Maharam algebras are complete Boolean algebras carrying a positive continuous submeasure. They were introduced and studied by Maharam [D. Maharam, An algebraic characterization of measure algebras, Ann. of Math. 48 154–167] in relation to Von Neumann’s problem on the characterization of measure algebras. The question whether every Maharam algebra is a measure algebra has been the main open problem in this area for around 60 years. It was finally resolved by Talagrand [M. Talagrand, Maharam’s problem, preprint, 31 pages, 2006] who (...) provided the first example of a Maharam algebra which is not a measure algebra. In this paper we survey some recent work on Maharam algebras in relation to the two conditions proposed by Von Neumann: weak distributivity and the countable chain condition. It turns out that by strengthening either one of these conditions one obtains a ZFC characterization of Maharam algebras. We also present some results on Maharam algebras as forcing notions showing that they share some of the well-known properties of measure algebras. (shrink)

Continuing [7], we here prove that the Chang Conjecture $(\aleph_3,\aleph_2) \Rightarrow (\aleph_2,\aleph_1)$ together with the Continuum Hypothesis, $2^{\aleph_0} = \aleph_1$ , implies that there is an inner model in which the Mitchell ordering is $\geq \kappa^{+\omega}$ for some ordinal $\kappa$.

We state some results related to the Silver Theorem.

Woodin has shown that if there is a measurable Woodin cardinal then there is, in an appropriate sense, a sharp for the Chang model. We produce, in a weaker sense, a sharp for the Chang model using only the existence of a cardinal \ having an extender of length \.

We show: There are pairs of universes V1V2 and there is a notion of forcing PV1 such that the change mentioned in the title occurs when going from V1[G] to V2[G] for a P-generic filter G over V2. We use forcing iterations with partial memories. Moreover, we implement highly transitive automorphism groups into the forcing orders.

We show how to construct Gitik’s short extenders gap-3 forcing using a morass, and that the forcing notion is of Prikry type.

We show that Gitik’s short extender gap-2 forcing is of Prikry type.

We say that κ is μ-hypermeasurable for a cardinal μ≥κ+ if there is an embedding j:V→M with critical point κ such that HV is included in M and j>μ. Such a j is called a witnessing embedding.Building on the results in [7], we will show that if V satisfies GCH and F is an Easton function from the regular cardinals into cardinals satisfying some mild restrictions, then there exists a cardinal-preserving forcing extension V* where F is realised on all V-regular (...) cardinals and moreover, all F-hypermeasurable cardinals κ, where F>κ+, with a witnessing embedding j such that either j=κ+ or j≥F, are turned into singular strong limit cardinals with cofinality ω. This provides some partial information about the possible structure of a continuum function with respect to singular cardinals with countable cofinality.As a corollary, this shows that the continuum function on a singular strong limit cardinal κ of cofinality ω is virtually independent of the behaviour of the continuum function below κ, at least for continuum functions which are simple in that 2α{α+,α++} for every cardinal α below κ ≥F or j=κ+). (shrink)

We prove equiconsistency results concerning gaps between a singular strong limit cardinal κ of cofinality 0 and its power under assumptions that 2κ=κ+δ+1 for δ<κ and some weak form of the Singular Cardinal Hypothesis below κ. Together with the previous results this basically completes the study of consistency strength of the various gaps between such κ and its power under GCH type assumptions below.

We show that under certain large cardinal requirements there is a generic extension in which the power function behaves differently on different stationary classes. We achieve this by doing an Easton support iteration of the Radin on extenders forcing.

The purpose of the present paper is to present new methods of blowing up the power of a singular cardinal κ of cofinality ω. New PCF configurations are obtained. The techniques developed here will be used in a subsequent paper to construct a model with a countable set which pcf has cardinality ℵ1.

The paper is concerned with methods for blowing power of singular cardinals using short extenders. Thus, for example, starting with κ of cofinality ω with {α<κ oα+n} cofinal in κ for every n<ω we construct a cardinal preserving extension having the same bounded subsets of κ and satisfying 2κ=κ+δ+1 for any δ<1.

Our aim is to show that it is impossible to find a bound for the power of the first fixed point of the aleph function.

We give an application of the extender based Radin forcing to cardinal arithmetic. Assuming κ is a large enough cardinal we construct a model satisfying 2κ = κ⁺ⁿ together with 2λ = λ⁺ⁿ for each cardinal λ < κ, where 0 < n < ω. The cofinality of κ can be set arbitrarily or κ can remain inaccessible. When κ remains an inaccessible, Vκ is a model of ZFC satisfying 2λ = λ+n for all cardinals λ.

We determine exact consistency strengths for various failures of the Singular Cardinals Hypothesis (SCH) in the setting of the Zermelo-Fraenkel axiom system ZF without the Axiom of Choice (AC). By the new notion of parallel Prikry forcing that we introduce, we obtain surjective failures of SCH using only one measurable cardinal, including a surjective failure of Shelah's pcf theorem about the size of the power set of $\aleph _{\omega}$ . Using symmetric collapses to $\aleph _{\omega}$ , $\aleph _{\omega _{1}}$ , (...) or $\aleph _{\omega _{2}}$ , we show that injective failures at $\aleph _{\omega}$ , $\aleph _{\omega _{1}}$ , or $\aleph _{\omega _{2}}$ can have relatively mild consistency strengths in terms of Mitchell orders of measurable cardinals. Injective failures of both the aforementioned theorem of Shelah and Silver's theorem that GCH cannot first fail at a singular strong limit cardinal of uncountable cofinality are also obtained. Lower bounds are shown by core model techniques and methods due to Gitik and Mitchell. (shrink)