Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-25T06:09:55.088Z Has data issue: false hasContentIssue false
  • English
  • Français

Some recent developments in higher recursion theory1

Published online by Cambridge University Press:  12 March 2014

Sy D. Friedman
Sy D. Friedman*
Affiliation:
Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
Get access Rights & Permissions [Opens in a new window]

Abstract

In recent years higher recursion theory has experienced a deep interaction with other areas of logic, particularly set theory (fine structure, forcing, and combinatorics) and infinitary model theory. In this paper we wish to illustrate this interaction by surveying the progress that has been made in two areas: the global theory of the κ-degrees and the study of closure ordinals.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 48 , Issue 3 , September 1983 , pp. 629 - 642
Copyright
Copyright © Association for Symbolic Logic 1983

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

1

This paper accompanies an invited lecture given by the author at the Spring 1981 Meeting of the ASL in Milwaukee, Wisconsin. The preparation of this paper was supported by NSF Grant # MCS 7906084.

References

REFERENCES

Barwise, J. [69], Infinitary logic and admissible sets, this Journal, vol. 34, pp. 226–252.Google Scholar
Barwise, J. [75], Admissible sets and structures, Springer-Verlag, New York.Google Scholar
David, R. [82], Some applications of Jensen's coding theorem, Annals of Mathematical Logic, vol. 22, pp. 177–196.CrossRefGoogle Scholar
Devlin, K. and Jensen, R. [74], Marginalia to a theorem of Silver, ⊨ ISILC Logic Conference (Kiel, 1974), Lecture Notes in Mathematics, vol. 499, Springer-Verlag, New York, pp. 115–142.Google Scholar
Friedman, H. and Jensen, R. [68], Note on admissible ordinals, The Syntax and Semantics of Infinitary Language, Lecture Notes in Mathematics, vol. 72, Springer-Verlag, New York, pp. 77–79.CrossRefGoogle Scholar
Friedman, S. [81A], Steel forcing and Barwise compactness, Annals of Mathematical Logic, vol. 22, pp. 31–46.CrossRefGoogle Scholar
Friedman, S. [81B], Negative solutions to Post's problem, II, Annals of Mathematics, ser. 2, vol. 113, pp. 25–43.CrossRefGoogle Scholar
Friedman, S. [82], Negative solutions to Post's problem. III, in preparation.Google Scholar
Friedman, S. [81C], Uncountable admissibles, I: Forcing, Transactions of the American Mathematical Society, vol. 270 (1982), pp. 61–73.Google Scholar
Friedman, S. [81D], Uncountable admissibles, II: Compactness, Israel Journal of Mathematics, vol. 40, pp. 129–149.CrossRefGoogle Scholar
Friedman, S. [81E], Model theory for , Proceedings of the 1980/81 Logic Year at Jerusalem, to appear.Google Scholar
Hodes, H. [80], Jumping through the transfinite: The master code hierarchy of Turing degrees, this Journal, vol. 45, pp. 204–220.Google Scholar
Jensen, R. [72A], The fine structure of the constructible hierarchy, Annals of Mathematical Logic, vol. 4, pp. 229–308.CrossRefGoogle Scholar
Jensen, R. [72B], Forcing over admissible sets, handwritten notes by K. Devlin.Google Scholar
Jensen, R. [75], Coding the universe by a real, unpublished manuscript.Google Scholar
Jensen, R. and Solovay, R. [70], Some applications of almost disjoint sets. Mathematical Logic and Foundations of Set Theory (International Colloquium, Jerusalem, 1968), North-Holland, Amsterdam, 1970, pp. 84–104.Google Scholar
Kanamori, A. [80], Perfect-set forcing for uncountable cardinals, Annals of Mathematical Logic, vol. 19, pp. 97–114.CrossRefGoogle Scholar
Keisler, H.J. [71], Model theory for infinitary logic, North-Holland, Amsterdam.Google Scholar
Magidor, M., Shelah, S. and Stavi, J. [81], Countably decomposable admissible sets, Annals of Mathematical Logic (to appear).Google Scholar
Nadel, M. [74], Scott sentences and admissible sets, Annals of Mathematical Logic, vol. 7, pp. 267–294.CrossRefGoogle Scholar
Sacks, G. [76], Countable admissible ordinals and hyperdegrees, Advances in Mathematics, vol. 20, pp. 213–262.CrossRefGoogle Scholar
Sacks, G. and Slaman, T. [82], Inadmissible forcing, to appear.Google Scholar
Shore, R. [77], α-recursion theory, Handbook of Mathematical Logic, North-Holland, Amsterdam, pp. 653–680.Google Scholar
Silver, J. [74], On the singular cardinals problem, Proceedings of the International Congress of Mathematicians (Vancouver, 1974), Canadian Mathematical Congress, Montreal, 1975, vol. 1, pp. 265–268.Google Scholar
Simpson, S. [77], Degrees of unsolvability: A survey of results, Handbook of Mathematical Logic, North-Holland, Amsterdam, pp. 631–652.Google Scholar
Simpson, S. [80], The hierarchy based on the jump operator, The Kleene Symposium (Madison, Wisconsin, 1978), North-Holland, Amsterdam, 1980, pp. 267–276.CrossRefGoogle Scholar
Steel, J. [78], Forcing with tagged trees, Annals of Mathematical Logic, vol. 15, pp. 55–74.CrossRefGoogle Scholar