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Generic absoluteness for Σ1 formulas and the continuum problem

Published online by Cambridge University Press:  31 March 2017

Zoé Chatzidakis
Affiliation:
Université de Paris VII (Denis Diderot)
Peter Koepke
Affiliation:
Rheinische Friedrich-Wilhelms-Universität Bonn
Wolfram Pohlers
Affiliation:
Westfälische Wilhelms-Universität Münster, Germany
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Publisher: Cambridge University Press
Print publication year: 2006

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References

[1] D., Asperó, Bounded Forcing Axioms and the Continuum, Ph.D. thesis, U. Barcelona, Barcelona, 2000.
[2] D., Asperó, A maximal bounded forcing axiom, The Journal of Symbolic Logic, vol. 67 (2002), no. 1, pp. 130–142.Google Scholar
[3] D., Asperó, Bounded forcing axioms and the size of the continuum, Logic Colloquium 2000, Lecture Notes in Logic, vol. 19, ASL, Urbana, IL, 2005, pp. 211–227.
[4] D., Asperó, On a convenient property for [γ] ℵ0, Submitted.
[5] D., Asperó, and J., Bagaria, Bounded forcing axioms and the continuum, Annals of Pure and Applied Logic, vol. 109 (2001), no. 3, pp. 179–203.
[6] D., Asperó and P., Welch, Bounded Martin's maximum, weak Erdʺos cardinals, and ψAC, The Journal of Symbolic Logic, vol. 67 (2002), no. 3, pp. 1141–1152.Google Scholar
[7] J., Bagaria, Bounded forcing axioms as principles of generic absoluteness, Archive for Mathematical Logic, vol. 39 (2000), no. 6, pp. 393–401.Google Scholar
[8] J., Baumgartner and A., Taylor, Saturation properties of ideals in generic extensions. I, Transactions of the American Mathematical Society, vol. 270 (1982), no. 2, pp. 557–574.Google Scholar
[9] M., Bekkali, Topics in Set Theory, Lecture Notes in Mathematics, vol. 1476, Springer-Verlag, Berlin, 1991.
[10] O., Deiser and H. D., Donder, Canonical functions, non-regular ultrafilters and Ulam's problem on ω1,The Journal of Symbolic Logic, vol. 68 (2003), no. 3, pp. 713–739.Google Scholar
[11] K. J., Devlin and S., Shelah, A weak version of ♣ which follows from 2ℵ0 2 ℵ1, Israel Journal of Mathematics, vol. 29 (1978), no. 2-3, pp. 239–247.Google Scholar
[12] Q., Feng and T., Jech, Projective stationary sets and a strong reflection principle Journal of the London Mathematical Society. Second Series, vol. 58 (1998), no. 2, pp. 271–283.
[13] M., Foreman, M., Magidor, and S., Shelah, Martin's maximum, saturated ideals, and nonregular ultrafilters. I Annals of Mathematics. Second Series, vol. 127 (1988), no. 1, pp. 1–47.Google Scholar
[14] S., Fuchino, On potential embedding and versions of Martin's axiom Notre Dame Journal of Formal Logic, vol. 33 (1992), no. 4, pp. 481–492.
[15] F., Galvin, T., Jech, and M., Magidor, An ideal game The Journal of Symbolic Logic, vol. 43 (1978), no. 2, pp. 284–292.
[16] M., Gitik, Nonsplitting subset of Pκ (κ+), The Journal of Symbolic Logic, vol. 50 (1985), no. 4, pp. 881–894.Google Scholar
[17] M., Goldstern and S., Shelah, The bounded proper forcing axiom The Journal of Symbolic Logic, vol. 60 (1995), no. 1, pp. 58–73.
[18] J. D., Hamkins, A simple maximality principle The Journal of Symbolic Logic, vol. 68 (2003), no. 2, pp. 527–550.
[19] T., Jech, Set Theory, Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1997.
[20] T., Jech and S., Shelah, A note on canonical functions Israel Journal of Mathematics, vol. 68 (1989), no. 3, pp. 376–380.
[21] D., Kueker, Countable approximations and Löwenheim-Skolem theorems Annals of Pure and Applied Logic, vol. 11 (1977), no. 1, pp. 57–103.
[22] K., Kunen, Set Theory: An Introduction to Independence Proofs, Studies in Logic and the Foundations of Mathematics, vol. 102, North-Holland Publishing Co., Amsterdam, 1980.
[23] P., Larson, The size of T Archive for Mathematical Logic, vol. 39 (2000), no. 7, pp. 541–568.
[24] D., Martin and R., Solovay, Internal Cohen extensions Annals of Pure and Applied Logic, vol. 2 (1970), no. 2, pp. 143–178.
[25] T., Miyamoto, A note on weak segments of PFA Proceedings of the Sixth Asian Logic Conference (Beijing, 1996), World Sci. Publishing, River Edge, NJ, 1998, pp. 175–197.
[26] R., Schindler, Coding into K by reasonable forcing Transactions of the American Mathematical Society, vol. 353 (2001), no. 2, pp. 479–489.
[27] S., Shelah, Semiproper forcing axiom implies Martin maximum but not PFA+, The Journal of Symbolic Logic, vol. 52 (1987), no. 2, pp. 360–367.
[28] S., Shelah Forcing axiom failure for ƛ ℵ1, [Sh 784], Available at http://arxiv.org/abs/math.LO/0112286.
[29] J., Stavi and J., Väänänen, Reflection principles for the continuum Logic and Algebra, Contemporary Mathematics, vol. 302, AMS, Providence, RI, 2002, pp. 59–84.Google Scholar
[30] S., Todorčevič, Oscillations of real numbers Logic Colloquium '86 (Hull, 1986), Studies in Logic and the Foundations of Mathematics, vol. 124, North-Holland, Amsterdam, 1988, pp. 325–331.
[31] S., Todorčevič, Partition Problems in Topology, Contemporary Mathematics, vol. 84, American Mathematical Society, Providence, RI, 1989.Google Scholar
[32] S., Todorčevič, Generic absoluteness and the continuum Mathematical Research Letters, vol. 9 (2002), no. 4, pp. 465–471.
[33] S., Todorčevič, Localized reflection and fragments of PFA Set Theory (Piscataway, NJ, 1999), Dimacs Series, vol. 58, AMS, Providence, RI, 2002, pp. 135–148.
[34] B., Veli čkovič, Forcing axioms and stationary sets Advances in Mathematics, vol. 94 (1992), no. 2, pp. 256–284.
[35] P. D., Welch, On unfoldable cardinals, ω-closed cardinals, and the beginning of the inner model hierarchy Archive for Mathematical Logic, vol. 43 (2004), no. 4, pp. 443–458.
[36] H., Woodin, The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal, de Gruyter Series in Logic and its Applications, vol. 1, Walter de Gruyter & Co., Berlin, 1999.

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