Skip to main content
Log in

On unfoldable cardinals, ω-closed cardinals, and the beginning of the inner model hierarchy

  • Published:
Archive for Mathematical Logic Aims and scope Submit manuscript

Abstract.

Let κ be a cardinal, and let H κ be the class of sets of hereditary cardinality less than κ ; let τ (κ) > κ be the height of the smallest transitive admissible set containing every element of {κ}∪H κ . We show that a ZFC-definable notion of long unfoldability, a generalisation of weak compactness, implies in the core model K, that the mouse order restricted to H κ is as long as τ. (It is known that some weak large cardinal property is necessary for the latter to hold.) In other terms we delimit its strength as follows: Theorem Con(ZFC21 1-Determinacy) ⇒ ⇒Con(ZFC+V=K+∃ a long unfoldable cardinal ⇒ ⇒Con(ZFC+∀X(X # exists) + ‘‘\(\forall D \subseteq \omega_1 D\) is universally Baire ⇔ ∃rR(DL(r)))’’, and this is set-generically absolute). We isolate a notion of ω-closed cardinal which is weaker than an ω1-Erd\ os cardinal, and show that this bounds the first long unfoldable: Theorem Let κ be ω -closed. Then there is a long unfoldable ł<κ.

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

Access this article

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Baumgartner, J.E., Galvin, F.: Generalized Erdos cardinals and O #. Annals of Mathematical Logic, vol. 15, 289–313 (1979)

    Google Scholar 

  2. Dodd, A.J.: The core model, London Mathematuical Society Lecture Notes in Mathematics, vol. 61, Cambridge University Press, Cambridge, 1982

  3. Donder, H-D., Lewinski, J.-P.: Some principles related to Chang’s conjecture. Annals of Pure and Applied Logic 45, 39–101 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  4. Donder, H-D., Koepke, P.: On the consistency strength of ‘accessible’ Jónsson cardinals and of the weak chang conjecture. Annals of Pure and Applied Logic 25, 223–261 (1983)

    Article  Google Scholar 

  5. Donder, H-D., Jensen, R.B., Koppelberg, B.: Some applications of K. Set theory and Model theory (R.Jensen, A.Prestel, eds.), Springer Lecture Notes in Mathematics, vol. 872, Springer Verlag, 1981, pp. 55–97

  6. Feng, Q., Magidor, M., Woodin, W.H.: Universally Baire sets of reals. Set Theory of the Continuum (W. Just, H. Judah, W. H. Woodin, eds.) MSRI Publications, Springer Verlag, 1992

  7. Hamkins, J. D.: Unfoldable cardinals and the GCH. Journal for Symbolic Logic (to appear)

  8. Jech, T.: Set theory. Pure and Applied Mathematics, Academic Press, New York, 1978

  9. Todorč ević, S.: Sigma-one forcing absoluteness and the continuum, unpublished (April 2002)

  10. Villaveces, A.: Chains of elementary extensions of models of set theory. Journal of Symbolic Logic 63, 1116–1136 (1998)

    MathSciNet  MATH  Google Scholar 

  11. Welch, P. D.: Characterising subsets of ω1. Journal of Symbolic Logic 59 (4), 1420–1432 (1994)

    MATH  Google Scholar 

  12. ––– : Countable unions of simple sets in the core model. Journal of Symbolic Logic 61 (1), 293–312 (1996)

    Google Scholar 

  13. ––– : Determinacy in the difference hierarchy of co-analytic sets. Annals of Pure and Applied Logic 80, 69–108 (1996)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to P. D. Welch.

Additional information

Mathematics Subject Classification (2000): 03E45, 03E15, 03E55, 03E60

The author wishes to gratefully acknowledge support from Nato Grant PST.CLG 975324.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Welch, P. On unfoldable cardinals, ω-closed cardinals, and the beginning of the inner model hierarchy. Arch. Math. Logic 43, 443–458 (2004). https://doi.org/10.1007/s00153-003-0199-6

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00153-003-0199-6

Keywords

Navigation