Hostname: page-component-8448b6f56d-42gr6 Total loading time: 0 Render date: 2024-04-23T17:34:43.816Z Has data issue: false hasContentIssue false
  • English
  • Français

HAPPY AND MAD FAMILIES IN L(ℝ)

Published online by Cambridge University Press:  01 August 2018

ITAY NEEMAN  and
ZACH NORWOOD
ITAY NEEMAN
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF CALIFORNIA LOS ANGELES LOS ANGELES, CA 90095-1555, USAE-mail:ineeman@math.ucla.edu
ZACH NORWOOD
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF CALIFORNIA LOS ANGELES LOS ANGELES, CA 90095-1555, USAE-mail:znorwood@math.ucla.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove that, in the choiceless Solovay model, every set of reals is H-Ramsey for every happy family H that also belongs to the Solovay model. This gives a new proof of Törnquist’s recent theorem that there are no infinite mad families in the Solovay model. We also investigate happy families and mad families under determinacy, applying a generic absoluteness result to prove that there are no infinite mad families under $A{D^ + }$.

Keywords

03E3503E55mad familieshappy familiesSolovay modeldeterminacy
Type
Articles
Information
The Journal of Symbolic Logic , Volume 83 , Issue 2 , June 2018 , pp. 572 - 597
Copyright
Copyright © The Association for Symbolic Logic 2018 

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

Bagaria, J. and Di Prisco, C. A., Parameterized partition relations on the real numbers. Archive for Mathematical Logic, vol. 48 (2009), no. 2, pp. 201–226.CrossRefGoogle Scholar
Bartoszyński, T. and Scheepers, M., Filters and games. Proceedings of the American Mathematical Society, vol. 123 (1995), no. 8, pp. 2529–2534.CrossRefGoogle Scholar
Blass, A., Combinatorial cardinal characteristics of the continuum, Handbook of Set Theory (Foreman, M. and Kanamori, A., editors), vols. 1, 2, 3, Springer, Dordrecht, 2010, pp. 395–489.CrossRefGoogle Scholar
Caicedo, A. E. and Ketchersid, R., A trichotomy theorem in natural models of $A{D^ + }$., Set Theory and its Applications (Babinkostova, L., Caicedo, A. E., Geschke, S., and Scheepers, M., editors), Contemporary Mathematics, vol. 533, American Mathematical Society, Providence, RI, 2011, pp. 227–258.CrossRefGoogle Scholar
Chan, W. and Magidor, M., When an equivalence relation with all borel classes will be borel somewhere?, preprint, 2016, arXiv:1608.04913.Google Scholar
Di Prisco, C., Mijares, J. G., and Uzcátegui, C., Ideal games and Ramsey sets. Proceedings of the American Mathematical Society, vol. 140 (2012), no. 7, pp. 2255–2265.CrossRefGoogle Scholar
Di Prisco, C. A. and Todorcevic, S., Perfect-set properties in L(R)[U]. Advances in Mathematics, vol. 139 (1998), no. 2, pp. 240–259.CrossRefGoogle Scholar
Eisworth, T., Selective ultrafilters and $\omega \to {\left( \omega \right)^\omega }$. Proceedings of the American Mathematical Society, vol. 127 (1999), no. 10, pp. 3067–3071.CrossRefGoogle Scholar
Farah, I., Semiselective coideals. Mathematika, vol. 45 (1998), no. 1, pp. 79–103.CrossRefGoogle Scholar
Feng, Q., Magidor, M., and Woodin, H., Universally Baire sets of reals, Set Theory of the Continuum (Berkeley, CA, 1989) (Judah, H., Just, W., and Woodin, H., editors), Mathematical Sciences Research Institute Publications, vol. 26, Springer, New York, 1992, pp. 203–242.CrossRefGoogle Scholar
Foreman, M. and Magidor, M., Large cardinals and definable counterexamples to the continuum hypothesis. Annals of Pure and Applied Logic, vol. 76 (1995), no. 1, pp. 47–97.CrossRefGoogle Scholar
Horowitz, H. and Shelah, S., Mad families and non-meager filters, preprint, 2017, arXiv:1701.02806.Google Scholar
Horowitz, H. and Shelah, S., Can you take Toerquist’s inaccessible away?, to appear.Google Scholar
Ihoda, J. I. and Shelah, S., .${\rm{\Delta }}_2^1$-sets of reals. Annals of Pure and Applied Logic, vol. 42 (1989), no. 3, pp. 207–223.CrossRefGoogle Scholar
Larson, P. and Raghavan, D., Real games and strategically selective coideals, Sets and Computations (Friedman, S.-D., Raghavan, D., and Yang, Y., editors), Lecture Notes Series, Institute for Mathematical Sciences, National University of Singapore, vol. 33, World Scientific Publishing, Hackensack, NJ, 2018, pp. 45–54.Google Scholar
Mathias, A. R. D., A remark on rare filters, Infinite and Finite Sets (Colloquium, Keszthely, 1973; Dedicated to P. Erdős on his 60th Birthday) (Hajnal, A., Rado, R., and Sós, V. T., editors), vol. III, Colloquia mathematica Societatis János Bolyai, vol. 10, North-Holland, Amsterdam, 1975, pp. 1095–1097.Google Scholar
Mathias, A. R. D., Happy families. Annals of Pure and Applied Logic, vol. 12 (1977), no. 1, pp. 59–111.Google Scholar
Neeman, I. and Zapletal, J., Proper forcings and absoluteness in L(R). Commentationes Mathematicae Universitatis Carolinae, vol. 39 (1998), no. 2, pp. 281–301.Google Scholar
Neeman, I. and Zapletal, J., Proper forcing and L(ℝ), this Journal, vol. 66 (2001), no. 2, pp. 801–810.Google Scholar
Oxtoby, J. C., Measure and Category, second ed., Graduate Texts in Mathematics, vol. 2, Springer-Verlag, New York-Berlin, 1980, A survey of the analogies between topological and measure spaces.CrossRefGoogle Scholar
Sargsyan, G. and Steel, J., The mouse set conjecture for sets of reals, this Journal, vol. 80 (2015), no. 2, pp. 671–683.Google Scholar
Schindler, R.-D., Proper forcing and remarkable cardinals. II, this Journal, vol. 66 (2001), no. 3, pp. 1481–1492.Google Scholar
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. 1–56.CrossRefGoogle Scholar
Steel, J. R., Derived models associated to mice, Computational Prospects of Infinity. Part I. Tutorials (Chong, C., Feng, Q., Slaman, T., Hugh Woodin, W., and Yang, Y., editors), Lecture Notes Series, Institute for Mathematical Sciences, National University of Singapore, vol. 14, World Scientific Publishing, Hackensack, NJ, 2008, pp. 105–193.CrossRefGoogle Scholar
Steel, J. R., A theorem of woodin on mouse sets, Ordinal Definability and Recursion Theory. The Cabal Seminar. Volume III (Kechris, A., Löwe, B., and Steel, J., editors), Association for Symbolic Logic, La Jolla, CA, 2016, pp. 243–256.CrossRefGoogle Scholar
Talagrand, M., Compacts de fonctions mesurables et filtres non mesurables. Studia Mathematica, vol. 67 (1980), no. 1, pp. 13–43.CrossRefGoogle Scholar
Todorcevic, S., Introduction to Ramsey Spaces, Annals of Mathematics Studies, vol. 174, Princeton University Press, Princeton, NJ, 2010.Google Scholar
Tornquist, A., Definability and almost disjoint families, to appear, available at arXiv:1503.07577.Google Scholar
Zapletal, J., Forcing Idealized, Cambridge Tracts in Mathematics, vol. 174, Cambridge University Press, Cambridge, 2008.CrossRefGoogle Scholar