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EQUIVALENCE RELATIONS WHICH ARE BOREL SOMEWHERE

  • WILLIAM CHAN (a1)
Abstract

The following will be shown: Let I be a σ-ideal on a Polish space X so that the associated forcing of I + ${\bf{\Delta }}_1^1$ sets ordered by ⊆ is a proper forcing. Let E be a ${\bf{\Sigma }}_1^1$ or a ${\bf{\Pi }}_1^1$ equivalence relation on X with all equivalence classes ${\bf{\Delta }}_1^1$ . If for all $z \in {H_{{{\left( {{2^{{\aleph _0}}}} \right)}^ + }}}$ , z exists, then there exists an I + ${\bf{\Delta }}_1^1$ set CX such that EC is a ${\bf{\Delta }}_1^1$ equivalence relation.

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[1] Apter, A. W., Gitman, V., and Hamkins, J. D., Inner models with large cardinal features usually obtained by forcing . Archive for Mathematical Logic, vol. 51 (2012), no. 3–4, pp. 257283.
[2] Bagaria, J. and Friedman, S. D., Generic absoluteness . Proceedings of the XIth Latin American Symposium on Mathematical Logic (Mérida, 1998), vol. 108 (2001), pp. 313.
[3] Burgess, J. P., Descriptive set theory and infinitary languages , Set Theory, Foundations of Mathematics (Proceeding of Symposia, Belgrade, 1977), Matematički institut SANU (Nova Serija), Zbornik Radova, vol.2(10), 1977, pp. 930.
[4] Burgess, J. P., Effective enumeration of classes in a equivalence relation . Indiana University Mathematics Journal, vol. 28 (1979), no. 3, pp. 353364.
[5] Caicedo, A. E. and Schindler, R., Projective well-orderings of the reals . Archive for Mathematical Logic, vol. 45 (2006), no. 7, pp. 783793.
[6] Chan, W., The countable admissible ordinal equivalence relation. Annals of Pure and Applied Logic, vol. 168 (2016), pp. 12241246.
[7] Chan, W., Canonicalization by absoluteness, Notes.
[8] Chan, W. and Magidor, M., When an equivalence relation with all Borel classes will be Borel somewhere? 2016, arXiv e-prints.
[9] Clemens, J. D., Equivalence relations which reduce all Borel equivalance relations, Available at http://www.math.uni-muenster.de/u/jclemens/public/Papers/aboveBorel.pdf.
[10] Devlin, K. J., An introduction to the fine structure of the constructible hierarchy (results of Ronald Jensen (Ann. Math. Logic 4 (1972), 229–308; erratum, ibid. 4(1972), 443)) , Generalized Recursion Theory (Proceedings of Symposia, University of Oslo, Oslo, 1972), Studies in Logic and the Foundations of Mathematics, vol. 79, North-Holland, Amsterdam, 1974, pp. 123163.
[11] Drucker, O., Borel canonization of analytic sets with Borel sections, 2015, arXiv e-prints.
[12] Feng, Q., Magidor, M., and Woodin, H., Universally Baire sets of reals , Set Theory of the Continuum (Berkeley, CA, 1989), Mathematical Sciences Research Institute Publications, vol. 26, Springer, New York, 1992, pp. 203242.
[13] Friedman, S. D., Minimal coding . Annals of Pure and Applied Logic, vol. 41 (1989), no. 3, pp. 233297.
[14] Hamkins, J. D. and Hugh Woodin, W., The necessary maximality principle for c.c.c. forcing is equiconsistent with a weakly compact cardinal . Mathematical Logic Quarterly, vol. 51 (2005), no. 5, pp. 493498.
[15] Hjorth, G., Thin equivalence relations and effective decompositions, this Journal, vol. 58 (1993), no. 4, pp. 11531164.
[16] Jech, T., Set Theory, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003.
[17] Jensen, R. B., The fine structure of the constructible hierarchy . Annals of Mathematical Logic, vol. 4 (1972), pp. 229308; erratum, ibid. 4(1972), 443.
[18] Jensen, R., Definable sets of minimal degree . Studies in Logic and the Foundations of Mathematics, vol. 59 (1970), pp. 122128.
[19] Kanovei, V., Sabok, M., and Zapletal, J., Canonical Ramsey Theory on Polish Spaces, Cambridge Tracts in Mathematics, vol. 202, Cambridge University Press, Cambridge, 2013.
[20] Kechris, A. S., Measure and category in effective descriptive set theory . Annals of Mathematical Logic, vol. 5 (1972/73), pp. 337384.
[21] Kechris, A. S., Classical Descriptive Set Theory, Graduate Texts in Mathematics, vol. 156, Springer-Verlag, New York, 1995.
[22] Kechris, A. S. and Louveau, A., The classification of hypersmooth Borel equivalence relations . Journal of the American Mathematical Society, vol. 10 (1997), no. 1, pp. 215242.
[23] Kunen, K., Set Theory, Studies in Logic (London), vol. 34, College Publications, London, 2011.
[24] Mansfield, R. and Weitkamp, G., Recursive Aspects of Descriptive Set Theory, Oxford Logic Guides, vol. 11, The Clarendon Press, Oxford University Press, New York, 1985.
[25] Martin, D. A. and Solovay, R. M., A basis theorem for sets of reals . Annals of Mathematics (2), vol. 89 (1969), pp. 138159.
[26] Martin, D. A. and Solovay, R. M., Internal Cohen extensions . Annals of Mathematical Logic, vol. 2 (1970), no. 2, pp. 143178.
[27] Neeman, I. and Norwood, Z., Happy and mad families in L(ℝ), Available at http://www.math.ucla.edu/∼ineeman.
[28] Schindler, R., Set Theory, Universitext, Springer, Cham, 2014.
[29] Schindler, R. and Zeman, M., Fine structure , Handbook of Set Theory (Foreman, M. and Kanamori, A., editors), Springer, Dordrecht, 2010, pp. 605656.
[30] Schindler, R.-D., Proper forcing and remarkable cardinals. II, this Journal, vol. 66 (2001), no. 3, pp. 1481–1492.
[31] Shelah, S., Can you take Solovay’s inaccessible away? Israel Journal of Mathematics, vol. 48 (1984), no. 1, pp. 147.
[32] Solovay, R. M., A model of set-theory in which every set of reals is Lebesgue measurable . Annals of Mathematics (2), vol. 92 (1970), pp. 156.
[33] Zapletal, J., Descriptive set theory and definable forcing . Memoirs of the American Mathematical Society, vol. 167 (2004), no. 793.
[34] Zapletal, J., Forcing Idealized, Cambridge Tracts in Mathematics, vol. 174, Cambridge University Press, Cambridge, 2008.
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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
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