Skip to main content Accessibility help
×
Home

Jónsson cardinals, Erdős cardinals, and the core model

  • W. J. Mitchell (a1)

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 L κ[A] is a Jónsson algebra. The proof of this result. Lemma 2.5, is very short and entirely elementary.

Copyright

References

Hide All
[1] Baumgartner, James, Ineffability properties of cardinals II, Logic, foundations of mathematics, and computer theory (Butts, and Hintikka, , editors), D. Reidel, 1977, pp. 87106.
[2] Dodd, Anthony and Jensen, Ronald B., The covering lemma for K, Annals of Mathematical Logic, vol. 20 (1981), pp. 4375.
[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. 5597.
[4] Kunen, Ken, Some applications of iterated ultraproducts in set theory, Annals of Mathematical Logic, vol. 2 (1970), pp. 71125.
[5] Martin, Donald A. and Steel, John R., Iteration trees, Journal of the American Mathematical Society, vol. 7 (1994), no. 1, pp. 173.
[6] Mathias, Adrian, Solovay, Robert, and Woodin, W. Hugh, The consistency strength of the axiom of determinacy, in preparation.
[7] Mitchell, William J., Ramsey cardinals and constructibility, this Journal, vol. 44 (1979), no. 2, pp. 260266.
[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. 219255.
[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.
[10] Steel, John R., The core model iterability problem, Association for Symbolic Logic Notes in Logic, no. 8, Springer-Verlag, 1996.

Related content

Powered by UNSILO

Jónsson cardinals, Erdős cardinals, and the core model

  • W. J. Mitchell (a1)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.