We obtain from the consistency of the existence of a measurable cardinal the consistency of “small” upper bounds on the cardinality of a large class of Lindelöf spaces whose singletons are Gδ sets. Call a topological space in which each singleton is a Gδ set a points Gδ space. A.V. Arhangel’skii proved that any points Gδ Lindelöf space must have cardinality less than the least measurable cardin...

We consider a method of representing projective sets by a particular type of union of Borel sets, assuming AD. We prove a generalization of the theorem that a set is S^ iff it is the union of a>x Borel sets. The Axiom of Determinacy (AD) is always assumed. Theorem 1 (Sierpinski, Solovay, Moschovakis [8, 7D.10]). Let A c com. A is Ej iff A is the union of cox Borel sets. The purpose of this pape...

ABSRACT In his book Proper Forcing (1982) Shelah introduced three classes of forcings (complete, proper, and semi-proper) and proved a strong iteration theorem for each of them: The first two are closed under countable support iterations. The latter is closed under revised countable support iterations subject to certain standard restraints. These theorems have been heavily used in modern set th...

We generalize the notion of a fat subset of a regular cardinal κ to a fat subset of Pκ(X), where κ ⊆ X. Suppose μ < κ, μ<μ = μ, 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 [H(λ)]μ which are internally club but not internally approachable. Suppose μ is an infinite cardinal. A set N is internally approach...

We investigate the pressing down game and its relation to the Banach Mazur game. In particular we show: Consistently relative to a supercompact, there is a nowhere precipitous normal ideal I on א2 such that player nonempty wins the pressing down game of length א1 on I even if player empty starts. For the proof, we construct a forcing notion to force the following: There is normal, nowhere preci...

The Levy-Solovay Theorem [LevSol67] limits the kind of large cardinal embeddings that can exist in a small forcing extension. Here I announce a generalization of this theorem to a broad new class of forcing notions. One consequence is that many of the forcing iterations most commonly found in the large cardinal literature create no new weakly compact cardinals, measurable cardinals, strong card...

After forcing which admits a very low gap—and this includes many of the forcing iterations, such as the Laver preparation, which are commonly found in the large cardinal context—every embedding j : V [G] → M [j(G)] in the extension which satisfies a mild closure condition is the lift of an embedding j : V → M in the ground model. In particular, every ultrapower embedding in the extension lifts ...

Continuing (Fuchino et al. in Arch Math Log, 2020. https://doi.org/10.1007/s00153-020-00730-x), we study the Strong Downward Lowenheim–Skolem Theorems (SDLSs) of stationary logic and their variations. In Fuchino (2020) it has been shown that SDLS for ordinary with weak second-order parameters $$\textsf {SDLS}({\mathcal {L}}^{\aleph _0}_{stat},{<}\,\aleph _2)$$ down to $${<}\,\aleph _2$$ is eq...

It is possible to control to a large extent, via semiproper forcing, the parameters (β0, β1) measuring the guessing density of the members of any given antichain of stationary subsets of ω1 (assuming the existence of an inaccessible limit of measurable cardinals). Here, given a pair (β0, β1) of ordinals, we will say that a stationary set S ⊆ ω1 has guessing density (β0, β1) if β0 = γ(S) and β1 ...

We address some question raised in Magidor-Sinapova [7] paper. Suppose V ⊆ W , κ is regular in V , but change its cofinality in W . Are there ”nice” witnesses for such change? This is a basic question and a lot of work was done around it (probably the most prominent are Prikry forcing, Jensen and Dodd-Jensen Covering Lemmas, Mitchell Covering Lemmas). Some ZFC results were proved in Dzamonia-Sh...