Displaying similar documents to “Generic extensions of models of ZFC”

Propositional extensions of L ω 1 ω

Richard Gostanian, Karel Hrbacek

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CONTENTS0. Preliminaries....................................................................... 71. Adding propositional connectives to L ω 1 ω ............... 82. The propositional part of L ω 1 ω (S)............................. 103. The operation S and the Boolean algebra B S ............... 114. General model-theoretic properties of L ω 1 ω (S)...... 175. Hanf number computations...................................................... 226. Negative results for L ω 1 ω (S)...........................................

Coherent ultrafilters and nonhomogeneity

Jan Starý (2015)

Commentationes Mathematicae Universitatis Carolinae

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We introduce the notion of a coherent P -ultrafilter on a complete ccc Boolean algebra, strengthening the notion of a P -point on ω , and show that these ultrafilters exist generically under 𝔠 = 𝔡 . This improves the known existence result of Ketonen [On the existence of P -points in the Stone-Čech compactification of integers, Fund. Math. 92 (1976), 91–94]. Similarly, the existence theorem of Canjar [On the generic existence of special ultrafilters, Proc. Amer. Math. Soc. 110 (1990), no. 1,...

Preservation of properties of a map by forcing

Akira Iwasa (2022)

Commentationes Mathematicae Universitatis Carolinae

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Let f : X Y be a continuous map such as an open map, a closed map or a quotient map. We study under what circumstances f remains an open, closed or quotient map in forcing extensions.

Cardinal sequences of length < ω₂ under GCH

István Juhász, Lajos Soukup, William Weiss (2006)

Fundamenta Mathematicae

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Let (α) denote the class of all cardinal sequences of length α associated with compact scattered spaces (or equivalently, superatomic Boolean algebras). Also put λ ( α ) = s ( α ) : s ( 0 ) = λ = m i n [ s ( β ) : β < α ] . We show that f ∈ (α) iff for some natural number n there are infinite cardinals λ i > λ > . . . > λ n - 1 and ordinals α , . . . , α n - 1 such that α = α + + α n - 1 and f = f f . . . f n - 1 where each f i λ i ( α i ) . Under GCH we prove that if α < ω₂ then (i) ω ( α ) = s α ω , ω : s ( 0 ) = ω ; (ii) if λ > cf(λ) = ω, λ ( α ) = s α λ , λ : s ( 0 ) = λ , s - 1 λ i s ω - c l o s e d i n α ; (iii) if cf(λ) = ω₁, λ ( α ) = s α λ , λ : s ( 0 ) = λ , s - 1 λ i s ω - c l o s e d a n d s u c c e s s o r - c l o s e d i n α ; (iv) if cf(λ) > ω₁, λ ( α ) = α λ . This yields a complete characterization of the classes (α) for all...

Characterizing the powerset by a complete (Scott) sentence

Ioannis Souldatos (2013)

Fundamenta Mathematicae

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This paper is part II of a study on cardinals that are characterizable by a Scott sentence, continuing previous work of the author. A cardinal κ is characterized by a Scott sentence ϕ if ϕ has a model of size κ, but no model of size κ⁺. The main question in this paper is the following: Are the characterizable cardinals closed under the powerset operation? We prove that if β is characterized by a Scott sentence, then 2 β + β is (homogeneously) characterized by a Scott sentence, for all 0 <...

On preimages of ultrafilters in ZF

Horst Herrlich, Paul Howard, Kyriakos Keremedis (2016)

Commentationes Mathematicae Universitatis Carolinae

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We show that given infinite sets X , Y and a function f : X Y which is onto and n -to-one for some n , the preimage of any ultrafilter of Y under f extends to an ultrafilter. We prove that the latter result is, in some sense, the best possible by constructing a permutation model with a set of atoms A and a finite-to-one onto function f : A ω such that for each free ultrafilter of ω its preimage under f does not extend to an ultrafilter. In addition, we show that in there exists an ultrafilter compact...

On sentences provable in impredicative extensions of theories

Zygmunt Ratajczyk

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CONTENTS0. Introduction.......................................................................... 51. Preliminaries............................................................................... 72. Basic facts to be used in the sequel....................................... 113. Predicates OD(.,.) and CL(.,.).................................................... 174. Predicate Sels............................................................................. 185. Strong n 1 -collection...........................................................

Balcar's theorem on supports

Lev Bukovský (2018)

Commentationes Mathematicae Universitatis Carolinae

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In A theorem on supports in the theory of semisets [Comment. Math. Univ. Carolinae 14 (1973), no. 1, 1–6] B. Balcar showed that if σ D M is a support, M being an inner model of ZFC, and 𝒫 ( D σ ) M = r ` ` σ with r M , then r determines a preorder " " of D such that σ becomes a filter on ( D , ) generic over M . We show that if the relation r is replaced by a function 𝒫 ( D σ ) M = f - 1 ( σ ) , then there exists an equivalence relation " " on D and a partial order on D / such that D / is a complete Boolean algebra, σ / is a generic filter and [ f ( u ) ] = - ( u / ) for...

Combinatorics of dense subsets of the rationals

B. Balcar, F. Hernández-Hernández, M. Hrušák (2004)

Fundamenta Mathematicae

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We study combinatorial properties of the partial order (Dense(ℚ),⊆). To do that we introduce cardinal invariants , , , , , describing properties of Dense(ℚ). These invariants satisfy ≤ ℚ ≤ ℚ ≤ ℚ ≤ ℚ ≤ ℚ . W e c o m p a r e t h e m w i t h t h e i r a n a l o g u e s i n t h e w e l l s t u d i e d B o o l e a n a l g e b r a ( ω ) / f i n . W e s h o w t h a t ℚ = p , ℚ = t a n d ℚ = i , w h e r e a s ℚ > h a n d ℚ > r a r e b o t h s h o w n t o b e r e l a t i v e l y c o n s i s t e n t w i t h Z F C . W e a l s o i n v e s t i g a t e c o m b i n a t o r i c s o f t h e i d e a l n w d o f n o w h e r e d e n s e s u b s e t s o f , . I n p a r t i c u l a r , w e s h o w t h a t non(M)=min||: ⊆ Dense(R) ∧ (∀I ∈ nwd(R))(∃D ∈ )(I ∩ D = ∅) and cof(M) = min||: ⊆ Dense(ℚ) ∧ (∀I ∈ nwd)(∃D ∈ )(I ∩ = ∅). We use these facts to show that cof(M) ≤ i, which improves a result of S. Shelah.

On ordinals accessible by infinitary languages

Saharon Shelah, Pauli Väisänen, Jouko Väänänen (2005)

Fundamenta Mathematicae

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Let λ be an infinite cardinal number. The ordinal number δ(λ) is the least ordinal γ such that if ϕ is any sentence of L λ ω , with a unary predicate D and a binary predicate ≺, and ϕ has a model ℳ with D , a well-ordering of type ≥ γ, then ϕ has a model ℳ ’ where D ' , ' is non-well-ordered. One of the interesting properties of this number is that the Hanf number of L λ ω is exactly δ ( λ ) . It was proved in [BK71] that if ℵ₀ < λ < κ a r e r e g u l a r c a r d i n a l n u m b e r s , t h e n t h e r e i s a f o r c i n g e x t e n s i o n , p r e s e r v i n g c o f i n a l i t i e s , s u c h t h a t i n t h e e x t e n s i o n 2λ = κ a n d δ ( λ ) < λ . W e i m p r o v e t h i s r e s u l t b y p r o v i n g t h e f o l l o w i n g : S u p p o s e < λ < θ κ a r e c a r d i n a l n u m b e r s s u c h t h a t λ < λ = λ ; ∙ cf(θ) ≥ λ⁺ and μ λ < θ whenever μ < θ; ∙ κ λ = κ . Then there...

On the solvability of systems of linear equations over the ring of integers

Horst Herrlich, Eleftherios Tachtsis (2017)

Commentationes Mathematicae Universitatis Carolinae

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We investigate the question whether a system ( E i ) i I of homogeneous linear equations over is non-trivially solvable in provided that each subsystem ( E j ) j J with | J | c is non-trivially solvable in where c is a fixed cardinal number such that c < | I | . Among other results, we establish the following. (a) The answer is ‘No’ in the finite case (i.e., I being finite). (b) The answer is ‘No’ in the denumerable case (i.e., | I | = 0 and c a natural number). (c) The answer in case that I is uncountable and c 0 is ‘No...

Logarithmically improved blow-up criterion for smooth solutions to the Leray- α -magnetohydrodynamic equations

Ines Ben Omrane, Sadek Gala, Jae-Myoung Kim, Maria Alessandra Ragusa (2019)

Archivum Mathematicum

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In this paper, the Cauchy problem for the 3 D Leray- α -MHD model is investigated. We obtain the logarithmically improved blow-up criterion of smooth solutions for the Leray- α -MHD model in terms of the magnetic field B only in the framework of homogeneous Besov space with negative index.

On the compositum of all degree d extensions of a number field

Itamar Gal, Robert Grizzard (2014)

Journal de Théorie des Nombres de Bordeaux

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We study the compositum k [ d ] of all degree d extensions of a number field k in a fixed algebraic closure. We show k [ d ] contains all subextensions of degree less than d if and only if d 4 . We prove that for d &gt; 2 there is no bound c = c ( d ) on the degree of elements required to generate finite subextensions of k [ d ] / k . Restricting to Galois subextensions, we prove such a bound does not exist under certain conditions on divisors of d , but that one can take c = d when d is prime. This question was inspired by work of...

Estimating composite functions by model selection

Yannick Baraud, Lucien Birgé (2014)

Annales de l'I.H.P. Probabilités et statistiques

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We consider the problem of estimating a function s on [ - 1 , 1 ] k for large values of k by looking for some best approximation of s by composite functions of the form g u . Our solution is based on model selection and leads to a very general approach to solve this problem with respect to many different types of functions g , u and statistical frameworks. In particular, we handle the problems of approximating s by additive functions, single and multiple index models, artificial neural networks, mixtures...

Resolvability in c.c.c. generic extensions

Lajos Soukup, Adrienne Stanley (2017)

Commentationes Mathematicae Universitatis Carolinae

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Every crowded space X is ω -resolvable in the c.c.c. generic extension V Fn ( | X | , 2 ) of the ground model. We investigate what we can say about λ -resolvability in c.c.c. generic extensions for λ > ω . A topological space is monotonically ω 1 -resolvable if there is a function f : X ω 1 such that { x X : f ( x ) α } d e n s e X for each α < ω 1 . We show that given a T 1 space X the following statements are equivalent: (1) X is ω 1 -resolvable in some c.c.c. generic extension; (2) X is monotonically ω 1 -resolvable; (3) X is ω 1 -resolvable in the Cohen-generic...