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Proper Forcing and Remarkable Cardinals

  • Ralf-Dieter Schindler (a1)

The present paper investigates the power of proper forcings to change the shape of the universe, in a certain well-defined respect. It turns out that the ranking among large cardinals can be used as a measure for that power. However, in order to establish the final result I had to isolate a new large cardinal concept, which I dubbed “remarkability.” Let us approach the exact formulation of the problem—and of its solution—at a slow pace.

Breathtaking developments in the mid 1980s found one of its culminations in the theorem, due to Martin, Steel, and Woodin, that the existence of infinitely many Woodin cardinals with a measurable cardinal above them all implies that AD, the axiom of determinacy, holds in the least inner model containing all the reals, L(ℝ) (cf. [6[, p. 91). One of the nice things about AD is that the theory ZF + AD + V = L(ℝ) appears as a choiceless “completion” of ZF in that any interesting question (in particular, about sets of reals) seems to find an at least attractive answer in that theory (cf., for example, [5] Chap. 6). (Compare with ZF + V = L!) Beyond that, AD is very canonical as may be illustrated as follows.

Let us say that L(ℝ) is absolute for set-sized forcings if for all posets P ∈ V, for all formulae ϕ, and for all ∈ ℝ do we have that

where is a name for the set of reals in the extension.

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[1] Bagaria, J. and Woodin, H., sets of reals, The Journal of Symbolic Logic, vol. 62 (1997), pp. 13791428.
[2] Beller, A., Jensen, R., and Welch, Ph., Coding the universe, Cambridge, 1982.
[3] Foreman, M. and Magidor, M., Large cardinals and definable counterexamples to the continuum hypothesis, Annals of Pure and Applied Logic, vol. 76 (1995), pp. 4797.
[4] Jensen, R. and Solovay, R., Some applications of almost disjoint sets, Mathematical logic and foundations of set theory (Bar-Hillel, , editor), North-Holland, 1970, pp. 84104.
[5] Kanamori, A., The higher infinite, Springer-Verlag, 1994.
[6] Martin, D. and Steel, J., A proof of projective determinacy, Journal of the American Mathematical Society, vol. 2 (1989), pp. 71125.
[7] Neeman, I. and Zapletal, J., Proper forcing and L(ℝ), preprint.
[8] Neeman, I. and Zapletal, J., Proper forcing and absoluteness in L(ℝ), Commentationes Mathematicae Universitatis Carolinae, vol. 39 (1998), pp. 281301.
[9] Schindler, R.-D., Coding into K by reasonable forcing, to appear in Transactions of the American Mathematical Society.
[10] Neeman, I. and Zapletal, J., Proper forcing and remarkable cardinals II, to appear in Journal of Symbolic Logic.
[11] Shelah, S., Proper forcing, Springer-Verlag, 1982.
[12] Steel, J., Core models with more Woodin cardinals, preprint.
[13] Steel, J., Inner models with many Woodin cardinals, Annals of Pure and Applied Logic, vol. 65 (1993), pp. 185209.
[14] Woodin, H., Lecture in the spring of 1990, notes taken by Burke, D. and Schimmerling, E..
[15] Woodin, H., The axiom of determinacy, forcing axioms, and the nonstationary ideal, de Gruyter, 1999.
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Bulletin of Symbolic Logic
  • ISSN: 1079-8986
  • EISSN: 1943-5894
  • URL: /core/journals/bulletin-of-symbolic-logic
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