Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-26T12:46:19.814Z Has data issue: false hasContentIssue false
  • English
  • Français

PFA implies ADL(ℝ)

Published online by Cambridge University Press:  12 March 2014

John R. Steel
John R. Steel*
Affiliation:
University of California at Berkeley, Department of Mathematics, Berkeley, CA 94720-3840, USA, E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Extract

In this paper we shall prove

Theorem 0.1. Suppose there is a singular strong limit cardinal κ such that □κ fails; then AD holds in L(R).

See [10] for a discussion of the background to this problem. We suspect that more work will produce a proof of the theorem with its hypothesis that κ is a strong limit weakened to ∀α < κ (αω < κ), and significantly more work will enable one to drop the hypothesis that K is a strong limit entirely. At present, we do not see how to carry out even the less ambitious project.

Todorcevic [23] has shown that if the Proper Forcing Axiom (PFA) holds, then □κ fails for all uncountable cardinals κ. Thus we get immediately:

It has been known since the early 90's that PFA implies PD, that PFA plus the existence of a strongly inaccessible cardinal implies ADL(ℝ) and that PFA plus a measurable yields an inner model of ADℝ containing all reals and ordinals. As we do here, these arguments made use of Tororcevic's work, so that logical strength is ultimately coming from a failure of covering for some appropriate core models.

In late 2000, A. S. Zoble and the author showed that (certain consequences of) Todorcevic's Strong Reflection Principle (SRP) imply ADL(ℝ). (See [22].) Since Martin's Maximum implies SRP, this gave the first derivation of ADL(ℝ) from an “unaugmented” forcing axiom.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 70 , Issue 4 , December 2005 , pp. 1255 - 1296
Copyright
Copyright © Association for Symbolic Logic 2005

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]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]Ketchersid, R., Toward ADℝ from the Continuum Hypothesis and an ω1 -dense ideal, Ph.D. thesis, Berkeley, 2000.Google Scholar
[3]Martin, D. A., The largest countable this, that, or the other, Cabal seminar 79–81 (Kechris, A. S., Martin, D. A., and Moscovakis, Y. N., editors), Lecture Notes in Mathematics, Springer-Verlag. 1983.Google Scholar
[4]Mitchell, W. J. and Schimmerling, E., Weak covering without countable closure, Mathematical Research Letters, vol. 2 (1995), no. 5, pp. 595–609.CrossRefGoogle Scholar
[5]Mitchell, W. J. and Steel, J. R., Fine structure and iteration trees, Lecture Notes in Logic, vol. 3, Springer-Verlag, 1994.CrossRefGoogle Scholar
[6]Neeman, I. and Steel, J. R., A weak Dodd-Jensen lemma, this Journal, vol. 64 (1999), pp. 1285–1294.Google Scholar
[7]Räsch, T. and Schindler, R. D., A new condensation principle, Archive for Mathematical Logic, vol. 44 (2005), pp. 159–166.CrossRefGoogle Scholar
[8]Schimmerling, E. and Steel, J. R., The maximality of the core model, Transactions of the American Mathematical Society, vol. 351 (1999), pp. 3119–3141.CrossRefGoogle Scholar
[9]Schindler, R. D., Proper forcing and remarkable cardinals, The Bulletin of Symbolic Logic, vol. 6 (2000), no. 2.CrossRefGoogle Scholar
[10]Schindler, R. D. and Steel, J. R., Problems in inner model theory, available at http://www.math.uni-muenster.de/math/inst/logik/org/staff/rds.Google Scholar
[11]Schindler, R. D. and Steel, J. R., The strength of AD, unpublished but available at http://www.math.uni-muenster.de/math/inst/logik/org/staff/rds.Google Scholar
[12]Steel, J. R., Scales in L(ℝ), Cabal seminar 79–81 (Kechris, A. S., Martin, D. A., and Moscovakis, Y. N., editors), Lecture Notes in Mathematics, Springer-Verlag, 1983, pp. 107–156.CrossRefGoogle Scholar
[13]Steel, J. R.Inner models with many Woodin cardinals, Annals of Pure and Applied Logic, vol. 65 (1993), pp. 185–209.CrossRefGoogle Scholar
[14]Steel, J. R.Projectively wellordered inner models, Annals of Pure and Applied Logic, vol. 74 (1995), pp. 77–104.CrossRefGoogle Scholar
[15]Steel, J. R.The core model iterability problem, Lecture Notes in Logic, vol. 8, Springer-Verlag, 1996.CrossRefGoogle Scholar
[16]Steel, J. R.An outline of inner model theory, Handbook of set theory, to appear.Google Scholar
[17]Steel, J. R.A theorem of Woodin on mouse sets, available at http://www.math.berkeley.edu/~steel.Google Scholar
[18]Steel, J. R.Scales at the end of a weak gap. available at http://www.math.berkeley.edu/~steel.Google Scholar
[19]Steel, J. R.Scales in K(ℝ), available at http://www.math.berkeley.edu/~steel.Google Scholar
[20]Steel, J. R.The derived model theorem, available at http://www.math,berkeley.edu/~steel.Google Scholar
[21]Steel, J. R.Woodin's analysis of HODL(ℝ), available at http://www.math.berkeley.edu/~steel.Google Scholar
[22]Steel, J. R. and Zoble, A. S., The strong reflection principle implies ADLL(ℝ), in preparation.Google Scholar
[23]Tororcevic, S., A note on the proper forcing axiom, Contemporary Mathematics, vol. 95 (1984), pp. 209–218.CrossRefGoogle Scholar
[24]Woodin, W. H., The axiom of determinacy, forcing axioms, and the nonstationary ideal, De Gruyter, 1999.CrossRefGoogle Scholar