Hostname: page-component-848d4c4894-ttngx Total loading time: 0 Render date: 2024-05-04T12:53:17.728Z Has data issue: false hasContentIssue false
  • English
  • Français

Projective well-orderings and bounded forcing axioms

Published online by Cambridge University Press:  12 March 2014

Andrés Eduardo Caicedo
Andrés Eduardo Caicedo*
Affiliation:
Kurt Gödel Research Center For Mathematical Logic, (Formerly, Institut Für Formale Logik), Universität Wien, Währinger Strasse 25, A-1090 Wien, Austria, E-mail: caicedo@logic.univie.ac.at
Get access Rights & Permissions [Opens in a new window]

Abstract

In the absence of Woodin cardinals, fine structural inner models for mild large cardinal hypotheses admit forcing extensions where bounded forcing axioms hold and yet the reals are projectively well-ordered.

Keywords

Fine structureinner modelsstrong cardinalsΣ30-absolutenessforcingbounded forcing axiomsΨACwell-orderings
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 70 , Issue 2 , June 2005 , pp. 557 - 572
Copyright
Copyright © Association for Symbolic Logic 2005

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

REFERENCES

[1] Asperó, D. and Welch, P., Bounded Martin's Maximum, weak Erdös cardinals and ψ A C- this Journal, vol. 67 (2002), no. 3. pp. 1141–1152.Google Scholar
[2] Bagaria, J., Bounded forcing axioms as principles of generic absoulteness, Archive for Mathematical Logic, vol. 39 (2000), no. 6, pp. 393–401.Google Scholar
[3] Baldwin, S., Between strong and superstrong, this Journal, vol. 51 (1986), no. 3, pp. 547–559.Google Scholar
[4] Bartoszyński, T. and Judah, H., Set theory, On the structure of the real line, A K Peters, 1995.Google Scholar
[5] Caicedo, A., Simply definable well-orderings of the reals, Ph.D. Dissertation. Department of Mathematics, University of California, Berkeley, 2003.Google Scholar
[6] Caicedo, A. and Schindler, R., Projective well-orderings of the reals, Archire for Mathematical Logic , (to appear).Google Scholar
[7] Deiser, O. and Donder, D., Canonical functions, non-regular ultrafilters and Ulam's problem on ω1 , this Journal, vol. 68 (2003), no. 3, pp. 713–739.Google Scholar
[8] Donder, D. and Fuchs, U., Revised countable support iterations, Handbook of set theory (M. Foreman. A. Kanamori, and M. Magidor, editors), to appear.Google Scholar
[9] Foreman, M., Magidor, M., and Shelah, S., Martin's Maximum, saturated ideals, and nonregular ultrafilters. I, Annals of Mathematics, vol. 127 (1988), no. 1, pp. 1–47. (FMSh 240).Google Scholar
[10] Friedman, S., Fine structure and class forcing, Walter de Gruyter, 2000.Google Scholar
[11] Fuchino, S., Open coloring axiom and forcing axioms, unpublished manuscript. 2000.Google Scholar
[12] Gitik, M. and Shelah, S., On certain indestructibility of strong cardinals and a question of Hajnal, Archive for Mathematical Logic, vol. 28 (1989), no. 1, pp. 35–42, (GiSh 344).Google Scholar
[13] Goldstern, M. and Shelah, S., The bounded proper forcing axiom, this Journal, vol. 60 (1995), no. 1, pp. 58–73, (GoSh 507).Google Scholar
[14] Hamkins, J., A class of strong diamond principles, preprint.Google Scholar
[15] Harrington, L., Long projective wellorderings, Annals of Mathematical Logic, vol. 12 (1977), no. 1, pp. 1–24.Google Scholar
[16] Jech, T., Multiple forcing, Cambridge University Press, 1986.Google Scholar
[17] Jech, T., Set theory. The third millenium edition, revised and expanded, Springer-Verlag, 2003.Google Scholar
[18] Jensen, R., The fine structure of the constructible hierarchy, Annals of Mathematical Logic, vol. 4 (1972), pp. 229–308.Google Scholar
[19] Judah, H., Δ3 1-sets of reals, Set theory of the reals (Judah, H., editor), Barllan University, 1993, pp. 361–384.Google Scholar
[20] Larson, P., The stationary tower: Notes on a course by W. Hugh Woodin, American Mathematical Society, 2004.Google Scholar
[21] Larson, P. and Shelah, S., Bounding by canonical functions, with CH. Journal of Mathematical Logic, vol. 3 (2003), no. 2, pp. 193–215.CrossRefGoogle Scholar
[22] Laver, R., Making the supercompactness of k indestructible under k-directed closed forcing. Israel Journal of Mathematics, vol. 29 (1978), no. 4, pp. 385–388.Google Scholar
[23] Martin, D. and Solovay, R.. A basis theorem for Σ3 0 sets of reals, Annals of Mathematics (2), vol. 89 (1969), no. 1, pp. 138–159.Google Scholar
[24] Martin, D., Internal Cohen extensions, Annals of Mathematical Logic, vol. 2 (1970), no. 2. pp. 143–178.Google Scholar
[25] Martin, D. and Steel, J., Iteration trees, Journal of the American Mathematical Society, vol. 7 (1994), no. 1, pp. 1–73.Google Scholar
[26] Mitchell, W. and Steel, J., Fine structure and iteration trees, Springer-Verlag, 1994.CrossRefGoogle Scholar
[27] Schindler, R., Bounded Martin's Maximum and strong cardinals, preprint.Google Scholar
[28] Schindler, R., Steel, J., and Zeman, M., Deconstructing inner model theory, this Journal, vol. 67 (2002), no. 2. pp. 721–736.Google Scholar
[29] Schindler, R. and Zeman, M., Fine structure theory. Handbook of set theory (M. Foreman, A. Kanamori, and M. Magidor, editors), to appear.Google Scholar
[30] Steel, J., An outline of inner model theory, Handbook of set theory (M. Foreman, A. Kanamori, and M. Magidor, editors), to appear.Google Scholar
[31] Todorcevic, S., Partition problems in topology, American Mathematical Society, 1989.Google Scholar
[32] Woodin, H., The axiom of determinacy, forcing axioms, and the nonstationary ideal. Walter de Gruyter, 1999.Google Scholar