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PFA implies ADL(ℝ)

  • John R. Steel (a1)

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 AD 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.

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[1] A. Andretta , I. Neeman , and J. R. Steel , The domestic levels of Kc are iterable, Israel Journal of Mathematics, vol. 125 (2001), pp. 157201.

[3] D. A. Martin , The largest countable this, that, or the other, Cabal seminar 79–81 ( A. S. Kechris , D. A. Martin , and Y. N. Moscovakis , editors), Lecture Notes in Mathematics, Springer-Verlag. 1983.

[4] W. J. Mitchell and E. Schimmerling , Weak covering without countable closure, Mathematical Research Letters, vol. 2 (1995), no. 5, pp. 595609.

[5] W. J. Mitchell and J. R. Steel , Fine structure and iteration trees, Lecture Notes in Logic, vol. 3, Springer-Verlag, 1994.

[7] T. Räsch and R. D. Schindler , A new condensation principle, Archive for Mathematical Logic, vol. 44 (2005), pp. 159166.

[8] E. Schimmerling and J. R. Steel , The maximality of the core model, Transactions of the American Mathematical Society, vol. 351 (1999), pp. 31193141.

[9] R. D. Schindler , Proper forcing and remarkable cardinals, The Bulletin of Symbolic Logic, vol. 6 (2000), no. 2.

[12] J. R. Steel , Scales in L(ℝ), Cabal seminar 79–81 ( A. S. Kechris , D. A. Martin , and Y. N. Moscovakis , editors), Lecture Notes in Mathematics, Springer-Verlag, 1983, pp. 107156.

[13] J. R. Steel Inner models with many Woodin cardinals, Annals of Pure and Applied Logic, vol. 65 (1993), pp. 185209.

[14] J. R. Steel Projectively wellordered inner models, Annals of Pure and Applied Logic, vol. 74 (1995), pp. 77104.

[15] J. R. Steel The core model iterability problem, Lecture Notes in Logic, vol. 8, Springer-Verlag, 1996.

[24] W. H. Woodin , The axiom of determinacy, forcing axioms, and the nonstationary ideal, De Gruyter, 1999.

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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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