Quelques précisions sur la D.O.P. et la profondeur d'une théorie

1985 ◽  
Vol 50 (2) ◽  
pp. 316-330 ◽  
Author(s):  
D. Lascar
Keyword(s):  
Order Property ◽  
Countable Theory

AbstractWe give here alternative definitions for the notions that S. Shelah has introduced in recent papers: the dimensional order property and the depth of a theory. We will also give a proof that the depth of a countable theory, when denned, is an ordinal recursive in T.

Metaontology

2018 ◽  
Author(s):  
Uriah Kriegel
Keyword(s):  
Second Order ◽  
Order Property ◽  
First Order ◽  
First Approximation

Brentano’s theory of judgment serves as a springboard for his conception of reality, indeed for his ontology. It does so, indirectly, by inspiring a very specific metaontology. To a first approximation, ontology is concerned with what exists, metaontology with what it means to say that something exists. So understood, metaontology has been dominated by three views: (i) existence as a substantive first-order property that some things have and some do not, (ii) existence as a formal first-order property that everything has, and (iii) existence as a second-order property of existents’ distinctive properties. Brentano offers a fourth and completely different approach to existence talk, however, one which falls naturally out of his theory of judgment. The purpose of this chapter is to present and motivate Brentano’s approach.

On chains of relatively saturated submodels of a model without the order property

1991 ◽  
Vol 56 (1) ◽  
pp. 124-128 ◽  
Author(s):  
Rami Grossberg
Keyword(s):  
Existence Theorems ◽  
Order Property ◽  
Similarity Type

AbstractLet M be a given model with similarity type L = L(M), and let L′ be any fragment of L∣L(M∣+,ω of cardinality ∣L(M)∣. We call N ≺ ML′-relatively saturated iff for every B ⊆ N of cardinality less than ∥N∥ every L′-type over B which is realized in M is realized in N. We discuss the existence of such submodels.The following are corollaries of the existence theorems.(1) If M is of cardinality at least ℶω1, and fails to have the ω order property, then there exists N ≺ M which is relatively saturated in M of cardinality ℶω1.(2) Assume GCH. Let ψ ∈ Lω1, ω, and let L′ ⊆ Lω1, ω be a countable fragment containing ψ. If ∃χ > ℵ0 such that I(χ, ψ) < 2χ, then for every M ⊨ ψ and every cardinal λ < ∥M∥ of uncountable cofinality, M has an L′-relatively saturated submodel of cardinality λ.

Properties of Classes of Random Graphs

1994 ◽  
Vol 3 (4) ◽  
pp. 435-454 ◽  
Author(s):  
Neal Brand ◽  
Steve Jackson
Keyword(s):  
Random Graphs ◽  
Higher Order ◽  
Order Property ◽  
First Order ◽  
New Class ◽  
Vertex Transitive ◽  
Vertex Transitive Graphs ◽  
Almost All ◽  
Transitive Graphs

In [11] it is shown that the theory of almost all graphs is first-order complete. Furthermore, in [3] a collection of first-order axioms are given from which any first-order property or its negation can be deduced. Here we show that almost all Steinhaus graphs satisfy the axioms of almost all graphs and conclude that a first-order property is true for almost all graphs if and only if it is true for almost all Steinhaus graphs. We also show that certain classes of subgraphs of vertex transitive graphs are first-order complete. Finally, we give a new class of higher-order axioms from which it follows that large subgraphs of specified type exist in almost all graphs.

scholarly journals Crisp-determinization of weighted tree automata over strong bimonoids

2021 ◽  
Vol vol. 23 no. 1 (Automata, Logic and Semantics) ◽  
Author(s):  
Zoltán Fülöp ◽  
Dávid Kószó ◽  
Heiko Vogler
Keyword(s):  
Finite Order ◽  
Positive Answer ◽  
Order Property ◽  
Sufficient Condition ◽  
Tree Automata ◽  
Weighted Tree ◽  
Bottom Up ◽  
Initial Algebra ◽  
Initial Algebra Semantics ◽  
Weighted Tree Automata

We consider weighted tree automata (wta) over strong bimonoids and their initial algebra semantics and their run semantics. There are wta for which these semantics are different; however, for bottom-up deterministic wta and for wta over semirings, the difference vanishes. A wta is crisp-deterministic if it is bottom-up deterministic and each transition is weighted by one of the unit elements of the strong bimonoid. We prove that the class of weighted tree languages recognized by crisp-deterministic wta is the same as the class of recognizable step mappings. Moreover, we investigate the following two crisp-determinization problems: for a given wta ${\cal A}$, (a) does there exist a crisp-deterministic wta which computes the initial algebra semantics of ${\cal A}$ and (b) does there exist a crisp-deterministic wta which computes the run semantics of ${\cal A}$? We show that the finiteness of the Nerode algebra ${\cal N}({\cal A})$ of ${\cal A}$ implies a positive answer for (a), and that the finite order property of ${\cal A}$ implies a positive answer for (b). We show a sufficient condition which guarantees the finiteness of ${\cal N}({\cal A})$ and a sufficient condition which guarantees the finite order property of ${\cal A}$. Also, we provide an algorithm for the construction of the crisp-deterministic wta according to (a) if ${\cal N}({\cal A})$ is finite, and similarly for (b) if ${\cal A}$ has finite order property. We prove that it is undecidable whether an arbitrary wta ${\cal A}$ is crisp-determinizable. We also prove that both, the finiteness of ${\cal N}({\cal A})$ and the finite order property of ${\cal A}$ are undecidable.

Finitely axiomatizable ω-categorical theories and the Mazoyer hypothesis

2005 ◽  
Vol 70 (2) ◽  
pp. 460-472 ◽  
Author(s):  
David Lippel
Keyword(s):  
Order Property ◽  
Simple Theories ◽  
Definable Set

AbstractLet F be the class of complete, finitely axiomatizable ω-categorical theories. It is not known whether there are simple theories in F. We prove three results of the form: if T ∈ F has a sufficently well-behaved definable set J, then T is not simple. (In one case, we actually prove that T has the strict order property.) All of our arguments assume that the definable set J satisfies the Mazoyer hypothesis, which controls how an element of J can be algebraic over a subset of the model. For every known example in F, there is a definable set satisfying the Mazoyer hypothesis.

On first-order sentences without finite models

2004 ◽  
Vol 69 (2) ◽  
pp. 329-339 ◽  
Author(s):  
Marko Djordjević
Keyword(s):  
Binary Relation ◽  
Function Symbol ◽  
Complete Theory ◽  
Order Property ◽  
Unary Function ◽  
First Order ◽  
Finite Models

We will mainly be concerned with a result which refutes a stronger variant of a conjecture of Macpherson about finitely axiomatizable ω-categorical theories. Then we prove a result which implies that the ω-categorical stable pseudoplanes of Hrushovski do not have the finite submodel property.Let's call a consistent first-order sentence without finite models an axiom of infinity. Can we somehow describe the axioms of infinity? Two standard examples are:ϕ1: A first-order sentence which expresses that a binary relation < on a nonempty universe is transitive and irreflexive and that for every x there is y such that x < y.ϕ2: A first-order sentence which expresses that there is a unique x such that, (0) for every y, s(y) ≠ x (where s is a unary function symbol),and, for every x, if x does not satisfy (0) then there is a unique y such that s(y) = x.Every complete theory T such that ϕ1 ϵ T has the strict order property (as defined in [10]), since the formula x < y will have the strict order property for T. Let's say that if Ψ is an axiom of infinity and every complete theory T with Ψ ϵ T has the strict order property, then Ψ has the strict order property.Every complete theory T such that ϕ2 ϵ T is not ω-categorical. This is the case because a complete theory T without finite models is ω-categorical if and only if, for every 0 < n < ω, there are only finitely many formulas in the variables x1,…,xn, up to equivalence, in any model of T.

The Predicate of Existence

2019 ◽  
pp. 14-37
Author(s):  
Palle Yourgrau
Keyword(s):  
Bertrand Russell ◽  
Gottlob Frege ◽  
Order Property ◽  
Saul Kripke ◽  
Existential Quantifier ◽  
Modern Logic ◽  
First Order ◽  
David Kaplan ◽  
Theory Of Descriptions ◽  
Nathan Salmon

Kant famously declared that existence is not a (real) predicate. This famous dictum has been seen as echoed in the doctrine of the founders of modern logic, Gottlob Frege and Bertrand Russell, that existence isn’t a first-order property possessed by individuals, but rather a second-order property expressed by the existential quantifier. Russell in 1905 combined this doctrine with his new theory of descriptions and declared the paradox of nonexistence to be resolved without resorting to his earlier distinction between existence and being. In recent years, however, logicians and philosophers like Saul Kripke, David Kaplan, and Nathan Salmon have argued that there is no defensible reason to deny that existence is a property of individuals. Kant’s dictum has also been re-evaluated, the result being that the paradox of nonexistence has not, after all, disappeared. Yet it’s not clear how exactly Kripke et al. propose to resolve the paradox.

scholarly journals Pseudofinite difference fields

2019 ◽  
Vol 19 (02) ◽  
pp. 1950011
Author(s):  
Tingxiang Zou
Keyword(s):  
Finite Fields ◽  
Transcendence Degree ◽  
Order Property ◽  
Frobenius Automorphism ◽  
Difference Fields ◽  
Definable Sets ◽  
Partial Connection

We study a family of ultraproducts of finite fields with the Frobenius automorphism in this paper. Their theories have the strict order property and TP2. But the coarse pseudofinite dimension of the definable sets is definable and integer-valued. Moreover, we establish a partial connection between coarse dimension and transformal transcendence degree in these difference fields.

Why some people are excited by Vaught's conjecture

1985 ◽  
Vol 50 (4) ◽  
pp. 973-982 ◽  
Author(s):  
Daniel Lascar
Keyword(s):  
Model Theory ◽  
Continuum Hypothesis ◽  
Complete Theory ◽  
Scott Sentences ◽  
Countable Theory ◽  
The Continuum ◽  
Countable Models

§I. In 1961, R. L. Vaught ([V]) asked if one could prove, without the continuum hypothesis, that there exists a countable complete theory with exactly ℵ1 isomorphism types of countable models. The following statement is known as Vaught conjecture:Let T be a countable theory. If T has uncountably many countable models, then T hascountable models.More than twenty years later, this question is still open. Many papers have been written on the question: see for example [HM], [M1], [M2] and [St]. In the opinion of many people, it is a major problem in model theory.Of course, I cannot say what Vaught had in mind when he asked the question. I just want to explain here what meaning I personally see to this problem. In particular, I will not speak about the topological Vaught conjecture, which is quite another issue.I suppose that the first question I shall have to face is the following: “Why on earth are you interested in the number of countable models—particularly since the whole question disappears if we assume the continuum hypothesis?” The answer is simply that I am not interested in the number of countable models, nor in the number of models in any cardinality, as a matter of fact. An explanation is due here; it will be a little technical and it will rest upon two names: Scott (sentences) and Morley (theorem).

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