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Large continuum, oracles

Saharon Shelah — 2010

Open Mathematics

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.

On Monk’s questions

Saharon Shelah — 1996

Fundamenta Mathematicae

We deal with Boolean algebras and their cardinal functions: π-weight π and π-character πχ. We investigate the spectrum of π-weights of subalgebras of a Boolean algebra B. Next we show that the π-character of an ultraproduct of Boolean algebras may be different from the ultraproduct of the π-characters of the factors.

Borel sets with large squares

Saharon Shelah — 1999

Fundamenta Mathematicae

 For a cardinal μ we give a sufficient condition μ (involving ranks measuring existence of independent sets) for: μ if a Borel set B ⊆ ℝ × ℝ contains a μ-square (i.e. a set of the form A × A with |A| =μ) then it contains a 2 0 -square and even a perfect square, and also for μ ' if ψ L ω 1 , ω has a model of cardinality μ then it has a model of cardinality continuum generated in a “nice”, “absolute” way. Assuming M A + 2 0 > μ for transparency, those three conditions ( μ , μ and μ ' ) are equivalent, and from this we deduce that...

On a problem of Steve Kalikow

Saharon Shelah — 2000

Fundamenta Mathematicae

The Kalikow problem for a pair (λ,κ) of cardinal numbers,λ > κ (in particular κ = 2) is whether we can map the family of ω-sequences from λ to the family of ω-sequences from κ in a very continuous manner. Namely, we demand that for η,ν ∈ ω we have: η, ν are almost equal if and only if their images are. We show consistency of the negative answer, e.g., for ω but we prove it for smaller cardinals. We indicate a close connection with the free subset property and its variants.

Strong covering without squares

Saharon Shelah — 2000

Fundamenta Mathematicae

Let W be an inner model of ZFC. Let κ be a cardinal in V. We say that κ-covering holds between V and W iff for all X ∈ V with X ⊆ ON and V ⊨ |X| < κ, there exists Y ∈ W such that X ⊆ Y ⊆ ON and V ⊨ |Y| < κ. Strong κ-covering holds between V and W iff for every structure M ∈ V for some countable first-order language whose underlying set is some ordinal λ, and every X ∈ V with X ⊆ λ and V ⊨ |X| < κ, there is Y ∈ W such that X ⊆ Y ≺ M and V ⊨ |Y| < κ.   We prove that if κ is V-regular,...

Cellularity of free products of Boolean algebras (or topologies)

Saharon Shelah — 2000

Fundamenta Mathematicae

The aim this paper is to present an answer to Problem 1 of Monk [10], [11]. We do this by proving in particular that if μ is a strong limit singular cardinal, θ = ( 2 c f ( μ ) ) + and 2 μ = μ + then there are Boolean algebras 𝔹 1 , 𝔹 2 such that c ( 𝔹 1 ) = μ , c ( 𝔹 2 ) < θ b u t c ( 𝔹 1 * 𝔹 2 ) = μ + . Further we improve this result, deal with the method and the necessity of the assumptions. In particular we prove that if 𝔹 is a ccc Boolean algebra and μ ω λ = c f ( λ ) 2 μ then 𝔹 satisfies the λ-Knaster condition (using the “revised GCH theorem”).

On what I do not understand (and have something to say): Part I

Saharon Shelah — 2000

Fundamenta Mathematicae

This is a non-standard paper, containing some problems in set theory I have in various degrees been interested in. Sometimes with a discussion on what I have to say; sometimes, of what makes them interesting to me, sometimes the problems are presented with a discussion of how I have tried to solve them, and sometimes with failed tries, anecdotes and opinions. So the discussion is quite personal, in other words, egocentric and somewhat accidental. As we discuss many problems, history and side references...

Reflection implies the SCH

Saharon Shelah — 2008

Fundamenta Mathematicae

We prove that, e.g., if μ > cf(μ) = ℵ₀ and μ > 2 and every stationary family of countable subsets of μ⁺ reflects in some subset of μ⁺ of cardinality ℵ₁, then the SCH for μ⁺ holds (moreover, for μ⁺, any scale for μ⁺ has a bad stationary set of cofinality ℵ₁). This answers a question of Foreman and Todorčević who get such a conclusion from the simultaneous reflection of four stationary sets.

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