Wyniki wyszukiwania - Biblioteka Nauki

1

The spectrum of characters of ultrafilters on ω

100%

Shelah S.

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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.

2

Categoricity of theories in $L_{κ*,ω}$, when κ* is a measurable cardinal. Part 2

100%

Shelah S.

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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.

3

Large continuum, oracles

100%

Shelah S.

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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.

4

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

100%

Shelah S.

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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.

5

Zero-one laws for graphs with edge probabilities decaying with distance. Part I

100%

Shelah S.

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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..

6

The combinatorics of reasonable ultrafilters

100%

Shelah S.

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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.

7

On strong measure zero subsets of $^{κ}2$

63%

Halko A. , Shelah S.

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nr 3

219-229

EN

We study the generalized Cantor space $^{κ}2$ and the generalized Baire space $^{κ}κ$ as analogues of the classical Cantor and Baire spaces. We equip $^{κ}κ$ with the topology where a basic neighborhood of a point η is the set {ν: (∀j < i)(ν(j) = η(j))}, where i < κ. We define the concept of a strong measure zero set of $^{κ}2$. We prove for successor $κ = κ^{<κ}$ that the ideal of strong measure zero sets of $^{κ}2$ is $𝔟_{κ}$-additive, where ${𝔟}_{κ}$ is the size of the smallest unbounded family in $^{κ}κ$, and that the generalized Borel conjecture for $^{κ}2$ is false. Moreover, for regular uncountable κ, the family of subsets of $^{κ}2$ with the property of Baire is not closed under the Suslin operation. These results answer problems posed in [2].

8

Borel completeness of some ℵ₀-stable theories

63%

Laskowski M. , Shelah S.

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nr 1

1-46

EN

We study ℵ₀-stable theories, and prove that if T either has eni-DOP or is eni-deep, then its class of countable models is Borel complete. We introduce the notion of λ-Borel completeness and prove that such theories are λ-Borel complete. Using this, we conclude that an ℵ₀-stable theory satisfies $I_{∞,ℵ₀}(T,λ) = 2^{λ}$ for all cardinals λ if and only if T either has eni-DOP or is eni-deep.

9

P-NDOP and P-decompositions of $ℵ_{ϵ}$-saturated models of superstable theories

63%

Shelah S. , Laskowski M.

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nr 1

47-81

EN

Given a complete, superstable theory, we distinguish a class P of regular types, typically closed under automorphisms of ℭ and non-orthogonality. We define the notion of P-NDOP, which is a weakening of NDOP. For superstable theories with P-NDOP, we prove the existence of P-decompositions and derive an analog of the first author's result in Israel J. Math. 140 (2004). In this context, we also find a sufficient condition on P-decompositions that implies non-isomorphic models. For this, we investigate natural structures on the types in P ∩ S(M) modulo non-orthogonality.

10

Possible cardinalities of maximal abelian subgroups of quotients of permutation groups of the integers

63%

Shelah S. , Steprāns J.

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nr 3

197-235

EN

If G is a group then the abelian subgroup spectrum of G is defined to be the set of all κ such that there is a maximal abelian subgroup of G of size κ. The cardinal invariant A(G) is defined to be the least uncountable cardinal in the abelian subgroup spectrum of G. The value of A(G) is examined for various groups G which are quotients of certain permutation groups on the integers. An important special case, to which much of the paper is devoted, is the quotient of the full symmetric group by the normal subgroup of permutations with finite support. It is shown that, if we use G to denote this group, then A(G) ≤ 𝔞. Moreover, it is consistent that A(G) ≠ 𝔞. Related results are obtained for other quotients using Borel ideals.

11

On partial orderings having precalibre-ℵ₁ and fragments of Martin's axiom

63%

Bagaria J. , Shelah S.

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nr 2

181-197

EN

We define a countable antichain condition (ccc) property for partial orderings, weaker than precalibre-ℵ₁, and show that Martin's axiom restricted to the class of partial orderings that have the property does not imply Martin's axiom for σ-linked partial orderings. This yields a new solution to an old question of the first author about the relative strength of Martin's axiom for σ-centered partial orderings together with the assertion that every Aronszajn tree is special. We also answer a question of J. Steprāns and S. Watson (1988) by showing that, by a forcing that preserves cardinals, one can destroy the precalibre-ℵ₁ property of a partial ordering while preserving its ccc-ness.

12

A partial order where all monotone maps are definable

63%

Goldstern M. , Shelah S.

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tom 152

|

nr 3

255-265

EN

It is consistent that there is a partial order (P,≤) of size $ℵ_1$ such that every monotone function f:P → P is first order definable in (P,≤).

13

The distributivity numbers of finite products of P(ω)/fin

63%

Shelah S. , Spinas O.

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nr 1

81-93

EN

Generalizing [ShSp], for every n < ω we construct a ZFC-model where ℌ(n), the distributivity number of r.o.$(P(ω)/fin)^n$, is greater than ℌ(n+1). This answers an old problem of Balcar, Pelant and Simon (see [BaPeSi]). We also show that both Laver and Miller forcings collapse the continuum to ℌ(n) for every n < ω, hence by the first result, consistently they collapse it below ℌ(n).

14

Planting Kurepa trees and killing Jech-Кunen trees in a model by using one inaccessible cardinal

63%

Shelah S. , Jin R.

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nr 3

287-296

EN

By an $ω_1$- tree we mean a tree of power $ω_1$ and height $ω_1$. Under CH and $2^{ω_{1}} > ω_2$ we call an $ω_1$-tree a Jech-Kunen tree if it has κ-many branches for some κ strictly between $ω_1$ and $2^{ω_{1}}$. In this paper we prove that, assuming the existence of one inaccessible cardinal, (1) it is consistent with CH plus $2^{ω_{1}} > ω_2$ that there exist Kurepa trees and there are no Jech-Kunen trees, which answers a question of [Ji2], (2) it is consistent with CH plus $2^{ω_{1}} = ω_4$ that there only exist Kurepa trees with $ω_{3}$-many branches, which answers another question of [Ji2].

15

Generalized E-algebras via λ-calculus I

63%

Göbel R. , Shelah S.

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nr 2

155-181

EN

An R-algebra A is called an E(R)-algebra if the canonical homomorphism from A to the endomorphism algebra $End_{R}A$ of the R-module $_{R}A$, taking any a ∈ A to the right multiplication $a_{r} ∈ End_{R}A$ by a, is an isomorphism of algebras. In this case $_{R}A$ is called an E(R)-module. There is a proper class of examples constructed in [4]. E(R)-algebras arise naturally in various topics of algebra. So it is not surprising that they were investigated thoroughly in the last decades; see [3, 5, 7, 8, 10, 13, 14, 15, 18, 19]. Despite some efforts ([14, 5]) it remained an open question whether proper generalized E(R)-algebras exist. These are R-algebras A isomorphic to $End_{R}A$ but not under the above canonical isomorphism, so not E(R)-algebras. This question was raised about 30 years ago (for R = ℤ) by Schultz [21] (see also Vinsonhaler [24]). It originates from Problem 45 in Fuchs [9], that asks for a characterization of the rings A for which $A ≅ End_{ℤ}A$ (as rings). We answer Schultz's question, thus contributing a large class of rings for Fuchs' Problem 45 which are not E-rings. Let R be a commutative ring with an element p ∈ R such that the additive group R⁺ is p-torsion-free and p-reduced (equivalently p is not a zero-divisor and $⋂_{n∈ω} pⁿR = 0$). As explained in the introduction we assume that either $|R| < 2^{ℵ₀}$ or R⁺ is free (see Definition 1.1). The main tool is an interesting connection between λ-calculus (used in theoretical computer science) and algebra. It seems reasonable to divide the work into two parts; in this paper we work in V = L (Gödel's universe) where stronger combinatorial methods make the final arguments more transparent. The proof based entirely on ordinary set theory (the axioms of ZFC) will appear in a subsequent paper [12]. However the general strategy will be the same, but the combinatorial arguments will utilize a prediction principle that holds under ZFC.

16

Partial choice functions for families of finite sets

63%

Hall E. , Shelah S.

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nr 3

207-216

EN

Let m ≥ 2 be an integer. We show that ZF + "Every countable set of m-element sets has an infinite partial choice function" is not strong enough to prove that every countable set of m-element sets has a choice function, answering an open question from . (Actually a slightly stronger result is obtained.) The independence result in the case where m = p is prime is obtained by way of a permutation (Fraenkel-Mostowski) model of ZFA, in which the set of atoms (urelements) has the structure of a vector space over the finite field $𝔽_{p}$. The use of atoms is then eliminated by citing an embedding theorem of Pincus. In the case where m is not prime, suitable permutation models are built from the models used in the prime cases.

17

Potential isomorphism and semi-proper trees

51%

Hellsten A. , Hyttinen T. , Shelah S.

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tom 175

|

nr 2

127-142

EN

We study a notion of potential isomorphism, where two structures are said to be potentially isomorphic if they are isomorphic in some generic extension that preserves stationary sets and does not add new sets of cardinality less than the cardinality of the models. We introduce the notion of weakly semi-proper trees, and note that there is a strong connection between the existence of potentially isomorphic models for a given complete theory and the existence of weakly semi-proper trees. We show that the existence of weakly semi-proper trees is consistent relative to ZFC by proving the existence of weakly semi-proper trees under certain cardinal arithmetic assumptions. We also prove the consistency of the non-existence of weakly semi-proper trees assuming the consistency of some large cardinals.

18

More on the Ehrenfeucht-Fraisse game of length ω₁

51%

Hyttinen T. , Shelah S. , Vaananen J.

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tom 175

|

nr 1

79-96

EN

By results of [9] there are models 𝔄 and 𝔅 for which the Ehrenfeucht-Fraïssé game of length ω₁, $EFG_{ω₁}(𝔄,𝔅)$, is non-determined, but it is consistent relative to the consistency of a measurable cardinal that no such models have cardinality ≤ ℵ₂. We now improve the work of [9] in two ways. Firstly, we prove that the consistency strength of the statement "CH and $EFG_{ω₁}(𝔄,𝔅)$ is determined for all models 𝔄 and 𝔅 of cardinality ℵ₂" is that of a weakly compact cardinal. On the other hand, we show that if $2^{ℵ₀} < 2^{ℵ₃}$, T is a countable complete first order theory, and one of (i) T is unstable, (ii) T is superstable with DOP or OTOP, (iii) T is stable and unsuperstable and $2^{ℵ₀} ≤ ℵ₃$, holds, then there are 𝓐,ℬ ⊨ T of power ℵ₃ such that $EFG_{ω₁}(𝓐,ℬ)$ is non-determined.

19

The linear refinement number and selection theory

51%

Machura M. , Shelah S. , Tsaban B.

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nr 1

15-40

EN

The linear refinement number 𝔩𝔯 is the minimal cardinality of a centered family in $[ω]^{ω}$ such that no linearly ordered set in $([ω]^{ω},⊆ *)$ refines this family. The linear excluded middle number 𝔩𝔵 is a variation of 𝔩𝔯. We show that these numbers estimate the critical cardinalities of a number of selective covering properties. We compare these numbers to the classical combinatorial cardinal characteristics of the continuum. We prove that 𝔩𝔯 = 𝔩𝔵 = 𝔡 in all models where the continuum is at most ℵ₂, and that the cofinality of 𝔩𝔯 is uncountable. Using the method of forcing, we show that 𝔩𝔯 and 𝔩𝔵 are not provably equal to 𝔡, and rule out several potential bounds on these numbers. Our results solve a number of open problems.

20

Examples for Souslin forcing

51%

Judah H. , Rosłanowski A. , Shelah S.

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tom 144

|

nr 1

23-42

EN

We give several examples of Souslin forcing notions. For instance, we show that there exists a proper analytical forcing notion without ccc and with no perfect set of incompatible elements, we give an example of a Souslin ccc partial order without the Knaster property, and an example of a totally nonhomogeneous Souslin forcing notion.