List of open problems in inner model theory

Edited by Ralf Schindler and John Steel.
Right now, we solicit contributions to this list. The following provides you with information on which problems have been submitted so far. By clicking on "postscript" or "pdf" next to its statement you may retrieve a ps- or a pdf-file, respectively, with a description of the problem you're interested in. The numbering is provisorial and agrees with the order in which submissions were received. In your problem description you may refer to one of the problems from the following list as "problem n," where n is its provisorial number. This site should list at least all the references from the descriptions of our problems. Go ahead and use it as an inner model theory bibliography. -- As of Feb 06, 02, the unofficial list runs as follows.

  1. Prove the existence of the core model in the theory BGC + "there is no inner model with a Woodin cardinal." postscript
  2. Show that if -UBH holds, then there is a nontame premouse. postscript
  3. Does BMM + "every set of reals in L(R) is Lebesgue measurable" imply PD ? postscript
  4. Does the existence of a singular strong limit cardinal \kappa such that \square_\kappa fails imply that AD holds in L(R) ? postscript
  5. Suppose that there is no inner model with a Woodin cardinal. Let \kappa >= \aleph_1, and suppose that X^# exists for every X being a subset of \kappa. Is there a cone C of subsets of \kappa such that for all X, Y \in C do we have that, setting \tau = min(\kappa^{+K^L[X]},\kappa^{+K^L[Y]}), K^L[X]|\tau = K^L[Y]|\tau ? postscript
  6. Suppose that W, V are inner models of set theory with the same cardinals, V=W[r] for some real r, W is a model of the GCH, but CH fails in V. Must 0-dagger exist ? postscript
  7. Suppose that \Omega is measurable and that there is no inner model with a Woodin cardinal. Let K denote Steel's core model.
    (a) Suppose that \kappa < \Omega is measurable and 2^\kappa = \lambda > \kappa^+. How large should \kappa be in K ? Is then o(\kappa) >= \lambda in K ?
    (b) Suppose that \kappa < \Omega is a strong limit cardinal with cf(\kappa) = \delta, \omega_1 < \delta < \kappa, and that 2^\kappa >= \lambda > \kappa^+, where \lambda is not the successor of a cardinal of cofinality less than \kappa. Is then o(\kappa) >= \lambda + \delta in K ? postscript
  8. Let M be a model of ZFC + K exists. Does the iterability of K^M imply the iterability of M ? postscript
  9. Let n be an integer. Suppose that V is closed under M_n^#, but there is no inner model with n+1 Woodin cardinals. Is K \Sigma^1_{n+3} correct ? postscript
  10. Suppose that \Omega is measurable and that there is no inner model with a Woodin cardinal. Let K denote Steel's core model. Does the following hold in K ? There a sequence (S_i^h : 0 < i < OR, h \in 2) such that S_i^h always consists of points of cofinality \omega_1 which are smaller than \omega_{i+1} and for all limit ordinals \lambda and all f : \lambda -> 2 we have that (S_i^{f(i)} : i<\lambda) is mutually stationary if and only if we can split \lambda into a finite partition of intervals so that f is constant on each of these intervals.
  11. Suppose that there is no inner model with a Woodin cardinal. Let x be a real such that x^## exists. Show that there is a proper initial segment of K^{L[x^#]} which iterates past K^{L[x]}.


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