arXiv subjects
Publications and source records attributed to Ralf Schindler.
It is shown that if BMM (= Bounded Martin's Maximum) holds then each set is contained in an inner model with a strong cardinal. This answers a question that has been asked by various people. It follows that BMM has a much larger consistency strength than BSPFA (= the Bounded Semi-proper Forcing Axiom).
It is shown that if there is a measurable cardinal above n Woodin cardinals and M_{n+1}^# doesn't exist then K exists. K is not fully iterable, though, but only iterable with respect to stacks of certain trees living between the Woodin cardinals. However, it is still true that if M is an omega-closed iterate of V then K^M is an iterate of K.
We prove the following two results. Theorem A: Let alpha be a limit ordinal. Suppose that 2^{|alpha|} aleph_{|alpha|^+}. Then for all n< omega and for all bounded X subset aleph_{|alpha|^+}, M_n^#(X) exists. Theorem B: Let kappa be a singular cardinal of uncountable cofinality. If {alpha<kappa| 2^alpha=alpha^+} is stationary as well as co-stationary then for all n< omega and for all bounded X subset kappa, M_n^#(X) exists. Theorem A answers a question of Gitik and Mitchell, and Theorem B yields a lower bound for an assertion discussed in Gitik, M., Introduction to Prikry type forcing notions, in: Handbook of set theory, Foreman, Kanamori, Magidor (see Problem 4 there). The proofs of these theorems combine pcf theory with core model theory. Along the way we establish some ZFC results in cardinal arithmetic, motivated by Silver's theorem and we obtain results of core model theory, motivated by the task of building a ``stable core model.'' Both sets of results are of independent interest.
Suppose that there is a measurable cardinal. If \aleph_ωis a strong limit cardinal, but the power of \aleph_ωis bigger than \aleph_{ω_1}, then there is an inner model with a Woodin cardinal. Modulo the need of the measurable cardinal this answers a question of Gitik and Mitchell.
Steel and Welch have shown that K is Σ^1_3 correct if the reals are closed under sharps but 0^\pistol doesn't exist. We'll give a simple and purely combinatorial proof of the following: K is Σ^1_3 correct if the reals are closed under sharps, there is no inner model with a Woodin cardinal, K exists, and * holds. Here, * is an assertion which can easily be verified if 0^\pistol doesn't exist. We conjecture that * holds outright. (* is denoted by \clubsuit in the paper.)
We present a simple and purely combinatorial proof of Steel's result according to which the core model is Σ^1_3 correct under the appropriate hypotheses.
We state a version of the P=?NP problem for infinite time Turing machines. It is observed that P not= NP for this version.
For an uncountable regular cardinal κwe let \nabla_κ(A) be the statement that A \subset κand for all regular θ> κ, the set of all X \in [θ]^<κsuch that X \cap κ\in κand otp(X \cap OR) is a cardinal in L[A \cap X \cap κ] is stationary. We had shown earlier that \nabla_{ω_1}(A) can hold in a generic extension of L. We now prove that \nabla_{ω_2}(A) can hold in a semi-proper generic extension of L, whereas \nabla_{ω_3}(0) is equivalent with the existence of 0^#.
Extending a result of Foreman and Magidor we prove that in the core model for almost linear iterations the following holds. There is a sequence (S^n_α: n<ω,α>0) such that each individual S^n_αis a stationary subset of \aleph_{α+1} consisting of points of cofinality ω_1, and for all limits λand for all f:λ-> ωdo we have that (S^{f(α)}_α: α<λ) is mutually stationary if and only if the range of f is finite.
Answering a question which was around for some while we prove that if π: V -> M is such that M is transitive and closed under ω-sequences then the core model of M is an iterate of the core model of V.
We prove in the theory "ZFC + there is no inner model with a Woodin cardinal" that there is a universal weasel. This shows in particular that one doesn't have to assume the existence of large cardinals in V to prove the existence of some such weasel.
We indicate that an argument of da Costa and Doria in fact proves P=NP. This observation makes their argument appear dubious. We isolate a weak version of one of their lemmas which would already prove P=NP. We point out that even this weak version is probably false. In fact, a generalized form of this weak version is provably false.
Assuming that there is no inner model with a strong cardinal, the following is shown: any subset of ω_1 can be made Δ^1_3 (in the codes) by a reasonable set-forcing; there is a reasonable set-generic extension with a Δ^1_3 well-ordering of its reals; 2-step Σ^1_4 absoluteness fails w.r.t. set-sized reasonable forcings.
We isolate a new large cardinal concept, "remarkability." Consistencywise, remarkable cardinals are between ineffable and omega-Erdos cardinals. They are characterized by the existence of "0^sharp-like" embeddings; however, they relativize down to L. It turns out that the existence of a remarkable cardinal is equiconsistent with L(R) absoluteness under proper forcings. In particular, said absoluteness does not imply Pi^1_1 determinacy.
It is shown that if every projective set of reals is Lebesgue measurable and has the property of Baire, if every projective set in the plane has a projective uniformization, and if Steel's K exists, then J^K_{ω_1} \models "there are infinitely many strong cardinals." This is best possible, by a recent result of Steel.
Suppose that there's no transitive model of ZFC + there's a strong cardinal, and let K denote the core model. It is shown that if δhas the tree property then δ^{+K} = δ^+ and δis weakly compact in K.