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1
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Embedding Cohen algebras using pcf theory

100%
Shelah S.
Fundamenta Mathematicae
|
2000
|
tom 166
|
nr 1-2
83-86
EN
Using a theorem from pcf theory, we show that for any singular cardinal ν, the product of the Cohen forcing notions on κ, κ < ν, adds a generic for the Cohen forcing notion on $ν^+$.
2
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The combinatorics of reasonable ultrafilters

100%
Shelah S.
Fundamenta Mathematicae
|
2006
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tom 192
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nr 1
1-23
EN
We are interested in generalizing part of the theory of ultrafilters on ω to larger cardinals. Here we set the scene for further investigations introducing properties of ultrafilters in strong sense dual to being normal.
3
Content available remote

On full Suslin trees

100%
Shelah S.
Colloquium Mathematicum
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1999
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tom 79
|
nr 1
1-7
4
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Strong covering without squares

100%
Shelah S.
Fundamenta Mathematicae
|
2000
|
tom 166
|
nr 1-2
87-107
EN
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, $κ^+_V = κ^+_W$, and we have both κ-covering and $κ^+$-covering between W and V, then strong κ-covering holds. Next we show that we can drop the assumption of $κ^+$-covering at the expense of assuming some more absoluteness of cardinals and cofinalities between W and V, and that we can drop the assumption that $κ^+_W = κ^+_V$ and weaken the $κ^+$-covering assumption at the expense of assuming some structural facts about W (the existence of certain square sequences).
5
Content available remote

Large continuum, oracles

100%
Shelah S.
Open Mathematics
|
2010
|
tom 8
|
nr 2
213-234
EN
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.
6
Content available remote

σ-Entangled linear orders and narrowness of products of Boolean algebras

100%
Shelah S.
Fundamenta Mathematicae
|
1997
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tom 153
|
nr 3
199-275
EN
We investigate σ-entangled linear orders and narrowness of Boolean algebras. We show existence of σ-entangled linear orders in many cardinals, and we build Boolean algebras with neither large chains nor large pies. We study the behavior of these notions in ultraproducts.
7
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Cellularity of free products of Boolean algebras (or topologies)

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Shelah S.
Fundamenta Mathematicae
|
2000
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tom 166
|
nr 1-2
153-208
EN
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, $θ = (2^{cf(μ)})^+$ and $2^μ = μ^+$ then there are Boolean algebras $\mathbb{B}_1,\mathbb{B}_2$ such that $c(\mathbb{B}_1) = μ, c(\mathbb{B}_2) < θ but c(\mathbb{B}_1*\mathbb{B}_2)=μ^+$. Further we improve this result, deal with the method and the necessity of the assumptions. In particular we prove that if $\mathbb{B}$ is a ccc Boolean algebra and $μ^{ℶ_ω} ≤ λ = cf(λ) ≤ 2^μ$ then $\mathbb{B}$ satisfies the λ-Knaster condition (using the "revised GCH theorem").
8
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The spectrum of characters of ultrafilters on ω

100%
Shelah S.
Colloquium Mathematicum
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2008
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tom 111
|
nr 2
213-220
EN
We show the consistency of the statement: "the set of regular cardinals which are the characters of ultrafilters on ω is not convex". We also deal with the set of π-characters of ultrafilters on ω.
9
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Covering of the null ideal may have countable cofinality

100%
Shelah S.
Fundamenta Mathematicae
|
2000
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tom 166
|
nr 1-2
109-136
EN
We prove that it is consistent that the covering number of the ideal of measure zero sets has countable cofinality.
10
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Zero-one laws for graphs with edge probabilities decaying with distance. Part II

100%
Shelah S.
Fundamenta Mathematicae
|
2005
|
tom 185
|
nr 3
211-245
EN
Let Gₙ be the random graph on [n] = {1,...,n} with the probability of {i,j} being an edge decaying as a power of the distance, specifically the probability being $p_{|i-j|} = 1/|i-j|^{α}$, where the constant α ∈ (0,1) is irrational. We analyze this theory using an appropriate weight function on a pair (A,B) of graphs and using an equivalence relation on B∖A. We then investigate the model theory of this theory, including a "finite compactness". Lastly, as a consequence, we prove that the zero-one law (for first order logic) holds.
11
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Categoricity of theories in $L_{κ*,ω}$, when κ* is a measurable cardinal. Part 2

100%
Shelah S.
Fundamenta Mathematicae
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2001
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tom 170
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nr 1-2
165-196
EN
We continue the work of [2] and prove that for λ successor, a λ-categorical theory T in $L_{κ*,ω}$ is μ-categorical for every μ ≤ λ which is above the $(2^{LS(T)})⁺$-beth cardinal.
12
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Borel sets with large squares

100%
Shelah S.
Fundamenta Mathematicae
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1999
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tom 159
|
nr 1
1-50
EN
 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 $2^{ℵ_0}$-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 $MA + 2^{ℵ_0} > μ$ for transparency, those three conditions ($⊕_μ$, $⊗_μ$ and $⊗'_μ$) are equivalent, and from this we deduce that e.g. $∧_{α < ω_1}[ 2^{ℵ_0}≥ ℵ_α ⇒ ￢ ⊗_{ℵ_α}]$, and also that $min{μ: ⊗_μ}$, if $ < 2^{ℵ_0}$, has cofinality $ℵ_1$.   We also deal with Borel rectangles and related model-theoretic problems.
13
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On a problem of Steve Kalikow

100%
Shelah S.
Fundamenta Mathematicae
|
2000
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tom 166
|
nr 1-2
137-151
EN
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 $ℵ_ω$ but we prove it for smaller cardinals. We indicate a close connection with the free subset property and its variants.
14
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The null ideal restricted to some non-null set may be ℵ₁-saturated

100%
Shelah S.
Fundamenta Mathematicae
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2003
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tom 179
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nr 2
97-129
EN
Our main result is that possibly some non-null set of reals cannot be divided into uncountably many non-null sets. We also deal with a non-null set of real, the graph of any function from which is null, and deal with our iterations somewhat more generally.
15
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Content available

On Monk’s questions

100%
Shelah S.
Fundamenta Mathematicae
|
1996
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tom 151
|
nr 1
1-19
EN
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.
16
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A polarized partition relation and failure of GCH at singular strong limit

100%
Shelah S.
Fundamenta Mathematicae
|
1998
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tom 155
|
nr 2
153-160
EN
The main result is that for λ strong limit singular failing the continuum hypothesis (i.e. $2^{λ} > λ^{+}$), a polarized partition theorem holds.
17
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Zero-one laws for graphs with edge probabilities decaying with distance. Part I

100%
Shelah S.
Fundamenta Mathematicae
|
2002
|
tom 175
|
nr 3
195-239
EN
Let Gₙ be the random graph on [n] = {1,...,n} with the possible edge {i,j} having probability $p_{|i-j|} = 1/|i-j|^α$ for j ≠ i, i+1, i-1 with α ∈ (0,1) irrational. We prove that the zero-one law (for first order logic) holds..
18
Content available remote

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

100%
Shelah S.
Fundamenta Mathematicae
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2000
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tom 166
|
nr 1-2
1-82
EN
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 are erratic, usually kept to a minimum ("see ..." means: see the references there and possibly the paper itself). The base were lectures in Rutgers, Fall '97, and reflect my knowledge then. The other half, [122], concentrating on model theory, will subsequently appear. I thank Andreas Blass and Andrzej Rosłanowski for many helpful comments.
19
Content available remote

Reflection implies the SCH

100%
Shelah S.
Fundamenta Mathematicae
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2008
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tom 198
|
nr 2
95-111
EN
We prove that, e.g., if μ > cf(μ) = ℵ₀ and $μ > 2^{ℵ₀}$ 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.
20
Content available remote

When a first order T has limit models

100%
Shelah S.
Colloquium Mathematicum
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2012
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tom 126
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nr 2
187-204
EN
We sort out to a large extent when a (first order complete theory) T has a superlimit model in a cardinal λ. Also we deal with related notions of being limit.
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