Our aim is to show the consistency of 2κ ≥ κ+θ, for any θ < κ starting with a singular cardinal κ of cofinality ω so that ∀n < ω∃α < κ(o(α) = α). Dropping cofinalities technics are used for this purpose. 1 א1-gap and infinite drops in cofinalities Let κ be a singular cardinal of cofinality ω such that for each γ < κ and n < ω there is α, γ < α < κ, such that o(α) = α. We fix a sequence of cardi...

The continuum function F on regular cardinals is known to have great freedom; if α, β are regular cardinals, then F needs only obey the following two restrictions: (1) cf(F (α)) > α, (2) α < β → F (α) ≤ F (β). However, if we wish to preserve measurable cardinals in the generic extension, new restrictions must be put on F . We say that κ is F (κ)-hypermeasurable if there is an elementary embeddi...

We improve the geometric properties of SLE(κ; ~ ρ) processes derived in an earlier paper, which are then used to obtain more results about the duality of SLE. We find that for κ ∈ (4, 8), the boundary of a standard chordal SLE(κ) hull stopped on swallowing a fixed x ∈ R \ {0} is the image of some SLE(16/κ; ~ ρ) trace started from a random point. Using this fact together with a similar propositi...

narrowing ABSTRACT NARROWING u∈ Onv (g) & (l=r ⇐ C) << R A & κ = mgu ({g|u,l}) ––––––––––––––––––––––––––––––––––––––– g ⇒κ (C, g[r]u)κ  abstract semantics of a goal g ∆(g)= {κ| g ⇒κ* true} (decidable)  the abstract semantics approximates the concrete semantics ∀ σ ∈ O(g). ∃ κ ∈ ∆(g). κ ≤ σ  unsatisfiability analysis ∆(g)=∅ ⇒ g es unsatisfiable  abstract basic narrowing is compositional ∆(g...

We generalize ∇(A), which was introduced in [Sch∞], to larger cardinals. For a regular cardinal κ > א0 we denote by ∇κ(A) the statement that A ⊆ κ and for all regular θ > κ, {X ∈ [Lθ[A] ] <κ : X ∩ κ ∈ κ ∧ otp(X ∩ Ord) ∈ Card} is stationary in [Lθ[A] ] . It was shown in [Sch∞] that ∇א1(A) can hold in a set-generic extension of L. We here prove that ∇א2(A) can hold in a set-generic extension of L...

Strengthening a result of Amir Leshem [7], we prove that the consistency strength of holding GCH together with definable tree property for all successors of regular cardinals is precisely equal to the consistency strength of existence of proper class many Π 1 reflecting cardinals. Moreover it is proved that if κ is a supercompact cardinal and λ > κ is measurable, then there is a generic extensi...

A question of Woodin asks if κ is strongly compact and GCH holds below κ, then must GCH hold everywhere? One variant of this question asks if κ is strongly compact and GCH fails at every regular cardinal δ < κ, then must GCH fail at some regular cardinal δ ≥ κ? Another variant asks if it is possible for GCH to fail at every limit cardinal less than or equal to a strongly compact cardinal κ. We ...

We reprove Gitik’s theorem that if the GCH holds and o(κ) = κ + 1 then there is a generic extension in which κ is still measurable and there is a closed unbounded subset C of κ such that every ν ∈ C is inaccessible in the ground model. Unlike the forcing used by Gitik, the iterated forcing Rλ+1 used in this paper has the property that if λ is a cardinal less then κ then Rλ+1 can be factored in ...

Three central combinatorial properties in set theory are the tree property, the approachability property and stationary reflection. We prove the mutual independence of these properties by showing that any of their eight Boolean combinations can be forced to hold at κ, assuming that κ = κ and there is a weakly compact cardinal above κ. If in addition κ is supercompact then we can force κ to be א...

Biswapped networks of the form Bsw(G) have recently been proposed as interconnection networks to be implemented as optical transpose interconnection systems. We provide a systematic construction of κ + 1 vertexdisjoint paths joining any two distinct vertices in Bsw(G), where κ ≥ 1 is the connectivity of G. In doing so, we obtain an upper bound of max{2∆(G)+5,∆κ(G)+∆(G)+2} on the (κ+1)-diameter ...