We generalize the notion of a fat subset of a regular cardinal κ to a fat subset of , where κ ⊆ X. Suppose μ < κ, , and κ is supercompact. Then there is a generic extension in which κ = μ⁺⁺, and for all regular λ ≥ μ⁺⁺, there are stationarily many N in which are internally club but not internally approachable.
We introduce a new method which combines Prikry forcing with an iteration between the Prikry points. Using our method we prove from large cardinals that it is consistent that the tree property holds at ℵₙ for n ≥ 2, is strong limit and .
It is proved that ideal-based forcings with the side condition method of Todorcevic (1984) add no random reals. By applying Judah-Repický's preservation theorem, it is consistent with the covering number of the null ideal being ℵ₁ that there are no S-spaces, every poset of uniform density ℵ₁ adds ℵ₁ Cohen reals, there are only five cofinal types of directed posets of size ℵ₁, and so on. This extends the previous work of Zapletal (2004).
Starting from large cardinals we construct a pair V₁⊆ V₂ of models of ZFC with the same cardinals and cofinalities such that GCH holds in V₁ and fails everywhere in V₂.
We prove that if there is a model of set-theory which contains no first countable, locally compact, scattered, countably paracompact space , whose Tychonoff square is a Dowker space, then there is an inner model which contains a measurable cardinal.
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.
A 1984 problem of S. Z. Ditor asks whether there exists a lattice of cardinality ℵ₂, with zero, in which every principal ideal is finite and every element has at most three lower covers. We prove that the existence of such a lattice follows from either one of two axioms that are known to be independent of ZFC, namely (1) Martin’s Axiom restricted to collections of ℵ₁ dense subsets in posets of precaliber ℵ₁, (2) the existence of a gap-1 morass. In particular, the existence of such a lattice is consistent...
We answer a question of Darji and Keleti by proving that there exists a compact set C₀ ⊂ ℝ of measure zero such that for every perfect set P ⊂ ℝ there exists x ∈ ℝ such that (C₀+x) ∩ P is uncountable. Using this C₀ we answer a question of Gruenhage by showing that it is consistent with ZFC (as it follows e.g. from ) that less than many translates of a compact set of measure zero can cover ℝ.
We construct two models for the level by level equivalence between strong compactness and supercompactness in which if κ is λ supercompact and λ ≥ κ is regular, we are able to determine exactly the number of normal measures carries. In the first of these models, carries many normal measures, the maximal number. In the second of these models, carries many normal measures, except if κ is a measurable cardinal which is not a limit of measurable cardinals. In this case, κ (and hence also )...
We construct three models containing exactly one supercompact cardinal in which level by level inequivalence between strong compactness and supercompactness holds. In the first two models, below the supercompact cardinal κ, there is a non-supercompact strongly compact cardinal. In the last model, any suitably defined ground model Easton function is realized.