We prove a Mathias-type criterion for the Magidor iteration of Prikry forcings.

It is known that there is a close relation between Prikry forcing and the iteration of ultrapowers: If U is a normal ultrafilter on a measurable cardinal κ and 〈Mn, jm,n | m ≤ n ≤ ω〉 is the iteration of ultrapowers of V by U, then the sequence of critical points 〈j0,n | n ∈ ω〉 is a Prikry generic sequence over Mω. In this paper we generalize this for normal precipitous filters.

Applying the seed concept to Prikry tree forcing P μ , I investigate how well P μ preserves the maximality property of ordinary Prikry forcing and prove that P μ Prikry sequences are maximal exactly when μ admits no non-canonical seeds via a finite iteration. In particular, I conclude that if μ is a strongly normal supercompactness measure, then P μ Prikry sequences are maximal, thereby proving, for a large class of measures, a conjecture of W. Hugh Woodin's.

If $U_\alpha$ is a length $\omega_1$ sequence of normal ultrafilters on a measurable cardinal $\kappa$ that is increaing w.r.t. the Mitchel order, then the intersection of the $\omega_1$ first iterated ultrapowers of the universe is a Magidor generic extension of the $\omega_1$th iterated ultrapower.

We investigate iterating the construction of $C^{*}$, the L-like inner model constructed using first order logic augmented with the “cofinality $\omega $ ” quantifier. We first show that $\left (C^{*}\right )^{C^{*}}=C^{*}\ne L$ is equiconsistent with $\mathrm {ZFC}$, as well as having finite strictly decreasing sequences of iterated $C^{*}$ s. We then show that in models of the form $L[U]$ we get infinite decreasing sequences of length $\omega $, and that an inner model with a measurable cardinal is required for that.

We improve the upper bound for the consistency strength of stationary reflection at successors of singular cardinals.

In this paper, we answer a question asked in Koepke et al. regarding a Mathias criteria for Tree-Prikry forcing. Also we will investigate Prikry forcing using various filters. For completeness and self inclusion reasons, we will give proofs of many known theorems.

We give a new characterization of λ-supercompact cardinal κ in terms of -Solovay pairs. We give some applications of -Solovay pairs.

Ifκis a measurable cardinal, let us say that a measure onκis aκ-complete nonprincipal ultrafilter onκ. IfUis a measure onκ, letjUbe the canonical elementary embedding ofVinto its Ultrapower UltU. Ifxis a set, say thatUmovesxwhenjU≠x; say thatκmovesxwhen some measure onκmovesx. Recall Kunen's lemma : “Every ordinal is moved only by finitely many measurable cardinals.” Kunen's proof and Fleissner's proof are essentially nonconstructive.The following proposition can be proved by using elementary facts about iterated ultrapowers.Proposition.Let ‹Un: n ∈ ω› be a sequence of measures (...) on a strictly increasing sequence ‹κn: n ∈ ω› of measurable cardinals. Let U = ‹ Wα: α < ω2›, where Wωm + n= Um. Then, for each θ inUltU,if E is the support of θ inUltU,then, for all m ∈ ω, Ummoves θ iff E ∩ [ωm, ω)≠ ∅. (shrink)