On order-types of models

1972 ◽  
Vol 37 (1) ◽  
pp. 69-70 ◽  
Author(s):  
Wilfrid Hodges
Keyword(s):  
Order Type ◽  
Linear Ordering ◽  
Infinite Field ◽  
Strong Limit ◽  
First Order ◽  
Order Types ◽  
Order Language ◽  
Language L ◽  
New Feature

Let T be a theory in a first-order language L. Let L have a predicate ν0 ≺ ν1 such that in every model of T, the interpretation of ≺ is a linear ordering with infinite field. The order-type of this ordering will be called the order-type of the model .Several recent theorems have the following form: if T has a model of order-type ξ then T has a model of order-type ζ (see [1]). We shall add one to the list. The new feature of our result is that the order-type ζ may be in a sense “opposite” to ξ. Silver's Theorem 2.24 of [3] is a corollary of Theorem 1 below.Theorem 1. Let κ be a strong limit number (i.e. μ < κ implies 2μ < κ). Suppose λ < κ, and suppose that for every cardinal μ < κ, T has a model with where the order-type of contains no descending well-ordered sequences of length λ. Then for every cardinal μ ≥ the cardinality ∣L∣ of the language L, T has models and such that(a) the field of is the union of ≤ ∣L∣ well-ordered (inversely well-ordered) parts;(b) .The proof is by Ehrenfeucht-Mostowski models; we presuppose [2].

scholarly journals ALGEBRAIC LINEAR ORDERINGS

2011 ◽  
Vol 22 (02) ◽  
pp. 491-515 ◽  
Author(s):  
S. L. BLOOM ◽  
Z. ÉSIK
Keyword(s):  
Order Type ◽  
Linear Ordering ◽  
Type Result ◽  
First Order ◽  
Linear Orderings ◽  
Context Free Language ◽  
Categorical Algebra ◽  
Context Free ◽  
Free Language

An algebraic linear ordering is a component of the initial solution of a first-order recursion scheme over the continuous categorical algebra of countable linear orderings equipped with the sum operation and the constant 1. Due to a general Mezei-Wright type result, algebraic linear orderings are exactly those isomorphic to the linear ordering of the leaves of an algebraic tree. Using Courcelle's characterization of algebraic trees, we obtain the fact that a linear ordering is algebraic if and only if it can be represented as the lexicographic ordering of a deterministic context-free language. When the algebraic linear ordering is a well-ordering, its order type is an algebraic ordinal. We prove that the Hausdorff rank of any scattered algebraic linear ordering is less than ωω. It follows that the algebraic ordinals are exactly those less than ωωω.

Interpolation theorems for Lk,k2+

1978 ◽  
Vol 43 (3) ◽  
pp. 535-549 ◽  
Author(s):  
Ruggero Ferro
Keyword(s):  
Natural Deduction ◽  
Interpolation Theorem ◽  
Positive Answer ◽  
Strong Limit ◽  
First Order ◽  
Limit Cardinal ◽  
Saturated Models ◽  
Order Language ◽  
Craig’S Interpolation Theorem

Chang, in [1], proves an interpolation theorem (Theorem I, remark b)) for a first-order language. The proof of Chang's theorem uses essentially nonsimple devices, like special and ω1-saturated models.In remark e) in [1], Chang asks if there is a simpler proof of his Theorem I.In [1], Chang proves also another interpolation theorem (Theorem II), which is not an extension of his Theorem I, but extends Craig's interpolation theorem to Lα+,ω languages with interpolant in Lα+,α where α is a strong limit cardinal of cofinality ω.In remark k) in [1], Chang asks if there is a generalization of both Theorems I and II in [1], or at least a generalization of both Theorem I in [1] and Lopez-Escobar's interpolation theorem in [7].Maehara and Takeuti, in [8], show that there is a completely different proof of Chang's interpolation Theorem I as a consequence of their interpolation theorems. The proofs of these theorems of Maehara and Takeuti are proof theoretical in character, involving the notion of cut-free natural deduction, and it uses devices as simple as those needed for the usual Craig's interpolation theorem. Hence this can be considered as a positive answer to Chang's question in remark e) in [1].

Topological Model Set Deformations in Logic Programming

1989 ◽  
Vol 12 (3) ◽  
pp. 357-399
Author(s):  
Aida Batarekh ◽  
V.S. Subrahmanian
Keyword(s):  
Logic Programming ◽  
Sufficient Conditions ◽  
Topological Spaces ◽  
Logic Programs ◽  
First Order ◽  
Order Language ◽  
Language L ◽  
General Logic ◽  
Model Set ◽  
Necessary And Sufficient

Given a first order language L, and a notion of a logic L w.r.t. L, we investigate the topological properties of the space of L-structures for L. We show that under a topology called the query topology which arises naturally in logic programming, the space of L-models (where L is a decent logic) of any sentence (set of clauses) in L may be regarded as a (closed, compact) T4-space. We then investigate the properties of maps from structures to structures. Our results allow us to apply various well-known results on the fixed-points of operators on topological spaces to the semantics of logic programming – in particular, we are able to derive necessary and sufficient topological conditions for the completion of covered general logic programs to be consistent. Moreover, we derive sufficient conditions guaranteeing the consistency of program completions, and for logic programs to be determinate. We also apply our results to characterize consistency of the unions of program completions.

An unsolved problem in the theory of constructive order types

1969 ◽  
Vol 33 (4) ◽  
pp. 565-567
Author(s):  
Alan G. Hamilton
Keyword(s):  
Recursive Function ◽  
Unsolved Problem ◽  
Order Type ◽  
Partial Answer ◽  
Linear Ordering ◽  
Partial Recursive Function ◽  
Order Types ◽  
The Difference ◽  
Definition Of

In the forthcoming monograph of Crossley [1] the question is raised whether the implication 2 + A = A ⇒ 1 + A = A is true for constructive order types. In this paper a partial answer to this question is given, in that a counterexample is constructed, using, however, not the definition of constructive order type (C.O.T.) given in [1], but an earlier one, namely that in Crossley [2]. The difference is that in [1] only orderings which can be imbedded in a standard dense r.e. ordering R by a partial recursive function are considered. The linear ordering constructed in this paper can be shown not to be such. The problem remains open in the case of orderings imbeddable in R.

Generalizing special Aronszajn trees

1974 ◽  
Vol 39 (4) ◽  
pp. 732-740 ◽  
Author(s):  
James H. Schmerl
Keyword(s):  
Model Theory ◽  
Initial Segment ◽  
Order Type ◽  
Linear Ordering ◽  
Property A ◽  
Transfer Property ◽  
First Order ◽  
Partition Property ◽  
Aronszajn Trees ◽  
Transfer Properties

In this paper we define by means of a partition property a decreasing sequence N = ‹Nα: α is an ordinal› of classes of ordinals. This property is a generalization of the nonexistence of special Aronszajn trees: the successor cardinal κ+ is in N0 iff there does not exist a special Aronszajn κ+-tree.The interest in the classes Nα stems from their applicability in model theory, in particular to that aspect of model theory dealing with ordered and two-cardinal models. A model is κ-like iff < is a linear ordering of A of cardinality κ but such that every proper initial segment has cardinality < κ. is α-ordered iff ≼ is a reflexive, linear ordering of some subset of A with order type α. The sequence N can be characterized by a first-order sentence σ in the following manner: The sentence σ has a κ-like α-ordered model iff κ ∉ Nα. This characterization will allow us to translate various independence statements regarding the sequence N to statements about the independence of transfer properties. We say that the transfer property κ → λ holds iff every first-order sentence which has a κ-like model also has a λ-like model. κ ⇸ λ is the negation of κ → λ.

The isomorphism property in nonstandard analysis and its use in the theory of Banach spaces

1974 ◽  
Vol 39 (4) ◽  
pp. 717-731 ◽  
Author(s):  
C. Ward Henson
Keyword(s):  
Nonstandard Analysis ◽  
Cardinal Number ◽  
Elementary Embedding ◽  
First Order ◽  
Saturation Properties ◽  
New Family ◽  
Theoretical Structure ◽  
Order Language ◽  
Language L ◽  
Elementary Topology

The basic setting of nonstandard analysis consists of a set-theoretical structure together with a map * from into another structure * of the same sort. The function * is taken to be an elementary embedding (in an appropriate sense) and is generally assumed to make * into an enlargement of [13]. The structures and * may be type-hierarchies as in [11] and [13] or they may be cumulative structures with ω levels as in [14]. The assumption that * is an enlargement of has been found to be the weakest hypothesis which allows for the familiar applications of nonstandard analysis in calculus, elementary topology, etc. Indeed, practice has shown that a smooth and useful theory can be achieved only by assuming also that * has some stronger properties such as the saturation properties first introduced in nonstandard analysis by Luxemburg [11].This paper concerns an entirely new family of properties, stronger than the saturation properties. For each cardinal number κ, * satisfies the κ-isomorphism property (as an enlargement of ) if the following condition holds:For each first order language L with fewer than κ nonlogical symbols, if and are elementarily equivalent structures for L whose domains, relations and functions are all internal (relative to * and ), then and are isomorphic.

PLATELET AGGREGATION DOES NOT CONFORM TO SIMPLE PARTICLE COLLISION THEORY

1987 ◽  
Author(s):  
Moideen P Jamaluddin
Keyword(s):  
Platelet Aggregation ◽  
Rate Equation ◽  
Rate Constants ◽  
Kinetic Scheme ◽  
Order Type ◽  
Second Order ◽  
Particle Collision ◽  
Collision Theory ◽  
Rate Law ◽  
First Order

Platelet aggregation kinetics, according to the particle collision theory, generally assumed to apply, ought to conform to a second order type of rate law. But published data on the time-course of ADP-induced single platelet recruitment into aggregates were found not to do so and to lead to abnormal second order rate constants much larger than even their theoretical upper bounds. The data were, instead, found to fit a first order type of rate law rather well with rate constants in the range of 0.04 - 0.27 s-1. These results were confirmed in our laboratory employing gelfiltered calf platelets. Thus a mechanism much more complex than hithertofore recognized, is operative. The following kinetic scheme was formulated on the basis of information gleaned from the literature.where P is the nonaggregable, discoid platelet, A the agonist, P* an aggregable platelet form with membranous protrusions, and P** another aggregable platelet form with pseudopods. Taking into account the relative magnitudes of the k*s and assuming aggregation to be driven by hydrophobic interaction between complementary surfaces of P* and P** species, a rate equation was derived for aggregation. The kinetic scheme and the rate equation could account for the apparent first order rate law and other empirical observations in the literature.

Weakly atomic-compact relational structures

1971 ◽  
Vol 36 (1) ◽  
pp. 129-140 ◽  
Author(s):  
G. Fuhrken ◽  
W. Taylor
Keyword(s):  
Abelian Group ◽  
Model Theory ◽  
Compact Abelian Group ◽  
Finite Subset ◽  
Relational Structure ◽  
Similarity Type ◽  
First Order ◽  
Relational Structures ◽  
Order Language ◽  
Weakly Atomic

A relational structure is called weakly atomic-compact if and only if every set Σ of atomic formulas (taken from the first-order language of the similarity type of augmented by a possibly uncountable set of additional variables as “unknowns”) is satisfiable in whenever every finite subset of Σ is so satisfiable. This notion (as well as some related ones which will be mentioned in §4) was introduced by J. Mycielski as a generalization to model theory of I. Kaplansky's notion of an algebraically compact Abelian group (cf. [5], [7], [1], [8]).

scholarly journals ON PRODUCTS OF ELEMENTARILY INDIVISIBLE STRUCTURES

2016 ◽  
Vol 81 (3) ◽  
pp. 951-971
Author(s):  
NADAV MEIR
Keyword(s):  
New Products ◽  
Elementary Substructure ◽  
First Order ◽  
Order Language ◽  
Image Position

AbstractWe say a structure ${\cal M}$ in a first-order language ${\cal L}$ is indivisible if for every coloring of its universe in two colors, there is a monochromatic substructure ${\cal M}\prime \subseteq {\cal M}$ such that ${\cal M}\prime \cong {\cal M}$. Additionally, we say that ${\cal M}$ is symmetrically indivisible if ${\cal M}\prime$ can be chosen to be symmetrically embedded in ${\cal M}$ (that is, every automorphism of ${\cal M}\prime$ can be extended to an automorphism of ${\cal M}$). Similarly, we say that ${\cal M}$ is elementarily indivisible if ${\cal M}\prime$ can be chosen to be an elementary substructure. We define new products of structures in a relational language. We use these products to give recipes for construction of elementarily indivisible structures which are not transitive and elementarily indivisible structures which are not symmetrically indivisible, answering two questions presented by A. Hasson, M. Kojman, and A. Onshuus.

scholarly journals Word order in Latin locative constructions: a corpus study in Caesar’s De bello gallico

2011 ◽  
Vol 64 (2) ◽  
Author(s):  
Stavros Skopeteas
Keyword(s):  
Word Order ◽  
Information Structure ◽  
Motion Verbs ◽  
Structural Domains ◽  
Corpus Study ◽  
First Order ◽  
Order Language ◽  
Free Word

AbstractClassical Latin is a free word order language, i.e., the order of the constituents is determined by information structure rather than by syntactic rules. This article presents a corpus study on the word order of locative constructions and shows that the choice between a Theme-first and a Locative-first order is influenced by the discourse status of the referents. Furthermore, the corpus findings reveal a striking impact of the syntactic construction: complements of motion verbs do not have the same ordering preferences with complements of static verbs and adjuncts. This finding supports the view that the influence of discourse status on word order is indirect, i.e., it is mediated by information structural domains.

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