Skip to main content
×
×
Home

GENERIC LARGE CARDINALS AND SYSTEMS OF FILTERS

  • GIORGIO AUDRITO (a1) and SILVIA STEILA (a2)
Abstract

We introduce the notion of ${\cal C}$ -system of filters, generalizing the standard definitions of both extenders and towers of normal ideals. This provides a framework to develop the theory of extenders and towers in a more general and concise way. In this framework we investigate the topic of definability of generic large cardinals properties.

Copyright
References
Hide All
[1] Audrito, G., Generic large cardinals and absoluteness, Ph.D. thesis, University of Torino, 2016. Available at http://hdl.handle.net/2318/1557477.
[2] Claverie, B., Ideals, ideal extenders and forcing axioms, Ph.D. thesis, Münster (Westfalen), 2010.
[3] Cody, B. and Cox, S., Indestructibility of generically strong cardinals . Fundamenta Mathematicae, vol. 232 (2016), pp. 131149.
[4] Cox, S. and Viale, M., Martin’s maximum and tower forcing . Israel Journal of Mathematics, vol. 197 (2013), no. 1, pp. 347376.
[5] Cummings, J., Iterated forcing and elementary embeddings , Handbook of Set Theory (Foreman, M. and Kanamori, A., editors), Springer, Dordrecht, 2010, pp. 775883.
[6] Dimonte, V. and Friedman, S.-D., Rank-into-rank hypotheses and the failure of GCH . Archive for Mathematical Logic, vol. 53 (2014), no. 3–4, pp. 351366.
[7] Dimonte, V. and Wu, L., A general tool for consistency results related to I1 . European Journal of Mathematics, vol. 2 (2016), no. 2, pp. 474492.
[8] Foreman, M., Ideals and generic elementary embeddings , Handbook of Set Theory (Foreman, M. and Kanamori, A., editors), Springer, Dordrecht, 2010, pp. 8851147.
[9] Foreman, M., Calculating quotient algebras of generic embeddings . Israel Journal of Mathematics, vol. 193 (2013), no. 1, pp. 309341.
[10] Hamkins, J. D., Kirmayer, G., and Perlmutter, N. L., Generalizations of the Kunen inconsistency . Annals of Pure and Applied Logic, vol. 163 (2012), no. 12, pp. 18721890.
[11] Jech, T., Set Theory, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003.
[12] Kakuda, Y., On a condition for cohen extensions which preserve precipitous ideals, this JOURNAL, vol. 46 (1981), no. 02, pp. 296300.
[13] Kanamori, A., The Higher Infinite, second ed., Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003.
[14] Koellner, P., Very large cardinals, preprint, 2015.
[15] Larson, P. B., The Stationary Tower, University Lecture Series, vol. 32, American Mathematical Society, Providence, RI, 2004.
[16] Magidor, M., Precipitous ideals and sets . Israel Journal of Mathematics, vol. 35 (1980), no. 1–2, pp. 109134.
[17] Suzuki, A., Non-existence of generic elementary embeddings into the ground model . Tsukuba Journal of Mathematics, vol. 22 (1998), no. 2, pp. 343347.
[18] Viale, M., Category forcings, MM+++ , and generic absoluteness for the theory of strong forcing axioms . Journal of the American Mathematical Society, vol. 29 (2016), no. 3, pp. 675728.
[19] Viale, M., Audrito, G., Carroy, R., and Steila, S., Notes on iterated forcing and tower forcing, preprint, 2017.
[20] Viale, M., Audrito, G., and Steila, S., A boolean algebraic approach to semiproper iterations, preprint, 2014, arXiv:1402.1714.
[21] Woodin, W. H., Suitable extender models II: Beyond ω-huge . Journal of Mathematical Logic, vol. 11 (2011), no. 2, pp. 115436.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Keywords