Iterations of V and the core model

Answering a question which was around for some while we prove that if \pi : V -> M is such that M is transitive and closed under \omega-sequences then the core model of M is an iterate of the core model of V.Comment: 3 page

Coding into K by reasonable forcing

Assuming that there is no inner model with a strong cardinal, the following is shown: any subset of \omega_1 can be made \Delta^1_3 (in the codes) by a reasonable set-forcing; there is a reasonable set-generic extension with a \Delta^1_3 well-ordering of its reals; 2-step \Sigma^1_4 absoluteness fails w.r.t. set-sized reasonable forcings

A note on an alleged proof of the relative consistency of P=NP with PA

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.Comment: 4 page

Core models in the presence of Woodin cardinals

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.Comment: 6 page

Cardinal arithmetic and Woodin cardinals

Suppose that there is a measurable cardinal. If \aleph_\omega is a strong limit cardinal, but the power of \aleph_\omega is bigger than \aleph_{\omega_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.Comment: 6 page

Weak covering and the tree property

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 \delta has the tree property then \delta^{+K} = \delta^+ and \delta is weakly compact in K

A simple proof of \Sigma^1_3 correctness of K

We present a simple and purely combinatorial proof of Steel's result according to which the core model is \Sigma^1_3 correct under the appropriate hypotheses.Comment: 5 page

More on mutual stationarity

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_\alpha : n0) such that each individual S^n_\alpha is a stationary subset of \aleph_{\alpha+1} consisting of points of cofinality \omega_1, and for all limits \lambda and for all f:\lambda -> \omega do we have that (S^{f(\alpha)}_\alpha : \alpha<\lambda) is mutually stationary if and only if the range of f is finite.Comment: 7 page

A universal weasel without large cardinals in 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.Comment: 8 page

L(R) absoluteness under proper 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.Comment: 15 page