Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-30T23:30:18.276Z Has data issue: false hasContentIssue false
  • English
  • Français

The fine structure of real mice

Published online by Cambridge University Press:  12 March 2014

Daniel W. Cunningham
Daniel W. Cunningham*
Affiliation:
Mathematics Department, State University of New York, College at Buffalo, Buffalo, NY 14222, USA E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Before one can construct scales of minimal complexity in the Real Core Model, K(ℝ), one needs to develop the fine-structure theory of K (ℝ). In this paper, the fine structure theory of mice, first introduced by Dodd and Jensen, is generalized to that of real mice. A relative criterion for mouse iterability is presented together with two theorems concerning the definability of this criterion. The proof of the first theorem requires only fine structure; whereas, the second theorem applies to real mice satisfying AD and follows from a general definability result obtained by abstracting work of John Steel on L(ℝ). In conclusion, we discuss several consequences of the work presented in this paper relevant to two issues: the complexity of scales in K(ℝ)and the strength of the theory ZF + AD + ┐DCℝ.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 63 , Issue 3 , September 1998 , pp. 937 - 994
Copyright
Copyright © Association for Symbolic Logic 1998

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]Cunningham, D. W., Is there a set of reals not in K(ℝ), Annals of Pure and Applied Logic, vol. 92 (1998), pp. 161–210.CrossRefGoogle Scholar
[2]Cunningham, D. W., Scales and the fine structure of K(ℝ), in preparation.Google Scholar
[3]Cunningham, D. W., The real core model and its scales, Annals of Pure and Applied Logic, vol. 72 (1995), pp. 213–289.CrossRefGoogle Scholar
[4]Dodd, A. J., The core model, London Mathematical Lecture Note Series, Cambridge University Press, 1982.CrossRefGoogle Scholar
[5]Dodd, A. J. and Jensen, R. B., The core model, Annals of Mathematical Logic, vol. 20 (1981), pp. 43–75.CrossRefGoogle Scholar
[6]Jech, T., Set theory, Academic Press, New York, 1980.Google Scholar
[7]Moschovakis, Y. N., Descriptive set theory, North-Holland, Amsterdam, 1980.Google Scholar
[8]Steel, J. R., Scales in L(ℝ), Cabal seminar 79–81, Lecture Notes in Mathematics, no. 1019, Springer-Verlag, Berlin, 1983.Google Scholar