We define and study two classes of uncountable ⊆*-chains: Hausdorff towers and Suslin towers. We discuss their existence in various models of set theory. Some of the results and methods are used to provide examples of indestructible gaps not equivalent to a Hausdorff gap. We also indicate possible ways of developing a structure theory for towers based on classification of their Tukey types.
We show that, consistently, for some regular cardinals θ <λ, there exists a Boolean algebra 𝔹 such that |𝔹| = λ⁺ and for every subalgebra 𝔹'⊆ 𝔹 of size λ⁺ we have Depth(𝔹') = θ.
In an attempt to extend the property of being supercompact but not HOD-supercompact to a proper class of indestructibly supercompact cardinals, a theorem is discovered about a proper class of indestructibly supercompact cardinals which reveals a surprising incompatibility. However, it is still possible to force to get a model in which the property of being supercompact but not HOD-supercompact holds for the least supercompact cardinal κ₀, κ₀ is indestructibly supercompact, the strongly compact and...
We show that assuming the consistency of a supercompact cardinal with a measurable cardinal above it, it is possible for to be measurable and to carry exactly τ normal measures, where is any regular cardinal. This contrasts with the fact that assuming AD + DC, is measurable and carries exactly three normal measures. Our proof uses the methods of [6], along with a folklore technique and a new method due to James Cummings.
We present a new forcing notion combining diagonal supercompact Prikry forcing with interleaved extender based forcing. We start with a supercompact cardinal κ. In the final model the cofinality of κ is ω, the singular cardinal hypothesis fails at κ, and GCH holds below κ. Moreover we define a scale at κ which has a stationary set of bad points in the ground model.
Questa è la prima parte di una articolo espositivo dedicato ai teoremi di assolutezza, un argomento che sta assumendo un’importanza via via più grande in teoria degli insiemi. In questa prima parte vedremo come le questioni di teoria dei numeri non siano influenzate da assunzioni insiemistiche quali l’assioma di scelta o l’ipotesi del continuo.
Questa è la seconda parte dell’articolo espositivo [A]. Qui vedremo come siapossibile utilizzare il forcinge gli assiomi forti dell’infinito per dimostrare nuovi teoremi sui numeri reali.
Countable products of finite discrete spaces with more than one point and ideals generated by Marczewski-Burstin bases (assigned to trimmed trees) are examined, using machinery of base tree in the sense of B. Balcar and P. Simon. Applying Kulpa-Szymanski Theorem, we prove that the covering number equals to the additivity or the additivity plus for each of the ideals considered.
Given a topological space ⟨X,T⟩ ∈ M, an elementary submodel of set theory, we define to be X ∩ M with topology generated by U ∩ M:U ∈ T ∩ M. We prove that if is homeomorphic to ℝ, then . The same holds for arbitrary locally compact uncountable separable metric spaces, but is independent of ZFC if “local compactness” is omitted.
We answer several questions of D. Monk by showing that every maximal family of pairwise incomparable elements of 𝒫(ω)/fin has size continuum, while it is consistent with the negation of the Continuum Hypothesis that there are maximal subtrees of both 𝒫(ω) and 𝒫(ω)/fin of size ω₁.
Foreman (2013) proved a Duality Theorem which gives an algebraic characterization of certain ideal quotients in generic extensions. As an application he proved that generic supercompactness of ω₁ is preserved by any proper forcing. We generalize portions of Foreman's Duality Theorem to the context of generic extender embeddings and ideal extenders (as introduced by Claverie (2010)). As an application we prove that if ω₁ is generically strong, then it remains so after adding any number of Cohen subsets...
We show the relative consistency of the existence of two strongly compact cardinals κ₁ and κ₂ which exhibit indestructibility properties for their strong compactness, together with level by level equivalence between strong compactness and supercompactness holding at all measurable cardinals except for κ₁. In the model constructed, κ₁'s strong compactness is indestructible under arbitrary κ₁-directed closed forcing, κ₁ is a limit of measurable cardinals, κ₂'s strong compactness is indestructible...
We construct a model in which there is a strong cardinal κ whose strongness is indestructible under κ-strategically closed forcing and in which level by level equivalence between strong compactness and supercompactness holds non-trivially.
We show that if a colouring c establishes ω₂ ↛ [(ω₁:ω)]² then c establishes this negative partition relation in each Cohen-generic extension of the ground model, i.e. this property of c is Cohen-indestructible. This result yields a negative answer to a question of Erdős and Hajnal: it is consistent that GCH holds and there is a colouring c:[ω₂]² → 2 establishing ω₂ ↛ [(ω₁:ω)]₂ such that some colouring g:[ω₁]² → 2 does not embed into c. It is also consistent that is arbitrarily large, and there...
We construct a model for the level by level equivalence between strong compactness and supercompactness with an arbitrary large cardinal structure in which the least supercompact cardinal κ has its strong compactness indestructible under κ-directed closed forcing. This is in analogy to and generalizes the author's result in Arch. Math. Logic 46 (2007), but without the restriction that no cardinal is supercompact up to an inaccessible cardinal.