scholarly journals Hanf number for Scott sentences of computable structures

2018 ◽  
Vol 57 (7-8) ◽  
pp. 889-907
Author(s):  
S. S. Goncharov ◽  
J. F. Knight ◽  
I. Souldatos
Keyword(s):  
Computable Structures ◽  
Hanf Number ◽  
Scott Sentences

scholarly journals Π10 classes and strong degree spectra of relations

2007 ◽  
Vol 72 (3) ◽  
pp. 1003-1018 ◽  
Author(s):  
John Chisholm ◽  
Jennifer Chubb ◽  
Valentina S. Harizanov ◽  
Denis R. Hirschfeldt ◽  
Carl G. Jockusch ◽  
...  
Keyword(s):  
Kolmogorov Complexity ◽  
Initial Segment ◽  
Truth Table ◽  
Linear Ordering ◽  
Computable Structures ◽  
Degree Spectra

AbstractWe study the weak truth-table and truth-table degrees of the images of subsets of computable structures under isomorphisms between computable structures. In particular, we show that there is a low c.e. set that is not weak truth-table reducible to any initial segment of any scattered computable linear ordering. Countable subsets of 2ω and Kolmogorov complexity play a major role in the proof.

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, <).

scholarly journals Some new computable structures of high rank

2018 ◽  
Vol 146 (7) ◽  
pp. 3097-3109 ◽  
Author(s):  
Matthew Harrison-Trainor ◽  
Gregory Igusa ◽  
Julia F. Knight
Keyword(s):  
High Rank ◽  
Computable Structures

Model theory under the axiom of determinateness

1985 ◽  
Vol 50 (3) ◽  
pp. 773-780
Author(s):  
Mitchell Spector
Keyword(s):  
Model Theory ◽  
Axiom Of Choice ◽  
Order Logic ◽  
First Order Logic ◽  
Technical Innovation ◽  
First Order ◽  
Partition Relations ◽  
Hanf Number ◽  
The Axiom Of Choice

AbstractWe initiate the study of model theory in the absence of the Axiom of Choice, using the Axiom of Determinateness as a powerful substitute. We first show that, in this context, is no more powerful than first-order logic. The emphasis then turns to upward Löwenhein-Skolem theorems; ℵ1 is the Hanf number of first-order logic, of , and of a strong fragment of , The main technical innovation is the development of iterated ultrapowers using infinite supports; this requires an application of infinite-exponent partition relations. All our theorems can be proven from hypotheses weaker than AD.

Types omitted in uncountable models of arithmetic

1975 ◽  
Vol 40 (3) ◽  
pp. 317-320 ◽  
Author(s):  
Julia F. Knight
Keyword(s):  
Completeness Theorem ◽  
Elementary Extension ◽  
Ramsey's Theorem ◽  
Definable Set ◽  
Ramsey’S Theorem ◽  
Hanf Number ◽  
Definable Sets ◽  
Omitting Types ◽  
Infinite Set ◽  
Models Of Arithmetic

In [4] it is shown that if the structure omits a type Σ, and Σ is complete with respect to Th(), then there is a proper elementary extension of which omits Σ. This result is extended in the present paper. It is shown that Th() has models omitting Σ in all infinite powers.A type is a countable set of formulas with just the variable ν occurring free. A structure is said to omit the type Σ if no element of satisfies all of the formulas of Σ. A type Σ, in the same language as a theory T, is said to be complete with respect to T if (1) T ∪ Σ is consistent, and (2) for every formula φ(ν) of the language of T (with just ν free), either φ or ¬φ is in Σ.The proof of the result of this paper resembles Morley's proof [5] that the Hanf number for omitting types is . It is shown that there is a model of Th() which omits Σ and contains an infinite set of indiscernibles. Where Morley used the Erdös-Rado generalization of Ramsey's theorem, a definable version of the ordinary Ramsey's theorem is used here.The “omitting types” version of the ω-completeness theorem ([1], [3], [6]) is used, as it was in Morley's proof and in [4]. In [4], satisfaction of the hypotheses of the ω-completeness theorem followed from the fact that, in , any infinite, definable set can be split into two infinite, definable sets.

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).

Recursive elements and constructive extensions of computable local integral domains

1973 ◽  
Vol 38 (2) ◽  
pp. 272-290 ◽  
Author(s):  
Glen H. Suter
Keyword(s):  
Prime Ideal ◽  
Local Ring ◽  
Integral Domain ◽  
Rational Numbers ◽  
Algebraic Structures ◽  
Computable Structures ◽  
Integral Domains ◽  
Local Integral ◽  
Algebraic Operations ◽  
Axiom Systems

With reservations, one can think of abstract algebra as the study of what consequences can be drawn from the axioms associated with certain concrete algebraic structures. Two important examples of such concrete algebraic structures are the integers and the rational numbers. The integers and the rational numbers have two properties which are not in general mirrored in the abstract axiom systems associated with them. That is, the integers and the rational numbers both have effectively computable metrics and their algebraic operations are effectively computable. The study of abstract algebraic systems which possess effectively computable algebraic operations has produced many interesting results. One can think of a computable algebraic structure as one whose elements have been labeled by the set of positive integers and whose operations are effectively computable. The formal definition of computable local integral domain will be given in §3. Some specific computable structures which have been studied are the integers, the rational numbers, and the rational numbers with p-adic valuation. Computable structures were studied in general by Rabin [12]. This paper concerns computable local integral domains as exemplified by the local integral domain Zp. Zp is the localization of the integers with respect to the maximal prime ideal generated by the positive prime p. We should note that the concept of local integral domain is not first order.Let the ordered pair (Q, M) stand for a local ring, where Q is the local ring and M is the unique maximal prime ideal of Q. Since most of my results are proving the existence of certain effective procedures, the assumption that Q has a principal maximal ideal M (rather than M has n generators) greatly simplifies many of the proofs.

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