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Around Logical Perfection

John A. Cruz Morales, Andrés Villaveces, Boris Zilber

Theoria > 87 > 4 > 971 - 985

In this article we present a notion of “logical perfection”. We first describe through examples a notion of logical perfection extracted from the contemporary logical concept of categoricity. Categoricity (in power) has become in the past half century a main driver of ideas in model theory, both mathematically (stability theory may be regarded as a way of approximating categoricity) and philosophically...

article

Completeness, Categoricity and Imaginary Numbers: The Debate on Husserl

Víctor Aranda

Bulletin of the Section of Logic > 2020 > 49 > 2

Husserl's two notions of "definiteness" enabled him to clarify the problem of imaginary numbers. The exact meaning of these notions is a topic of much controversy. A "definite" axiom system has been interpreted as a syntactically complete theory, and also as a categorical one. I discuss whether and how far these readings manage to capture Husserl's goal of elucidating the problem...

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The categoricity spectrum of large abstract elementary classes

Sebastien Vasey

Selecta Mathematica > 2019 > 25 > 5 > 1-51

The categoricity spectrum of a class of structures is the collection of cardinals in which the class has a single model up to isomorphism. Assuming that cardinal exponentiation is injective (a weakening of the generalized continuum hypothesis, GCH), we give a complete list of the possible categoricity spectrums of an abstract elementary class with amalgamation and arbitrarily large models. Specifically,...

article

Categoricity Results and Large Model Constructions for Second-Order ZF in Dependent Type Theory

Dominik Kirst, Gert Smolka

Journal of Automated Reasoning > 2019 > 63 > 2 > 415-438

We formalise second-order ZF set theory in the dependent type theory of Coq. Assuming excluded middle, we prove Zermelo’s embedding theorem for models, categoricity in all cardinalities, and the categoricity of extended axiomatisations fixing the number of Grothendieck universes. These results are based on an inductive definition of the cumulative hierarchy eliminating the need for ordinals and set-theoretic...

article

A Note on Carnap’s Result and the Connectives

Tristan Haze

Axiomathes > 2019 > 29 > 3 > 285-288

Carnap’s result about classical proof-theories not ruling out non-normal valuations of propositional logic formulae has seen renewed philosophical interest in recent years. In this note I contribute some considerations which may be helpful in its philosophical assessment. I suggest a vantage point from which to see the way in which classical proof-theories do, at least to a considerable extent, encode...

article

Structures Computable in Polynomial Time. II

P. E. Alaev

Algebra and Logic > 2018 > 56 > 6 > 429-442

We consider a new approach to investigating categoricity of structures computable in polynomial time. The approach is based on studying polynomially computable stable relations. It is shown that this categoricity is equivalent to the usual computable categoricity for computable Boolean algebras with computable set of atoms, and for computable linear orderings with computable set of adjacent pairs...

article

Shelah's eventual categoricity conjecture in universal classes: Part I

Sebastien Vasey

Annals of Pure and Applied Logic > 2017 > 168 > 9 > 1609-1642

We prove: Theorem 0.1Let K be a universal class. If K is categorical in cardinals of arbitrarily high cofinality, then K is categorical on a tail of cardinals.The proof stems from ideas of Adi Jarden and Will Boney, and also relies on a deep result of Shelah. As opposed to previous works, the argument is in ZFC and does not use the assumption of categoricity in a successor cardinal. The argument generalizes...

article

Uporządkowane struktury liczbowe

Antoni Chronowski

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam... > 2013 > 5 > 5-37

In this article we consider the ordered algebraic structures of thesystems of natural numbers, integers, rational and real numbers. We presentthe ordered algebra of natural numbers, the ordered ring of integers, and theordered fields of rational and real numbers. The main problem considered forordered number structures is the categoricity of these systems determined bya suitable isomorphism. First...

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Superstability from categoricity in abstract elementary classes

Will Boney, Rami Grossberg, Monica M. VanDieren, Sebastien Vasey

Annals of Pure and Applied Logic > 2017 > 168 > 7 > 1383-1395

Starting from an abstract elementary class with no maximal models, Shelah and Villaveces have shown (assuming instances of diamond) that categoricity implies a superstability-like property for nonsplitting, a particular notion of independence. We generalize their result as follows: given any abstract notion of independence for Galois (orbital) types over models, we derive that the notion satisfies...

article

Saturation and solvability in abstract elementary classes with amalgamation

Sebastien Vasey

Archive for Mathematical Logic > 2017 > 56 > 5-6 > 671-690

Theorem 0.1Let $$\mathbf {K}$$ K be an abstract elementary class (AEC) with amalgamation and no maximal models. Let $$\lambda > {LS}(\mathbf {K})$$ λ > LS ( K ) . If $$\mathbf {K}$$ K is categorical in $$\lambda $$ λ , then the model of cardinality $$\lambda $$ λ is Galois-saturated.This answers a question asked independently by Baldwin and Shelah...

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Symmetry in abstract elementary classes with amalgamation

Monica M. VanDieren, Sebastien Vasey

Archive for Mathematical Logic > 2017 > 56 > 3-4 > 423-452

This paper is part of a program initiated by Saharon Shelah to extend the model theory of first order logic to the non-elementary setting of abstract elementary classes (AECs). An abstract elementary class is a semantic generalization of the class of models of a complete first order theory with the elementary substructure relation. We examine the symmetry property of splitting (previously isolated...

article

Downward categoricity from a successor inside a good frame

Sebastien Vasey

Annals of Pure and Applied Logic > 2017 > 168 > 3 > 651-692

In the setting of abstract elementary classes (AECs) with amalgamation, Shelah has proven a downward categoricity transfer from categoricity in a successor and Grossberg and VanDieren have established an upward transfer assuming in addition a locality property for Galois types that they called tameness.We further investigate categoricity transfers in tame AECs. We use orthogonality calculus to prove...

article

Shelah’s eventual categoricity conjecture in universal classes: part II

Sebastien Vasey

Selecta Mathematica > 2017 > 23 > 2 > 1469-1506

We prove that a universal class categorical in a high-enough cardinal is categorical on a tail of cardinals. As opposed to other results in the literature, we work in ZFC, do not require the categoricity cardinal to be a successor, do not assume amalgamation, and do not use large cardinals. Moreover we give an explicit bound on the “high-enough” threshold:Theorem 0.1 Let $$\psi $$ ψ be a...

article

Building independence relations in abstract elementary classes

Sebastien Vasey

Annals of Pure and Applied Logic > 2016 > 167 > 11 > 1029-1092

We study general methods to build forking-like notions in the framework of tame abstract elementary classes (AECs) with amalgamation. We show that whenever such classes are categorical in a high-enough cardinal, they admit a good frame: a forking-like notion for types of singleton elements. Theorem 0.1Let K be a (<κ)-tame AEC with amalgamation. If κ=ℶκ>LS(K) and K is categorical in a λ>κ,...

article

Tameness, uniqueness triples and amalgamation

Adi Jarden

Annals of Pure and Applied Logic > 2016 > 167 > 2 > 155-188

We combine two approaches to the study of classification theory of AECs:(1)that of Shelah: studying non-forking frames without assuming the amalgamation property but assuming the existence of uniqueness triples and(2)that of Grossberg and VanDieren [6]: (studying non-splitting) assuming the amalgamation property and tameness.In [7] we derive a good non-forking λ+-frame from a semi-good non-forking...

article

Hilbert's axiomatic method and Carnap's general axiomatics

Michael Stöltzner

Studies in History and Philosophy of Science > 2015 > 53 > Complete > 12-22

This paper compares the axiomatic method of David Hilbert and his school with Rudolf Carnap's general axiomatics that was developed in the late 1920s, and that influenced his understanding of logic of science throughout the 1930s, when his logical pluralism developed. The distinct perspectives become visible most clearly in how Richard Baldus, along the lines of Hilbert, and Carnap and Friedrich Bachmann...

article

On Δ20-categoricity of equivalence relations

Rod Downey, Alexander G. Melnikov, Keng Meng Ng

Annals of Pure and Applied Logic > 2015 > 166 > 9 > 851-880

We investigate which computable equivalence structures are isomorphic relative to the Halting problem.

article

Axiomatizations of arithmetic and the first-order/second-order divide

Catarina Dutilh Novaes

Synthese > 2019 > 196 > 7 > 2583-2597

It is often remarked that first-order Peano Arithmetic is non-categorical but deductively well-behaved, while second-order Peano Arithmetic is categorical but deductively ill-behaved. This suggests that, when it comes to axiomatizations of mathematical theories, expressive power and deductive power may be orthogonal, mutually exclusive desiderata. In this paper, I turn to Hintikka’s (Philos Top 17(2):69–90,...

article

Upward Morley's theorem downward

Gábor Sági, Zalán Gyenis

Mathematical Logic Quarterly > 59 > 4-5 > 303 - 331

By a celebrated theorem of Morley, a theory T is ℵ₁‐categorical if and only if it is κ‐categorical for all uncountable κ. In this paper we are taking the first steps towards extending Morley's categoricity theorem “to the finite”. In more detail, we are presenting conditions, implying that certain finite subsets of certain ℵ₁‐categorical T have at most one n‐element model for each natural number

n

...

article

Characterizing all models in infinite cardinalities

Lauri Keskinen

Annals of Pure and Applied Logic > 2013 > 164 > 3 > 230-250

Fix a cardinal κ. We can ask the question: what kind of a logic L is needed to characterize all models of cardinality κ (in a finite vocabulary) up to isomorphism by their L-theories? In other words: for which logics L it is true that if any models A and B of cardinality κ satisfy the same L-theory then they are isomorphic?It is always possible to characterize models of cardinality κ by their Lκ+,κ+-theories,...

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Around Logical Perfection

Completeness, Categoricity and Imaginary Numbers: The Debate on Husserl

The categoricity spectrum of large abstract elementary classes

Categoricity Results and Large Model Constructions for Second-Order ZF in Dependent Type Theory

A Note on Carnap’s Result and the Connectives

Structures Computable in Polynomial Time. II

Shelah's eventual categoricity conjecture in universal classes: Part I

Uporządkowane struktury liczbowe

Superstability from categoricity in abstract elementary classes

Saturation and solvability in abstract elementary classes with amalgamation

Symmetry in abstract elementary classes with amalgamation

Downward categoricity from a successor inside a good frame

Shelah’s eventual categoricity conjecture in universal classes: part II

Building independence relations in abstract elementary classes

Tameness, uniqueness triples and amalgamation

Hilbert's axiomatic method and Carnap's general axiomatics

On Δ20-categoricity of equivalence relations

Axiomatizations of arithmetic and the first-order/second-order divide

Upward Morley's theorem downward

Characterizing all models in infinite cardinalities

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