We prove that if there is a model of set-theory which contains no first countable, locally compact, scattered, countably paracompact space $X$, 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 $cof\left(\right)<2^{\omega }$) that less than $2^{\omega }$ 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 $P_{\kappa }\left(\lambda \right)$ carries. In the first of these models, $P_{\kappa }\left(\lambda \right)$ carries $2^{2^{{\left[\lambda \right]}^{<\kappa }}}$ many normal measures, the maximal number. In the second of these models, $P_{\kappa }\left(\lambda \right)$ carries $2^{2^{{\left[\lambda \right]}^{<\kappa }}}$ 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 $P_{\kappa }\left(\kappa \right)$)...

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.

I prove that the statement that “every linear order of size $2^{\omega }$ can be embedded in $({\omega }^{\omega },\ll )$” is consistent with MA + ¬ wKH.

We force and construct a model in which GCH and level by level equivalence between strong compactness and supercompactness hold, along with certain additional “L-like” combinatorial principles. In particular, this model satisfies the following properties: (1) ${♢}_{\delta }$ holds for every successor and Mahlo cardinal δ. (2) There is a stationary subset S of the least supercompact cardinal κ₀ such that for every δ ∈ S, ${◻}_{\delta }$ holds and δ carries a gap 1 morass. (3) A weak version of ${◻}_{\delta }$ holds for every infinite cardinal...

We work towards establishing that if it is consistent that there is a supercompact cardinal then it is consistent that every locally compact perfectly normal space is paracompact. At a crucial step we use some still unpublished results announced by Todorcevic. Modulo this and the large cardinal, this answers a question of S. Watson. Modulo these same unpublished results, we also show that if it is consistent that there is a supercompact cardinal, it is consistent that every locally compact space...

Given an uncountable cardinal κ with $\kappa ={\kappa }^{<\kappa }$ and $2^{\kappa }$ regular, we show that there is a forcing that preserves cofinalities less than or equal to $2^{\kappa }$ and forces the existence of a well-order of H(κ⁺) that is definable over ⟨H(κ⁺),∈⟩ by a Σ₁-formula with parameters. This shows that, in contrast to the case "κ = ω", the existence of a locally definable well-order of H(κ⁺) of low complexity is consistent with failures of the GCH at κ. We also show that the forcing mentioned above introduces a Bernstein subset...

Assuming the continuum hypothesis there is an inseparable sequence of length ω₁ that contains no Lusin subsequence, while if Martin's Axiom and ¬ CH are assumed then every inseparable sequence (of length ω₁) is a union of countably many Lusin subsequences.

Under $M{A}_{{\omega }_1}$ every uncountable almost disjoint family is either anti-Luzin or has an uncountable Luzin subfamily. This fails under CH. Related properties are also investigated.