Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-09T06:27:43.635Z Has data issue: false hasContentIssue false

Forcing and reductibilities. II. Forcing in fragments of analysis

Published online by Cambridge University Press:  12 March 2014

Piergiorgio Odifreddi*
Affiliation:
University of Turin, TurinItaly

Extract

In Forcing and reducibilities, I [20] we reviewed various forcing techniques in the context of arithmetic. This second part deals with the same topics in the context of analysis. The numbering of sections is the continuation of the numbering of the first part. For the reader's convenience, we collect in an independent bibliography the papers which are referred to in this part.

Type
Research Article
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.)

References

BIBLIOGRAPHY

[1]Enderton, H.B. and Putnam, H., A note on the hyperarithmetical hierarchy, this Journal, vol. 35 (1970), pp. 429430.Google Scholar
[2]Feferman, S., Some applications of the notion of forcing and generic sets, Fundamenta Mathematicae, vol. 56 (1965), pp. 325345.CrossRefGoogle Scholar
[3]Feferman, S. and Spector, C., Incompleteness along paths in progressions of theories, this Journal, vol. 27 (1962), pp. 383390.Google Scholar
[4]Feiner, L, The strong homogeneity conjecture, this Journal, vol. 35 (1970), pp. 375377.Google Scholar
[5]Gandy, R.O., Proof of Mostowski's conjecture, L'Académie Polonaise des Sciences. Bulletin. Série des Sciences Mathématiques, vol. 8 (1960), pp. 571575.Google Scholar
[6]Gandy, R.O., Kreisel, G. and Tait, W.W., Set existence, L'Académie Polonaise des Sciences. Bulletin. Série des Sciences Mathématiques, vol. 8 (1960), pp. 577582.Google Scholar
[7]Gandy, R.O. and Sacks, G.E., A minimal hyperdegree, Fundamenta Mathematicae, vol. 61 (1967), pp. 215233.CrossRefGoogle Scholar
[8]Hinman, P.G., Some applications of forcing to hierarchy problems in arithmetic, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 15 (1969), pp. 341352.CrossRefGoogle Scholar
[9]Hodes, H.T., Jumping through the transfinite: the master code hierarchy of Turing degree, this Journal, vol. 45 (1980), pp. 204220.Google Scholar
[10]Jockusch, C., Recursiveness of initial segments of Kleene's , Fundamenta Mathematicae, vol. 87 (1975), pp. 161167.CrossRefGoogle Scholar
[11]Jockusch, C. and Simpson, S.G., A degree-theoretic definition of the ramified analytic hierarchy, Annals of Mathematical Logic, vol. 10 (1975), pp. 132.CrossRefGoogle Scholar
[12]Kleene, S.C., Hierarchies of number theoretic predicates, Bulletin of the American Mathematical Society, vol. 61 (1955), pp. 193213.CrossRefGoogle Scholar
[13]Kleene, S.C., Quantification of number theoretic functions, Compositia Mathematica, vol. 14 (1959), pp. 2340.Google Scholar
[14]Kreisel, G., Set-theoretic problems suggested by the notion of potential totability, Infinitistic methods, Pergamon Press, New York, 1961, pp. 103140.Google Scholar
[15]Kreisel, G., The axiom of choice and the class of hyperarithmetic functions, Indagationes Mathematicae, vol. 24 (1962), pp. 307319.CrossRefGoogle Scholar
[16]Kreisel, G. and Sacks, G.E., Metarecursive sets, this Journal, vol. 30 (1965), pp. 318338.Google Scholar
[17]Kripke, S., Transfinite recursion on admissible ordinals. I, II (Abstracts), this Journal, vol. 29 (1964), pp. 161162.Google Scholar
[18]Macintyre, J.M., Transfinite extensions of Friedberg's completeness criterion, this Journal, vol. 42 (1977), pp. 110.Google Scholar
[19]Nerode, A. and Shore, R.A., Reducibility orderings: Theories, definability and automorphisms, Annals of Mathematical Logic, vol. 18 (1980), pp. 6189.CrossRefGoogle Scholar
[20]Odifreddi, P.G., Forcing and reducibilities, this Journal, vol. 48 (1983), pp. 288310.Google Scholar
[21]Posner, D. and Robinson, R.W., Degrees joining to 0, this Journal, vol. 46 (1981), pp. 714722.Google Scholar
[22]Rogers, H., Theory of recursive functions and effective computability, McGraw-Hill, New York, 1967.Google Scholar
[23]Sacks, G. E., Forcing with perfect closed sets, Proceedings of Symposia in Pure Mathematics, vol. 13, Part I, American Mathematical Society, Providence, R.I., 1971, pp. 331355.Google Scholar
[24]Sacks, G. E., On the reducibility of sets, Advances in Mathematics, vol. 7 (1971), pp. 5782.CrossRefGoogle Scholar
[25]Sacks, G. E., RE sets higher up, Logic, foundations of mathematics and computability theory, (Butts, R. and Hintikka, J., Editors), Reidel, Dordrecht, 1977, pp. 173194.CrossRefGoogle Scholar
[26]Shore, R.A., The homogeneity conjecture, Proceedings of the National Academy of Sciences of the United States of America, vol. 76 (1979), pp. 42184219.CrossRefGoogle ScholarPubMed
[27]Shore, R.A., On homogeneity and definability in the first order theory of Turing degrees, this Journal, vol. 47 (1982), pp. 816.Google Scholar
[28]Simpson, S.G., Minimal covers and hyperdegrees, Transactions of the American Mathematical Society, vol. 209 (1975), pp. 4564.CrossRefGoogle Scholar
[29]Shore, R.A., First-order theory of the degrees of recursive unsolvability, Annals of Mathematics, vol. 105 (1977), pp. 121139.Google Scholar
[30]Spector, C., Recursive well-orderings, this Journal, vol. 20 (1955), pp. 151163.Google Scholar
[31]Spector, C., Hyperarithmetical quantifiers, Fundamenta Mathematicae, vol. 48 (1959), pp. 313320.CrossRefGoogle Scholar
[32]Tanaka, H., On analytic well-orderings, this Journal, vol. 35 (1970), pp. 198204.Google Scholar
[33]Thomason, S.K., The forcing method and the upper semilattice of hyperdegrees, Transactions of the American Mathematical Society, vol. 129 (1967), pp. 3857.CrossRefGoogle Scholar
[34]Thomason, S.K., On initial segments of hyperdegrees, this Journal, vol. 35 (1970), pp. 189197.Google Scholar