scholarly journals Minimal Theory of Bigravity: construction and cosmology

2021 ◽  
Vol 2021 (04) ◽  
pp. 015
Author(s):  
Antonio De Felice ◽  
François Larrouturou ◽  
Shinji Mukohyama ◽  
Michele Oliosi
Keyword(s):  
Minimal Theory

Omitting types in -minimal theories

1986 ◽  
Vol 51 (1) ◽  
pp. 63-74 ◽  
Author(s):  
David Marker
Keyword(s):  
Finite Union ◽  
Structure Theory ◽  
Real Closed Fields ◽  
Minimal Theory ◽  
Order Language ◽  
Closed Fields ◽  
Hanf Number ◽  
Definable Sets ◽  
Omitting Types ◽  
Binary Relation Symbol

Let L be a first order language containing a binary relation symbol <.Definition. Suppose ℳ is an L-structure and < is a total ordering of the domain of ℳ. ℳ is ordered minimal (-minimal) if and only if any parametrically definable X ⊆ ℳ can be represented as a finite union of points and intervals with endpoints in ℳ.In any ordered structure every finite union of points and intervals is definable. Thus the -minimal structures are the ones with no unnecessary definable sets. If T is a complete L-theory we say that T is strongly (-minimal if and only if every model of T is -minimal.The theory of real closed fields is the canonical example of a strongly -minimal theory. Strongly -minimal theories were introduced (in a less general guise which we discuss in §6) by van den Dries in [1]. Extending van den Dries' work, Pillay and Steinhorn (see [3], [4] and [2]) developed an extensive structure theory for definable sets in strongly -minimal theories, generalizing the results for real closed fields. They also established several striking analogies between strongly -minimal theories and ω-stable theories (most notably the existence and uniqueness of prime models). In this paper we will examine the construction of models of strongly -minimal theories emphasizing the problems involved in realizing and omitting types. Among other things we will prove that the Hanf number for omitting types for a strongly -minimal theory T is at most (2∣T∣)+, and characterize the strongly -minimal theories with models order isomorphic to (R, <).

Definability of types, and pairs of O-minimal structures

1994 ◽  
Vol 59 (4) ◽  
pp. 1400-1409 ◽  
Author(s):  
Anand Pillay
Keyword(s):  
Elementary Proof ◽  
Elementary Extension ◽  
Complete Model ◽  
Unary Predicate ◽  
Minimal Theory ◽  
Simple Set

AbstractLet T be a complete O-minimal theory in a language L. We first give an elementary proof of the result (due to Marker and Steinhorn) that all types over Dedekind complete models of T are definable. Let L* be L together with a unary predicate P. Let T* be the L*-theory of all pairs (N, M), where M is a Dedekind complete model of T and N is an ⅼMⅼ+-saturated elementary extension of N (and M is the interpretation of P). Using the definability of types result, we show that T* is complete and we give a simple set of axioms for T*. We also show that for every L*-formula ϕ(x) there is an L-formula ψ(x) such that T* ⊢ (∀x)(P(x) → (ϕ(x) ↔ ψ(x)). This yields the following result:Let M be a Dedekind complete model of T. Let ϕ(x, y) be an L-formula where l(y) – k. Let X = {X ⊂ Mk: for some a in an elementary extension N of M, X = ϕ(a, y)N ∩ Mk}. Then there is a formula ψ(y, z) of L such that X = {ψ(y, b)M: b in M}.

scholarly journals Phenomenology in minimal theory of massive gravity

2016 ◽  
Vol 2016 (04) ◽  
pp. 028-028 ◽  
Author(s):  
Antonio De Felice ◽  
Shinji Mukohyama
Keyword(s):  
Massive Gravity ◽  
Minimal Theory

scholarly journals Some two-cardinal results for O-minimal theories

1998 ◽  
Vol 63 (2) ◽  
pp. 543-548 ◽  
Author(s):  
Timothy Bays
Keyword(s):  
Minimal Theory ◽  
Chang’S Conjecture ◽  
Chang's Conjecture

AbstractWe examine two-cardinal problems for the class of O-minimal theories. We prove that an O-minimal theory which admits some (κ, λ) must admit every (κ′, λ′). We also prove that every “reasonable” variant of Chang's Conjecture is true for O-minimal structures. Finally, we generalize these results from the two-cardinal case to the δ-cardinal case for arbitrary ordinals δ.

Semi-minimal theories and categoricity

1975 ◽  
Vol 40 (3) ◽  
pp. 419-438 ◽  
Author(s):  
Daniel Andler
Keyword(s):  
Simple Proof ◽  
Minimal Theory ◽  
Two Stages ◽  
Countable Theory ◽  
Countable Models

The study of countable theories categorical in some uncountable power was initiated by Łoś and Vaught and developed in two stages. First, Morley proved (1962) that a countable theory categorical in some uncountable power is categorical in every uncountable power, a conjecture of Łoś. Second, Baldwin and Lachlan confirmed (1969) Vaught's conjecture that a countable theory categorical in some uncountable power has either one or countably many isomorphism types of countable models. That result was obtained by pursuing a line of research developed by Marsh (1966). For certain well-behaved theories, which he called strongly minimal, Marsh's method yielded a simple proof of Łoś's conjecture and settled Vaught's conjecture.In recent years efforts have been made to extend these results to uncountable theories. The generalized Łoś conjecture states that a theory T categorical in some power greater than ∣T∣ is categorical in every such power. It was settled by Shelah (1970). Shelah then raised the question of the models in power ∣T∣ = ℵα of a theory T categorical in ∣T∣+, conjecturing in [S3] that there are exactly ∣α∣ + ℵ0 such models, up to isomorphism. This conjecture provided the initial motivation for the present work. We define and study semi-minimal theories analogous in some ways to Marsh's strongly minimal (countable) theories. We describe the models of a semi-minimal theory T which contain an infinite indiscernible set. Besides throwing some light on Shelah's conjecture, our method gives simple proofs of the Łoś conjecture and of the Morley conjecture on categoricity in ∣T∣, in the case of a semi-minimal theory T. Other results as well as some examples are provided.

Can the APT be a theory of Multiple Grammars and a minimal theory of parsing?

2004 ◽  
Vol 7 (1) ◽  
pp. 40-41
Author(s):  
TOM ROEPER
Keyword(s):  
The Other ◽  
L2 Acquisition ◽  
Final State ◽  
Minimal Theory ◽  
Processing Theory ◽  
Computational Work ◽  
Lexical Items ◽  
L1 Acquisition

This essay by Truscott and Sharwood-Smith is a valiant attempt with a laudable goal. It seeks to show how different perspectives and disciplines can capture what is happening in acquisition and notably in L2 acquisition. Nonetheless, I think that the results are much closer to an elaborated grammatical theory than an elaborated processing theory (thus it seems less important to me than to the authors to choose one label over the other). Their essential, sensible idea is that where different grammars provide independent analyses of constructions, both are computed (even where the lexical items belong to only one grammar). Therefore conflict is experienced which produces computational demands. However this is pure grammar – just a more sophisticated situation where, as I like to argue, the L2 person has Multiple Grammars (Roeper, 1999), creating more computational work as well. In this respect, the situation is no different from L1 acquisition, where the course of acquisition involves generation and maintenance of multiple grammars, some of which are shed and some retained in the Final State.

scholarly journals DMP in strongly minimal sets

2007 ◽  
Vol 72 (3) ◽  
pp. 1019-1030 ◽  
Author(s):  
Assaf Hasson ◽  
Ehud Hrushovski
Keyword(s):  
Finite Cover ◽  
Minimal Set ◽  
Minimal Sets ◽  
Minimal Theory

AbstractWe construct a strongly minimal set which is not a finite cover of one with DMP. We also show that for a strongly minimal theory T, generic automorphisms exist iff T has DMP, thus proving a conjecture of Kikyo and Pillay.

Disquotationalism, Minimalism, and the Finite Minimal Theory

2004 ◽  
Vol 34 (1) ◽  
pp. 61-86 ◽  
Author(s):  
Jay Newhard
Keyword(s):  
Prima Facie ◽  
Truth Predicate ◽  
Minimal Theory ◽  
Paul Horwich ◽  
Minimalist Theory

Recently, Paul Horwich has developed the minimalist theory of truth, according to which the truth predicate does not express a Substantive property, though it may be used as a grammatical expedient. Minimalism shares these Claims with Quine's disquotationalism; it differs from disquotationalism primarily in holding that truth-bearers are propositions, rather than sentences. Despite potential ontological worries, allowing that propositions bear truth gives Horwich a prima facie response to several important objections to disquotationalism. In section I of this paper, disquotationalism is given a careful exegesis, in which seven known objections are traced to the disquotational Schema, and two new objections are raised. A version of disquotationalism which avoids two of the seven known objections is recommended. In section II, an examination of minimalism shows that it faces eight of the nine objections facing disquotationalism, plus a new objection.

On the Locality and Motivation of Move and Agree: An Even More Minimal Theory

2007 ◽  
Vol 38 (4) ◽  
pp. 589-644 ◽  
Author(s):  
Željko Bošković
Keyword(s):  
Driving Force ◽  
Current Approach ◽  
Activation Condition ◽  
Early Approach ◽  
Minimal Theory ◽  
Cyclic Movement ◽  
Feature Checking

The article proposes a new theory of successive-cyclic movement that reconciles the early and the current minimalist approaches to it. As in the early approach, there is no feature checking in intermediate positions of successive-cyclic movement. However, as in the current approach and unlike in early minimalism, successive-cyclic movement starts before the final target of movement enters the structure, and Form Chain is eliminated. The locality of Move and the locality of Agree are shown to be radically different, Agree being free from several mechanisms that constrain Move, namely, phases and the Activation Condition. However, there is no need to take phases to define locality domains of syntax or to posit the Activation Condition as an independent principle. They still hold empirically for Move as theorems. The Generalized EPP (the “I need a Spec” property of attracting heads) and the Inverse Case Filter are also dispensable. The traditional Case Filter, stated as a checking requirement, is the sole driving force of A-movement. More generally, Move is always driven by a formal inadequacy (an uninterpretable feature) of the moving element, while Agree is target driven. The system resolves a lookahead problem that arises under the EPP-driven movement approach, where the EPP diacritic indicating that X moves is placed on Y, not X, although X often needs to start moving before Y enters the structure.

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