Iteration of -complete forcing notions not collapsing .
Laminations, or How to Build a Quantum-Logic-Valued Model of Set Theory.
Large continuum, oracles
Our main theorem is about iterated forcing for making the continuum larger than ℵ2. We present a generalization of [2] which deal with oracles for random, (also for other cases and generalities), by replacing ℵ1,ℵ2 by λ, λ + (starting with λ = λ <λ > ℵ1). Well, we demand absolute c.c.c. So we get, e.g. the continuum is λ + but we can get cov(meagre) = λ and we give some applications. As in non-Cohen oracles [2], it is a “partial” countable support iteration but it is c.c.c.
Life in the Sacks model
Limits of families of measure algebras
Locally constant functions
Let X be a compact Hausdorff space and M a metric space. is the set of f ∈ C(X,M) such that there is a dense set of points x ∈ X with f constant on some neighborhood of x. We describe some general classes of X for which is all of C(X,M). These include βℕ, any nowhere separable LOTS, and any X such that forcing with the open subsets of X does not add reals. In the case where M is a Banach space, we discuss the properties of as a normed linear space. We also build three first countable Eberlein...
MAD families with strong combinatorial properties
In his paper in Fund. Math. 178 (2003), Miller presented two conjectures regarding MAD families. The first is that CH implies the existence of a MAD family that is also a σ-set. The second is that under CH, there is a MAD family concentrated on a countable subset. These are proved in the present paper.
Mycielski ideals generated by uncountable systems
Nonexistence of nonmolecular generic sets.
Non-Glimm–Effros equivalence relations at second projective level
A model is presented in which the equivalence relation xCy iff L[x]=L[y] of equiconstructibility of reals does not admit a reasonable form of the Glimm-Effros theorem. The model is a kind of iterated Sacks generic extension of the constructible model, but with an “ill“founded “length” of the iteration. In another model of this type, we get an example of a non-Glimm-Effros equivalence relation on reals. As a more elementary application of the technique of “ill“founded Sacks iterations, we obtain...
Norms on possibilities. II: More ccc ideals on .
On absolutely operations
On collections of almost disjoint families
On game ideals
On iterated forcing for successors of regular cardinals
We investigate the problem of when ≤λ-support iterations of < λ-complete notions of forcing preserve λ⁺. We isolate a property- properness over diamonds-that implies λ⁺ is preserved and show that this property is preserved by λ-support iterations. Our condition is a relative of that presented by Rosłanowski and Shelah in [2]; it is not clear if the two conditions are equivalent. We close with an application of our technology by presenting a consistency result on uniformizing colorings of ladder...
On the existence of a -closed dense subset
It is consistent with the axioms of set theory that there are two co-dense partial orders, one of them -closed and the other one without a -closed dense subset.
On the injectivity of Boolean algebras
The functor taking global elements of Boolean algebras in the topos of sheaves on a complete Boolean algebra is shown to preserve and reflect injectivity as well as completeness. This is then used to derive a result of Bell on the Boolean Ultrafilter Theorem in -valued set theory and to prove that (i) the category of complete Boolean algebras and complete homomorphisms has no non-trivial injectives, and (ii) the category of frames has no absolute retracts.
On Todorcevic orderings
The Todorcevic ordering 𝕋(X) consists of all finite families of convergent sequences in a given topological space X. Such an ordering was defined for the special case of the real line by S. Todorcevic (1991) as an example of a Borel ordering satisfying ccc that is not σ-finite cc and even need not have the Knaster property. We are interested in properties of 𝕋(X) where the space X is taken as a parameter. Conditions on X are given which ensure the countable chain condition and its stronger versions...
Open mappings on extremally disconnected compact spaces
Perfect sets and collapsing continuum
Under Martin’s axiom, collapsing of the continuum by Sacks forcing is characterized by the additivity of Marczewski’s ideal (see [4]). We show that the same characterization holds true if proving that under this hypothesis there are no small uncountable maximal antichains in . We also construct a partition of into perfect sets which is a maximal antichain in and show that -sets are exactly (subsets of) selectors of maximal antichains of perfect sets.