We analyze the notion of guessing model, a way to assign combinatorial properties to arbitrary regular cardinals. Guessing models can be used, in combination with inaccessibility, to characterize various large cardinals axioms, ranging from supercompactness to rank-to-rank embeddings. The majority of these large cardinals properties can be defined in terms of suitable elementary embeddings j : ...

In the context of large cardinals, the classical diamond principle 3κ is easily strengthened in natural ways. When κ is a measurable cardinal, for example, one might ask that a 3κ sequence anticipate every subset of κ not merely on a stationary set, but on a set of normal measure one. This is equivalent to the existence of a function ` ... κ → Vκ such that for any A ∈ H(κ) there is an embedding...

The authors show, by means of a finitary version ¤ λ,D of the combinatorial principle ¤ ∗ λ of [7], the consistency of the failure, relative to the consistency of supercompact cardinals, of the following: for all regular filters D on a cardinal λ, if Mi and Ni are elementarily equivalent models of a language of size ≤ λ, then the second player has a winning strategy in the Ehrenfeucht-Fräıssé g...

Given a cardinal κ that is λ-supercompact for some regular cardinal λ ≥ κ and assuming GCH, we show that one can force the continuum function to agree with any function F : [κ, λ] ∩ REG → CARD satisfying ∀α, β ∈ dom(F ) α < cf(F (α)) and α < β =⇒ F (α) ≤ F (β), while preserving the λ-supercompactness of κ from a hypothesis that is of the weakest possible consistency strength, namely, from the h...

Drosophila melanogaster telomeric DNA is known to comprise two domains: the terminal tract of retrotransposons (HeT-A, TART and TAHRE) and telomere-associated sequences (TAS). Chromosome tips are capped by a protein complex, which is assembled on the chromosome ends independently of the underlying terminal DNA sequences. To investigate the properties of these domains in salivary gland polytene ...

We establish that if it is consistent that there is a supercompact cardinal, then it is consistent that every locally compact, hereditarily normal space which does not include a perfect pre-image of ω1 is hereditarily paracompact. This is the fifth in a series of papers ([LTo], [L2], [FTT], [LT], [T1] being the logically previous ones) that establish powerful topological consequences in models ...

Relative to the existence of a supercompact cardinal with a measurable cardinal above it, we show that it is consistent for א1 to be regular and for אω1+1 to be measurable and to carry precisely τ normal measures, where τ ≥ אω1+2 is any regular cardinal. This extends the work of [2], in which the analogous result was obtained for אω+1 using the same hypotheses. In [2], models were constructed f...

The Singular Cardinal Hypothesis (SCH) asserts that if κ is any singular strong limit cardinal then 2 = κ. It is known to be consistent that the SCH fails: Prikry [Pr] obtains a model of ¬SCH from a model in which the GCH fails at a measurable cardinal κ, and Silver in turn (see [KM]) obtains the failure of the GCH at a measurable cardinal from a model for a κ-supercompact cardinal. Silver’s re...

In two recent papers [9, 10] we answered a question raised in the book by Eklof and Mekler [7, p. 455, Problem 12] under the set theoretical hypothesis of ♦א1 which holds in many models of set theory, respectively of the special continuum hypothesis (CH). The objects are reflexive modules over countable principal ideal domains R, which are not fields. Following H. Bass [1] an R-module G is refl...

We show that from a supercompact cardinal κ, there is a forcing extension V [G] that has a symmetric inner model N in which ZF + ¬AC holds, κ and κ are both singular, and the continuum function at κ can be precisely controlled, in the sense that the final model contains a sequence of distinct subsets of κ of length equal to any predetermined ordinal. We also show that the above situation can be...