A Hanf number for saturation and omission

John T. Baldwin; Saharon Shelah

Displaying similar documents to “A Hanf number for saturation and omission”

On normalization of proofs in set theory

Lars Hallnäs

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CONTENTSIntroduction..............................................................................................................................................5I. Naive set theory.....................................................................................................................................61. The formal system................................................................................................................................62. Inversion and reduction...

A Kalmár-style completeness proof for the logics of the hierarchy $𝕀^{n} ℙ^{k}$

Víctor Fernández (2023)

Commentationes Mathematicae Universitatis Carolinae

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The logics of the family $𝕀^{n} ℙ^{k}$ := ${I^{n} P^{k}}_{(n, k) \in ω^{2}}$ are formally defined by means of finite matrices, as a simultaneous generalization of the weakly-intuitionistic logic $I^{1}$ and of the paraconsistent logic $P^{1}$ . It is proved that this family can be naturally ordered, and it is shown a sound and complete axiomatics for each logic of the form $I^{n} P^{k}$ . The involved completeness proof showed here is obtained by means of a generalization of the well-known Kalmár’s method, usually applied for many-valued logics.

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 $ϕ_{ℳ}$ if $ϕ_{ℳ}$ 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 $ℵ_{β}$ is characterized by a Scott sentence, then $2^{ℵ_{β + β ₁}}$ is (homogeneously) characterized by a Scott sentence, for all 0 <...

Uncountable cardinals have the same monadic ∀₁¹ positive theory over large sets

Athanassios Tzouvaras (2004)

Fundamenta Mathematicae

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We show that uncountable cardinals are indistinguishable by sentences of the monadic second-order language of order of the form (∀X)ϕ(X) and (∃X)ϕ(X), for ϕ positive in X and containing no set-quantifiers, when the set variables range over large (= cofinal) subsets of the cardinals. This strengthens the result of Doner-Mostowski-Tarski [3] that (κ,∈), (λ,∈) are elementarily equivalent when κ, λ are uncountable. It follows that we can consistently postulate that the structures $(2^{κ}, {[2^{κ}]}^{> κ}, <)$ , $(2^{λ}, {[2^{λ}]}^{> λ}, <)$ are...

Completeness properties of classical theories of finite type and the normal form theorem

Peter Päppinghaus

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CONTENTSIntroduction........................................................................................................................................................................................................................50. Terminology and preliminaries......................................................................................................................................................................................121. The extent of cut elimination by...

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)...........................................

The number of $L_{\infty κ}$ -equivalent nonisomorphic models for κ weakly compact

Saharon Shelah, Pauli Vaisanen (2002)

Fundamenta Mathematicae

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For a cardinal κ and a model M of cardinality κ let No(M) denote the number of nonisomorphic models of cardinality κ which are $L_{\infty, κ}$ -equivalent to M. We prove that for κ a weakly compact cardinal, the question of the possible values of No(M) for models M of cardinality κ is equivalent to the question of the possible numbers of equivalence classes of equivalence relations which are Σ¹₁-definable over $V_{κ}$ . By [SV] it is possible to have a generic extension where the possible numbers of equivalence...

Counting models of set theory

Ali Enayat (2002)

Fundamenta Mathematicae

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Let T denote a completion of ZF. We are interested in the number μ(T) of isomorphism types of countable well-founded models of T. Given any countable order type τ, we are also interested in the number μ(T,τ) of isomorphism types of countable models of T whose ordinals have order type τ. We prove: (1) Suppose ZFC has an uncountable well-founded model and $κ \in ω \cup ℵ ₀, ℵ ₁, 2^{ℵ ₀}$ . There is some completion T of ZF such that μ(T) = κ. (2) If α <ω₁ and μ(T,α) > ℵ₀, then $μ (T, α) = 2^{ℵ ₀}$ . (3) If α < ω₁ and T ⊢ V ≠ OD,...

A topological duality for the $F$ -chains associated with the logic $C_{ω}$

Verónica Quiroga, Víctor Fernández (2017)

Mathematica Bohemica

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In this paper we present a topological duality for a certain subclass of the $F_{ω}$ -structures defined by M. M. Fidel, which conform to a non-standard semantics for the paraconsistent N. C. A. da Costa logic $C_{ω}$ . Actually, the duality introduced here is focused on $F_{ω}$ -structures whose supports are chains. For our purposes, we characterize every $F_{ω}$ -chain by means of a new structure that we will call (DCC) here. This characterization will allow us to prove the dual equivalence between the category...

Constructibility in Ackermann's set theory

C. Alkor

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CONTENTSIntroduction......................... 5Section I. Preliminaries............ 6 § 1. Notation..................... 6 § 2. Ackermann’s set theory and some extensions................. 7 § 3. Absoluteness............................................... 8 § 4. Ordinals................................................... 9 § 5. Reflection principles...................................... 10Section 2. The usual notion of constructibility.............. 11 § 1. General considerations about...

The decidability of some $ℵ_{0}$ -categorical theories

James H. Schmerl (1976)

Colloquium Mathematicae

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A generalization of a formalized theory of fields of sets on non-classical logics

Helena Rasiowa

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Contents Introduction.................................................................................................................................................. 3 § 1. System $𝒮$ of a propositional calculus...................................................................... 4 § 2. System $𝒮 *$ ..................................................................................................................... 5 § 3. $𝒮 *$ -algebras.....................................................................................................................

Existentially closed II₁ factors

Ilijas Farah, Isaac Goldbring, Bradd Hart, David Sherman (2016)

Fundamenta Mathematicae

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We examine the properties of existentially closed ( $^{ω}$ -embeddable) II₁ factors. In particular, we use the fact that every automorphism of an existentially closed ( $^{ω}$ -embeddable) II₁ factor is approximately inner to prove that Th() is not model-complete. We also show that Th() is complete for both finite and infinite forcing and use the latter result to prove that there exist continuum many nonisomorphic existentially closed models of Th().

More reflections on compactness

Lúcia R. Junqueira, Franklin D. Tall (2003)

Fundamenta Mathematicae

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We consider the question of when $X_{M} = X$ , where $X_{M}$ is the elementary submodel topology on X ∩ M, especially in the case when $X_{M}$ is compact.

The $ℒ_{n}^{m}$ -propositional calculus

Carlos Gallardo, Alicia Ziliani (2015)

Mathematica Bohemica

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T. Almada and J. Vaz de Carvalho (2001) stated the problem to investigate if these Łukasiewicz algebras are algebras of some logic system. In this article an affirmative answer is given and the $ℒ_{n}^{m}$ -propositional calculus, denoted by $ℓ_{n}^{m}$ , is introduced in terms of the binary connectives $\to$ (implication), $↠$ (standard implication), $\land$ (conjunction), $\lor$ (disjunction) and the unary ones $f$ (negation) and $D_{i}$ , $1 \leq i \leq n - 1$ (generalized Moisil operators). It is proved that $ℓ_{n}^{m}$ belongs to the class of standard systems...

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

Saharon Shelah (2001)

Fundamenta Mathematicae

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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 $(2^{L S (T)}) ⁺$ -beth cardinal.

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 $⟨ D^{ℳ}, ≺^{ℳ} ⟩$ a well-ordering of type ≥ γ, then ϕ has a model ℳ ’ where $⟨ D^{ℳ^{'}}, ≺^{ℳ^{'}} ⟩$ is non-well-ordered. One of the interesting properties of this number is that the Hanf number of $L_{λ ⁺ ω}$ is exactly $ℶ_{δ (λ)}$ . 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 $a n d δ (λ) < λ ⁺ ⁺ . W e i m p r o v e t h i s r e s u l t b y p r o v i n g t h e f o l l o w i n g : S u p p o s e ℵ ₀ < λ < θ \leq κ a r e c a r d i n a l n u m b e r s s u c h t h a t$ ∙ $λ^{< λ} = λ$ ; ∙ cf(θ) ≥ λ⁺ and $μ^{λ} < θ$ whenever μ < θ; ∙ $κ^{λ} = κ$ . Then there...

Generic extensions of models of ZFC

Lev Bukovský (2017)

Commentationes Mathematicae Universitatis Carolinae

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The paper contains a self-contained alternative proof of my Theorem in Characterization of generic extensions of models of set theory, Fund. Math. 83 (1973), 35–46, saying that for models $M \subseteq N$ of ZFC with same ordinals, the condition $A p r_{M, N} (κ)$ implies that $N$ is a $κ$ -C.C. generic extension of $M$ .

A note on star Lindelöf, first countable and normal spaces

Wei-Feng Xuan (2017)

Mathematica Bohemica

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A topological space $X$ is said to be star Lindelöf if for any open cover $𝒰$ of $X$ there is a Lindelöf subspace $A \subset X$ such that $St (A, 𝒰) = X$ . The “extent” $e (X)$ of $X$ is the supremum of the cardinalities of closed discrete subsets of $X$ . We prove that under $V = L$ every star Lindelöf, first countable and normal space must have countable extent. We also obtain an example under $MA + \neg CH$ , which shows that a star Lindelöf, first countable and normal space may not have countable extent.

Higher order spreading models

S. A. Argyros, V. Kanellopoulos, K. Tyros (2013)

Fundamenta Mathematicae

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We introduce higher order spreading models associated to a Banach space X. Their definition is based on ℱ-sequences ${(x_{s})}_{s \in ℱ}$ with ℱ a regular thin family and on plegma families. We show that the higher order spreading models of a Banach space X form an increasing transfinite hierarchy ${(ℳ_{ξ} (X))}_{ξ < ω ₁}$ . Each $ℳ_{ξ} (X)$ contains all spreading models generated by ℱ-sequences ${(x_{s})}_{s \in ℱ}$ with order of ℱ equal to ξ. We also study the fundamental properties of this hierarchy.

General operators binding variables in the interpreted modal calculus $\mathcal{MC}^{\nu}$

Aldo Bressan, Alberto Zanardo (1981)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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Si considera il calcolo modale interpretato $\mathcal{MC}^{\nu}$ , che è basato su un sistema di tipi con infiniti livelli, contiene descrizioni, ed è dotato di una semantica di tipo generale - v. [2], o [3], o [4], o [5]. In modo semplice e naturale si introducono in $\mathcal{MC}^{\nu}$ operatori vincolanti variabili, di tipo generale. Per teorie basate sul calcolo logico risultante $\mathcal{MC}^{\nu}$ vale un teorema di completezza, che si dimostra in modo immediato sulla base dell'estensione del teorema parziale di completezza stabilito...