Skip to main content
SpringerLink
Log in

Partition properties for simply definable colourings

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

We study partition properties for uncountable regular cardinals that arise by restricting partition properties defining large cardinal notions to classes of simply definable colourings. We show that both large cardinal assumptions and forcing axioms imply that there is a homogeneous closed unbounded subset of ω1 for every colouring of the finite sets of countable ordinals that is definable by a Σ1-formula that only uses the cardinal ω1 and real numbers as parameters. Moreover, it is shown that certain large cardinal properties cause analogous partition properties to hold at the given large cardinal and these implications yield natural examples of inaccessible cardinals that possess strong partition properties for Σ1-definable colourings and are not weakly compact. In contrast, we show that Σ1- definability behaves fundamentally different at ω2 by showing that various large cardinal assumptions and Martin’s Maximum are compatible with the existence of a colouring of pairs of elements of ω2 that is definable by a Σ1-formula with parameter ω2 and has no uncountable homogeneous set. Our results will also allow us to derive tight bounds for the consistency strengths of various partition properties for definable colourings. Finally, we use the developed theory to study the question whether certain homeomorphisms that witness failures of weak compactness at small cardinals can be simply definable.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. U. Abraham and S. Shelah, Forcing closed unbounded sets, Journal of Symbolic Logic 48 (1983), 643–657.

    Article  MathSciNet  Google Scholar 

  2. A. Andretta and L. Motto Ros, Souslin quasi-orders and bi-embeddability of uncountable structures, Memoirs of the American Mathematical Society, to appear.

  3. D. Asperó, Guessing and non-guessing of canonical functions, Annals of Pure and Applied Logic 146 (2007), 150–179.

    Article  MathSciNet  Google Scholar 

  4. J. Bagaria and R. Bosch, Generic absoluteness under projective forcing, Fundamenta Mathematicae 194 (2007), 95–120.

    Article  MathSciNet  Google Scholar 

  5. J. E. Baumgartner, Applications of the proper forcing axiom, in Handbook of Settheoretic Topology, North-Holland, Amsterdam, 1984, pp. 913–959.

    Chapter  Google Scholar 

  6. R. Bosch, Small definably-large cardinals, in Set Theory, Trends in Mathematics, Birkhäuser, Basel, 2006, pp. 55–82.

    Google Scholar 

  7. A. Eduardo Caicedo and B. Veličković, The bounded proper forcing axiom and well orderings of the reals, Mathematical Research Letters 13 (2006), 393–408.

    Article  MathSciNet  Google Scholar 

  8. J. Cummings, Iterated forcing and elementary embeddings, in Handbook of Set Theory. Vols. 1, 2, 3, Springer, Dordrecht, 2010, pp. 775–883.

    Chapter  Google Scholar 

  9. J. Cummings, S.- D. Friedman, M. Magidor, A. Rinot and D. Sinapova, Ordinal definable subsets of singular cardinals, Israel Journal of Mathematics 226 (2018), 781–804.

    Article  MathSciNet  Google Scholar 

  10. T. Dodd and R. B. Jensen, The covering lemma for K, Annals of Mathematical Logic 22 (1982), 1–30.

    Article  MathSciNet  Google Scholar 

  11. H.-D. Donder, R. B. Jensen and B. J. Koppelberg, Some applications of the core model, in Set Theory and Model Theory (Bonn, 1979), Lecture Notes in Mathematics, Vol. 872, Springer, Berlin–New York, 1981, pp. 55–97.

    Article  Google Scholar 

  12. H.-D. Donder and P. Koepke, On the consistency strength of “accessible” Jónsson cardinals and of the weak Chang conjecture, Annals of Pure and Applied Logic 25 (1983), 233–261.

    Article  MathSciNet  Google Scholar 

  13. P. Erd˝os and A. Hajnal, Some remarks concerning our paper “On the structure of setmappings”. Non-existence of a two-valued σ-measure for the first uncountable inaccessible cardinal, Acta Mathematica Academiae Scientiarum Hungaricae 13 (1962), 223–226.

    Article  MathSciNet  Google Scholar 

  14. V. Gitman, Ramsey-like cardinals, Journal of Symbolic Logic 76 (2011), 519–540.

    Article  MathSciNet  Google Scholar 

  15. V. Gitman and P. D. Welch, Ramsey-like cardinals II, Journal of Symbolic Logic 76 (2011), 541–560.

    Article  MathSciNet  Google Scholar 

  16. S. Grigorieff, Intermediate submodels and generic extensions in set theory, Annals of Mathematics 101 (1975), 447–490.

    Article  MathSciNet  Google Scholar 

  17. K. Hauser, Indescribable cardinals and elementary embeddings, Journal of Symbolic Logic 56 (1991), 439–457.

    Article  MathSciNet  Google Scholar 

  18. H. H. Hung and S. Negrepontis, Spaces homeomorphic to (2α) α, Bulletin of the American Mathematical Society 79 (1973), 143–146.

    Article  MathSciNet  Google Scholar 

  19. T. Jech, ω1can be measurable, Israel Journal of Mathematics 6 (1968), 363–367.

    Article  MathSciNet  Google Scholar 

  20. T. Jech, Set Theory, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003.

    Google Scholar 

  21. T. Jech, M. Magidor, W. J. Mitchell and K. Prikry, Precipitous ideals, Journal of Symbolic Logic 45 (1980), 1–8.

    Article  MathSciNet  Google Scholar 

  22. A. Kanamori, The Higher Infinite, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003.

    Google Scholar 

  23. K. Kunen, Set Theory, Studies in Logic and the Foundations of Mathematics, Vol. 102, North-Holland, Amsterdam–New York, 1980.

  24. P. B. Larson, The Stationary Tower, University Lecture Series, Vol. 32, American Mathematical Society, Providence, RI, 2004.

  25. P. B. Larson, Martin’s maximum and definability in H(ℵ2), Annals of Pure and Applied Logic 156 (2008), 110–122.

    Article  MathSciNet  Google Scholar 

  26. P. B. Larson, Forcing over models of determinacy, in Handbook of Set Theory. Vols. 1, 2, 3, Springer, Dordrecht, 2010, pp. 2121–2177.

    Chapter  Google Scholar 

  27. A. Leshem, On the consistency of the definable tree property on ℵ1, Journal of Symbolic Logic 65 (2000), 1204–1214.

    Article  MathSciNet  Google Scholar 

  28. P. Lücke, Σ11-definability at uncountable regular cardinals, Journal of Symbolic Logic 77 (2012), 1011–1046.

    Article  MathSciNet  Google Scholar 

  29. P. Lücke, R. Schindler and P. Schlicht, Σ1(κ)-definable subsets of H(κ+), Journal of Symbolic Logic 82 (2017), 1106–1131.

    Article  MathSciNet  Google Scholar 

  30. P. Lücke and P. Schlicht, Measurable cardinals and good Σ1(κ)-wellorderings, Mathematical Logic Quarterly 64 (2018), 207–217.

    Article  MathSciNet  Google Scholar 

  31. W. J. Mitchell and J. R. Steel, Fine Structure and Iteration Trees, Lecture Notes in Logic, Vol. 3, Springer-Verlag, Berlin, 1994.

  32. E. Schimmerling, A core model toolbox and guide, in Handbook of Set Theory. Vols. 1, 2, 3, Springer, Dordrecht, 2010, pp. 1685–1751.

    Chapter  Google Scholar 

  33. I. Sharpe and P. D. Welch, Greatly Erd˝os cardinals with some generalizations to the Chang and Ramsey properties, Annals of Pure and Applied Logic 162 (2011), 863–902.

    Article  MathSciNet  Google Scholar 

  34. S. Shelah, Set theory without choice: not everything on cofinality is possible, Archive for Mathematical Logic 36 (1997), 81–125.

    Article  MathSciNet  Google Scholar 

  35. J. R. Steel, Inner models with many Woodin cardinals, Annals of Pure and Applied Logic 65 (1993), 185–209.

    Article  MathSciNet  Google Scholar 

  36. J. R. Steel, An outline of inner model theory, in Handbook of Set Theory. Vols. 1, 2, 3, Springer, Dordrecht, 2010, pp. 1595–1684.

    Chapter  Google Scholar 

  37. S. Todorčević, Walks on Ordinals and their Characteristics, Progress in Mathematics, Vol. 263, Birkhäuser, Basel, 2007.

  38. M. Viale, Category forcings, MM+++, and generic absoluteness for the theory of strong forcing axioms, Journal of the American Mathematical Society 29 (2016), 675–728.

    Article  MathSciNet  Google Scholar 

  39. P. D. Welch, Stably measurable cardinals, Journal of Symbolic Logic, to appear.

Download references

Author information

Authors and Affiliations

  1. Mathematisches Institut, Rheinische Friedrich-Wilhelms-Universität Bonn, Endenicher Allee 60, 53115, Bonn, Germany

    Philipp Lücke

Authors
  1. Philipp Lücke
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Philipp Lücke.

Additional information

The author would like to thank Luca Motto Ros and Philip Welch for helpful discussions on the content of the paper. Moreover, the author would like to thank the anonymous referee for numerous suggestions and corrections.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lücke, P. Partition properties for simply definable colourings. Isr. J. Math. 236, 841–898 (2020). https://doi.org/10.1007/s11856-020-1993-0

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11856-020-1993-0

Search

Navigation