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A combinatorial forcing for coding the universe by a real when there are no sharps

Published online by Cambridge University Press:  12 March 2014

Saharon Shelah
The Hebrew University of Jerusalem, Department of Mathematics, Jerusalem, Israel, E-mail: Rutgers University, Department of Mathematics, New Brunswick, New Jersey 08903, E-mail:
Lee J. Stanley
Lehigh University, Department of Mathematics, Bethlehem, PA 18015, E-mail:


Assuming 0# does not exist, we present a combinatorial approach to Jensen's method of coding by a real. The forcing uses combinatorial consequences of fine structure (including the Covering Lemma, in various guises), but makes no direct appeal to fine structure itself.

Research Article
Copyright © Association for Symbolic Logic 1995

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[1]Beller, A., Jensen, R., and Welch, P., Coding the universe, London Mathematical Society Lecture Note Series, vol. 47, Cambridge University Press, Cambridge, 1982.CrossRefGoogle Scholar
[2]David, R., Some applications of Jensen's coding theorem, Annals of Mathematical Logic, vol. 22 (1982), pp. 177196.CrossRefGoogle Scholar
[3]David, R., reals, Annals of Pure and Applied Logic, vol. 23 (1982), pp. 121125.Google Scholar
[4]David, R., A functorial singleton, Advances in Mathematics, vol. 74 (1989), pp. 258268.CrossRefGoogle Scholar
[5]Friedman, S., A guide to ‘Coding the universe’ by Beller, Jensen, Welch, this Journal, vol. 50 (1985), pp. 10021019.Google Scholar
[6]Friedman, S., An immune partition of the ordinals, Recursion theory week; proceedings Oberwolfach 1984, Lecture Notes in Mathematics, vol 1121, (Ebbinghaus, H.-D., et al., editors), Springer-Verlag, Berlin, 1985, pp. 141147.Google Scholar
[7]Friedman, S., Strong coding, Annals of Pure and Applied Logic, vol. 35 (1987), pp. 1–98, 99122.CrossRefGoogle Scholar
[8]Friedman, S., Coding over a measurable cardinal, this Journal, vol. 54 (1989), pp. 11451159.Google Scholar
[9]Friedman, S., Minimal coding, Annals of Pure and Applied Logic, vol. 41 (1989), pp. 233297.CrossRefGoogle Scholar
[10]Friedman, S., The -singleton conjecture, Journal of the American Mathematical Society, vol. 3 (1990), pp. 771791.Google Scholar
[11]Friedman, S., A simpler proof of Jensen's coding theorem, Annals of Pure and Applied Logic (to appear).Google Scholar
[12]Friedman, S., A large Π set, absolute for set forcings, Proceedings of the American Mathematical Society, vol. 122 (1994), pp. 253256.Google Scholar
[13]Shelah, S., Proper forcing, Lecture Notes in Mathematics, vol 940, Springer-Verlag, Berlin, 1982.CrossRefGoogle Scholar
[14]Shelah, S., Cardinal arithmetic, Oxford University Press, Oxford (to appear).Google Scholar
[15]Shelah, S. and Stanley, L., Corrigendum to ‘Generalized Martin's axiom and Souslin's hypothesis for higher cardinals’, Israel Journal of Mathematics, vol. 53 (1986), pp. 304314.CrossRefGoogle Scholar
[16]Shelah, S. and Stanley, L., Coding and reshaping when there are no sharps, Set theory of the continuum, Mathematical Sciences Research Institute Publications, vol. 26 (Judah, H., et al.), Springer-Verlag, Berlin, 1992, pp. 407416.CrossRefGoogle Scholar
[17]Shelah, S. and Stanley, L., The combinatorics of combinatorial coding by a real, this Journal, vol. 60 (1994), pp. 3657.Google Scholar