The continuum function αmaps to2α on regular cardinals is known to have great freedom. Let us say that F is an Easton function iff for regular cardinals α and β, image and α<β→F≤F. The classic example of an Easton function is the continuum function αmaps to2α on regular cardinals. If GCH holds then any Easton function is the continuum function on regular cardinals of some cofinality-preserving extension V[G]; we say that F is realised in V[G]. However if we also wish (...) to preserve measurable cardinals, new restrictions must be put on F. We say that κ is F-hypermeasurable iff there is an elementary embedding j:V→M with critical point κ such that H)Vsubset of or equal toM; j will be called a witnessing embedding. We will show that if GCH holds then for any Easton function F there is a cofinality-preserving generic extension V[G] such that if κ, closed under F, is F-hypermeasurable in V and there is a witnessing embedding j such that j≥F, then κ will remain measurable in V[G]. (shrink)

There have been numerous results showing that a measurable cardinal κ can carry exactly α normal measures in a model of GCH, where a is a cardinal at most κ⁺⁺. Starting with just one measurable cardinal, we have [9] (for α = 1), [10] (for α = κ⁺⁺, the maximum possible) and [1] (for α = κ⁺, after collapsing κ⁺⁺) . In addition, under stronger large cardinal hypotheses, one can handle the remaining cases: [12] (starting with a measurable cardinal of (...) Mitchell order α ) , [2] (as in [12], but where κ is the least measurable cardinal and α is less than κ, starting with a measurable of high Mitchell order) and [11] (as in [12], but where κ is the least measurable cardinal, starting with an assumption weaker than a measurable cardinal of Mitchell order 2). In this article we treat all cases by a uniform argument, starting with only one measurable cardinal and applying a cofinality-preserving forcing. The proof uses κ-Sacks forcing and the "tuning fork" technique of [8]. In addition, we explore the possibilities for the number of normal measures on a cardinal at which the GCH fails. (shrink)

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 continue investigations of reasonable ultrafilters on uncountable cardinals defined in previous work by Shelah. We introduce stronger properties of ultrafilters and we show that those properties may be handled in λ-support iterations of reasonably bounding forcing notions. We use this to show that consistently there are reasonable ultrafilters on an inaccessible cardinal λ with generating systems of size less than $2^\lambda$ . We also show how ultrafilters generated by small systems can be killed by forcing notions which have enough (...) reasonable completeness to be iterated with λ-supports. (shrink)

The lottery preparation, a new general kind of Laver preparation, works uniformly with supercompact cardinals, strongly compact cardinals, strong cardinals, measurable cardinals, or what have you. And like the Laver preparation, the lottery preparation makes these cardinals indestructible by various kinds of further forcing. A supercompact cardinal κ, for example, becomes fully indestructible by <κ-directed closed forcing; a strong cardinal κ becomes indestructible by κ-strategically closed forcing; and a strongly compact cardinal κ becomes indestructible by, among others, the forcing to (...) add a Cohen subset to κ, the forcing to shoot a club Cκ avoiding the measurable cardinals and the forcing to add various long Prikry sequences. The lottery preparation works best when performed after fast function forcing, which adds a new completely general kind of Laver function for any large cardinal, thereby freeing the Laver function concept from the supercompact cardinal context. (shrink)

Assuming the existence of a weakly compact hypermeasurable cardinal we prove that in some forcing extension א ω is a strong limit cardinal and א ω+2 has the tree property. This improves a result of Matthew Foreman (see [2]).

An important technique in large cardinal set theory is that of extending an elementary embedding j: M → N between inner models to an elementary embedding j*: M[G] → N[G*] between generic extensions of them. This technique is crucial both in the study of large cardinal preservation and of internal consistency. In easy cases, such as when forcing to make the GCH hold while preserving a measurable cardinal (via a reverse Easton iteration of α-Cohen forcing for successor cardinals α), the (...) generic G* is simply generated by the image of G. But in difficult cases, such as in Woodin's proof that a hypermeasurable is sufficient to obtain a failure of the GCH at a measurable, a preliminary version of G* must be constructed (possibly in a further generic extension of M[G]) and then modified to provide the required G*. In this article we use perfect trees to reduce some difficult cases to easy ones, using fusion as a substitute for distributivity. We apply our technique to provide a new proof of Woodin's theorem as well as the new result that global domination at inaccessibles (the statement that d (κ) is less than 2κ for inaccessible κ, where d (κ) is the dominating number at κ) is internally consistent, given the existence of 0#. (shrink)