Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-25T02:43:31.148Z Has data issue: false hasContentIssue false
  • English
  • Français

THE ABC'S of Mice

Published online by Cambridge University Press:  15 January 2014

Ernest Schimmerling
Ernest Schimmerling*
Affiliation:
Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213-3890, USA, E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Extract

This is the introductory half of a lecture given at the Association for Symbolic Logic Annual Meeting in Philadelphia in March, 2001. Our goal is to give logicians and advanced students who are unfamiliar with inner model theory a taste of what the subject is about at a level where it is possible to say something meaningful about the intuitions and proofs. At the same time, we wish to avoid overwhelming the reader, so we will leave out many well known closely related theorems and give only short proof sketches. Neither the results nor the proofs found here are due to the author. We will begin near the beginning but still touch on some modern aspects of the theory. The proofs and historical notes that we leave out can be found in the books and articles listed in the references. Our approach to the theory of 0# is a combination of that taken by Dodd in his book and by Steel in his lectures to students. Dodd's introduction is one of the author's favorite essays on inner model theory; another is the introduction to Martin-Steel [6]. In Section 2, we review the theory of 0# in a form that touches on some of the techniques that have been and continue to be generalized. These combine what is commonly called iteration and fine structure. In Section 3, we give a superficial description of what comes after 0#. Those interested in learning about more recent developments in inner model theory are encouraged to see the articles Lowe-Steel [5] and Steel [8], both of which are currently available online at http://www.math.berkeley.edu/~ steel.

Type
Research Article
Information
Bulletin of Symbolic Logic , Volume 7 , Issue 4 , December 2001 , pp. 485 - 503
Copyright
Copyright © Association for Symbolic Logic 2001

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] Devlin, K., Constructibility, Perspectives in Mathematical Logic, Springer-Verlag, 1984.Google Scholar
[2] Dodd, A. J., The core model, London Mathematical Society Lecture Note Series, vol. 61, Cambridge University Press, 1982.CrossRefGoogle Scholar
[3] Jech, T., Set theory, Springer-Verlag, 1997.CrossRefGoogle Scholar
[4] Kanamori, A., The higher infinite, Perspectives in Mathematical Logic, Springer-Verlag, 1997.CrossRefGoogle Scholar
[5] Löwe, B. and Steel, J., An introduction to core model theory, Sets and proofs (Leeds, 1997), London Mathematical Society Lecture Note Series, vol. 258, Cambridge University Press, 1999, pp. 103–157.Google Scholar
[6] Martin, D. A. and Steel, J. R., Iteration trees, Journal of the American Mathematical Society, vol. 7 (1994), pp. 1–73.Google Scholar
[7] Mitchell, W. J. and Steel, J. R., Fine structure and iteration trees, Lecture Notes in Logic, vol. 3, Springer-Verlag, 1994.CrossRefGoogle Scholar
[8] Steel, J. R., An outline of inner model theory, to appear in the Handbook of Set Theory.Google Scholar
[9] Steel, J. R., Projectively well-ordered inner models, Annals of Pure and Applied Logic, vol. 74 (1995), pp. 77–104.CrossRefGoogle Scholar
[10] Steel, J. R., The core model iterability problem, Lecture Notes in Logic, vol. 8, Springer-Verlag, 1996.Google Scholar