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Scales in K(ℝ) at the end of a weak gap

  • J. R. Steel (a1)

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In this note we shall prove

Theorem 0.1. Letbe a countably ω-iterable-mouse which satisfies AD, and [α, β] a weak gap of. Supposeis captured by mice with iteration strategies inα. Let n be least such that ; then we have that believes that has the Scale Property.

This complements the work of [5] on the construction of scales of minimal complexity on sets of reals in K(ℝ). Theorem 0.1 was proved there under the stronger hypothesis that all sets definable over are determined, although without the capturing hypothesis. (See [5, Theorem 4.14].) Unfortunately, this is more determinacy than would be available as an induction hypothesis in a core model induction. The capturing hypothesis, on the other hand, is available in such a situation. Since core model inductions are one of the principal applications of the construction of optimal scales, it is important to prove 0.1 as stated.

Our proof will incorporate a number of ideas due to Woodin which figure prominently in the weak gap case of the core model induction. It relies also on the connection between scales and iteration strategies with the Dodd-Jensen property first discovered in [3]. Let be the pointclass at the beginning of the weak gap referred to in 0.1. In section 1, we use Woodin's ideas to construct a Γ-full a mouse having ω Woodin cardinals cofinal in its ordinals, together with an iteration strategy Σ which condenses well in the sense of [4, Def. 1.13]. In section 2, we construct the desired scale from and Σ.

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[1]Ketchersid, R. O., Toward AD from the Continuum Hypothesis and an ω1-dense ideal, Ph.D. thesis, University of California at Berkeley, 06 2000.
[2]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. 107156.
[3]Steel, J. R., HODL(ℝ) is a core model below Θ, The Bulletin of Symbolic Logic, vol. 1 (1995), pp. 7584.
[4]Steel, J. R., PFA implies ADL(ℝ), this Journal, vol. 70 (2005), no. 4, pp. 12551296.
[5]Steel, J. R., Scales in K(ℝ), The Cabal Seminar I; Games, Scales, and Suslin Cardinals (Kechris, A. S., Loewe, B., and Steel, J., editors), Lecture Notes in Logic, vol. 31, ASL and Cambridge University Press, 2008, pp. 176208.
[6]Steel, J. R., An outline of inner model theory, The handbook of set theory (Foreman, M., Kanamori, A., and Magidor, M., editors), to appear.
[7]Steel, J. R., A theorem of Woodin on mouse sets, available at http://www.math.berkeley.edu/~steel.
[8]Steel, J. R., Woodin's analysis of HODL(ℝ), available at http://www.math.berkeley.edu/~steel.

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Scales in K(ℝ) at the end of a weak gap

  • J. R. Steel (a1)

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