Hostname: page-component-76fb5796d-45l2p Total loading time: 0 Render date: 2024-04-27T09:00:40.865Z Has data issue: false hasContentIssue false
  • English
  • Français

Local Kc constructions

Published online by Cambridge University Press:  12 March 2014

J. R. Steel
J. R. Steel*
Affiliation:
Department of Mathematics, University of California, Berkeley, Berkeley, California 94720, USA. E-mail: steel@math.berkeley.edu
Get access Rights & Permissions [Opens in a new window]

Extract

The full-background-extender Kc -construction of [2] has the property that, if it does not break down and produces a final model , then

Ή is Woodin in V ⇒ Ή is Woodin in ,

for all Ή. It is natural to ask whether

κ is strong in V ⇒ κ is λ-strong in ,

for all κ, or even better,

κ is λ-strong in V ⇒ κ is λ-strong in .

As one might suspect, the more useful answer would be “yes”.

For the Kc-construction of [2], this question is open. The problem is that the construction of [2] is not local: because of the full-background-extender demand, it may produce mice projecting to ρ at stages much greater than ρ. Because of this, there is no reason to believe that if E is a λ-strong extender of V, then The natural proof only gives that if κ is Σ2-strong, then Σ, is strong in .

We do not know how to get started on this question, and suspect that in fact strong cardinals in V may fail to be strong in , if is the output of the construction of [2]. Therefore, we shall look for a modification of the construction of [2]. One might ask for a construction with output such that

(1) iteration trees on can be lifted to iteration trees on V,

(2) ∀δ(δ is Woodin ⇒ δ is Woodin in ), and

(3) (a) ↾κ(κ is a strong cardinal ⇒ κ is strong in ), and (b) ↾κ↾λ(Lim(λ) Λ κ is λ-strong ⇒ κ is λ-strong in ).

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 72 , Issue 3 , September 2007 , pp. 721 - 737
Copyright
Copyright © Association for Symbolic Logic 2007

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

[1] Andretta, A., Neeman, I., and Steel, J. R., The domestic levels of Kc are iterable, Israel Journal of Mathematics, vol. 125 (2001), pp. 157–201.CrossRefGoogle Scholar
[2] Mitchell, W. J. and Steel, J. R., Fine structure and iteration trees. Lecture Notes in Logic 3, Springer-Verlag, Berlin, 1994.CrossRefGoogle Scholar
[3] Neeman, I., unpublished manuscript.Google Scholar
[4] Neeman, I. and Steel, J. R., A weak Dodd-Jensen lemma, this Journal, vol. 64 (1999), pp. 1285–1294.Google Scholar
[5] Steel, J. R., The core model iterability problem, Lecture Notes in Logic 8, Springer-Verlag, Berlin, 1996.CrossRefGoogle Scholar
[6] Steel, J. R., Derived models associated to mice, Proceedings of the July 2005 Singapore workshop Computational prospects of infinity, to appear.Google Scholar
[7] Steel, J. R., An outline of inner model theory, Handbook of set theory, to appear.Google Scholar