Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-23T17:50:18.305Z Has data issue: false hasContentIssue false
  • English
  • Français

Jónsson cardinals, Erdős cardinals, and the core model

Published online by Cambridge University Press:  12 March 2014

W. J. Mitchell
W. J. Mitchell*
Affiliation:
Department of Mathematics, University of Florida, 358 Little Hall, P.O. Box 118105, Gainesville, Florida 32611-8105, USA E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We show that if there is no inner model with a Woodin cardinal and the Steel core model K exists, then every Jónsson cardinal is Ramsey in K, and every δ-Jónsson cardinal is δ5-Erdős in K.

In the absence of the Steel core model K we prove the same conclusion for any model L[] such that either V = L[] is the minimal model for a Woodin cardinal, or there is no inner model with a Woodin cardinal and V is a generic extension of L[].

The proof includes one lemma of independent interest: If V = L[A], where A ⊂ κ and κ is regular, then Lκ[A] is a Jónsson algebra. The proof of this result. Lemma 2.5, is very short and entirely elementary.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 64 , Issue 3 , September 1999 , pp. 1065 - 1086
Copyright
Copyright © Association for Symbolic Logic 1999

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

[1] Baumgartner, James, Ineffability properties of cardinals II, Logic, foundations of mathematics, and computer theory (Butts, and Hintikka, , editors), D. Reidel, 1977, pp. 87–106.Google Scholar
[2] Dodd, Anthony and Jensen, Ronald B., The covering lemma for K, Annals of Mathematical Logic, vol. 20 (1981), pp. 43–75.CrossRefGoogle Scholar
[3] Jensen, Ronald B., Some applications of the core model, Set theory and model theory (Jensen, Ronald B. and Prestel, A., editors), Lecture Notes in Mathematics, no. 872, Springer-Verlag, New York, 1981, pp. 55–97.CrossRefGoogle Scholar
[4] Kunen, Ken, Some applications of iterated ultraproducts in set theory, Annals of Mathematical Logic, vol. 2 (1970), pp. 71–125.Google Scholar
[5] Martin, Donald A. and Steel, John R., Iteration trees, Journal of the American Mathematical Society, vol. 7 (1994), no. 1, pp. 1–73.CrossRefGoogle Scholar
[6] Mathias, Adrian, Solovay, Robert, and Woodin, W. Hugh, The consistency strength of the axiom of determinacy, in preparation.Google Scholar
[7] Mitchell, William J., Ramsey cardinals and constructibility, this Journal, vol. 44 (1979), no. 2, pp. 260–266.Google Scholar
[8] Mitchell, William J., Schimmerling, E., and Steel, J. R., The covering lemma up to a Woodin cardinal, Annals of Pure and Applied Logic, vol. 84 (1997), no. 2, pp. 219–255.CrossRefGoogle Scholar
[9] Mitchell, William J. and Steel, John R., Fine structure and iteration trees, Lecture Notes in Logic, no. 3, Springer-Verlag, Berlin, 1994, 130 pp.CrossRefGoogle Scholar
[10] Steel, John R., The core model iterability problem, Association for Symbolic Logic Notes in Logic, no. 8, Springer-Verlag, 1996.CrossRefGoogle Scholar