO RAN, Vol. 2, Nauka, Novosibirsk (1982), pp. 4-12.Google Scholar
14. 14.
S. S. Goncharov, A. Yakhnis, and V. Yakhnis, “Some effectively infinite classes of enumerations,” Ann. Pure Appl. Log., 60, No. 3, 207-236 (1993).Google Scholar
15. 15.
S. S. Goncharov, “Positive numerations of families with one-to-one numerations,” Algebra Logika, 22, No. 5, 481-488 (1983).Google Scholar
16. 16.
P. Odifreddi, Classical Recursion Theory, Vol. 1, North-Holland, Amsterdam (1989).Google Scholar