A class of connected theories of order

1994 ◽  
Vol 59 (2) ◽  
pp. 534-542
Author(s):  
Alan S. Stern ◽  
Stanisław S. Świerczkowski
Keyword(s):  
Linear Order ◽  
Maximal Element ◽  
Disjoint Union ◽  
General Notion ◽  
Real Closed Fields ◽  
Ordered Fields ◽  
Closed Fields ◽  
Finite Sequences

In 1962 Jan Mycielski proposed a very general notion of interpretability [M1]. This led to the question whether a given theory could be interpreted in the disjoint union of two theories, without being interpretable in any of them. He argued that in such a case it would be presumably simpler to study each of these theories separately, and hence conjectured that this situation can never occur for any of the well-known theories of mathematics. This conjecture has now been verified for the following theories (see [MPS], [P], [S1, 2]): ELO (endless, i.e., without maximal element, linear order), Th(〈ℚ, ≤〉), Th(〈ω, ≤〉) and all sequential theories (those which can code finite sequences of elements of their models). The latter include PA, ZF, GB and Th(〈ω,+,·〉). In view of these confirmations it became ever more plausible that the conjecture is valid also for RCF (real closed fields), i.e., for Th(〈ℝ,≤,+,·,0,1〉). In the present paper we show that Mycielski's conjecture is valid for a class of theories which includes RCF and OF (ordered fields).We consider only theories with equality and without function symbols. Interpretations will be meant local, multidimensional, and with parameters, as defined in [M1], [M2] and surveyed in [MPS] (for a recent definition see also [S2]). We shall write T0 ≪ T1 to say that T0 is interpretable in T1 (or that T1 interprets T0), and this will mean that for every theorem α of T0 there is an interpretation of α in T1.

The undecidability of intuitionistic theories of algebraically closed fields and real closed fields

1973 ◽  
Vol 38 (1) ◽  
pp. 86-92 ◽  
Author(s):  
Dov M. Gabbay
Keyword(s):  
Classical Theory ◽  
Linear Order ◽  
Abelian Groups ◽  
Classical Model ◽  
Sufficient Conditions ◽  
Interesting Case ◽  
Real Closed Fields ◽  
Closed Fields ◽  
Intuitionistic Theory ◽  
Algebraically Closed Fields

Let T be a set of axioms for a classical theory TC (e.g. abelian groups, linear order, unary function, algebraically closed fields, etc.). Suppose we regard T as a set of axioms for an intuitionistic theory TH (more precisely, we regard T as axioms in Heyting's predicate calculus HPC).Question. Is TH decidable (or, more generally, if X is any intermediate logic, is TX decidable)? In [1] we gave sufficient conditions for the undecidability of TH. These conditions depend on the formulas of T (different axiomatization of the same TC may give rise to different TH) and on the classical model theoretic properties of TC (the method did not work for model complete theories, e.g. those of the title of the paper). For details see [1]. In [2] we gave some decidability results for some theories: The problem of the decidability of theories TH for a classically model complete TC remained open. An undecidability result in this direction, for dense linear order was obtained by Smorynski [4]. The cases of algebraically closed fields and real closed fields and divisible abelian groups are treated in this paper. Other various decidability results of the intuitionistic theories were obtained by several authors, see [1], [2], [4] for details.One more remark before we start. There are several possible formulations for an intuitionistic theory of, e.g. fields, that correspond to several possible axiomatizations of the classical theory. Other formulations may be given in terms of the apartness relation, such as the one for fields given by Heyting [5]. The formulations that we consider here are of interest as these systems occur in intuitionistic mathematics. We hope that the present methods could be extended to the (more interesting) case of Heyting's systems [5].

Ordered Fields, Real Closed Fields

1998 ◽  
pp. 7-21
Author(s):  
Jacek Bochnak ◽  
Michel Coste ◽  
Marie-Françoise Roy
Keyword(s):  
Real Closed Fields ◽  
Ordered Fields ◽  
Closed Fields

scholarly journals THE NON-AXIOMATIZABILITY OF O-MINIMALITY

2014 ◽  
Vol 79 (01) ◽  
pp. 54-59 ◽  
Author(s):  
ALEX RENNET
Keyword(s):  
Function Symbol ◽  
Real Closed Field ◽  
Closed Field ◽  
Real Closed Fields ◽  
Ordered Fields ◽  
Closed Fields

Abstract Fix a language extending the language of ordered fields by at least one new predicate or function symbol. Call an L-structure R pseudo-o-minimal if it is (elementarily equivalent to) an ultraproduct of o-minimal structures. We show that for any recursive list of L-sentences , there is a real closed field satisfying which is not pseudo-o-minimal. This shows that the theory To−min consisting of those -sentences true in all o-minimal -structures, also called the theory of o-minimality (for L), is not recursively axiomatizable. And, in particular, there are locally o-minimal, definably complete expansions of real closed fields which are not pseudo-o-minimal.

Alfred Tarski's elimination theory for real closed fields

1988 ◽  
Vol 53 (1) ◽  
pp. 7-19 ◽  
Author(s):  
Lou Van Den Dries
Keyword(s):  
Computer Science ◽  
Basic Structure ◽  
Elimination Theory ◽  
Real Closed Fields ◽  
Ordered Fields ◽  
Closed Fields ◽  
Logistic Support ◽  
Intermediate Value

Tarski made a fundamental contribution to our understanding of R, perhaps mathematics’ most basic structure. His theorem is the following.To any formula ϕ(X1, …, Xm) in the vocabulary {0, 1, +, ·, <} one can effectively associate two objects: (i) a quantifier free formula (X1, …, Xm) in(1) the same vocabulary, and (ii) a proofof the equivalence ϕ ↔ that uses only the axioms for real closed fields. (Reminder: real closed fields are ordered fields with the intermediate value property for polynomials.)Everything in (1) has turned out to be crucial: that arbitrary formulas are considered rather than just sentences, that the equivalence ϕ ↔ holds in all real closed fields rather than only in R; even the effectiveness of the passage from ϕ to has found good theoretical uses besides firing the imagination.We begin this survey with some history in §1. In §2 we discuss three other influential proofs of Tarski's theorem, and in §3 we consider some of the remarkable and totally unforeseen ways in which Tarski's theorem functions nowadays in mathematics, logic and computer science.I thank Ward Henson, and in particular Wilfrid Hodges without whose constant prodding and logistic support this article would not have been written.

Separately Nash and arc‐Nash functions over real closed fields

2020 ◽  
Author(s):  
Wojciech Kucharz ◽  
Krzysztof Kurdyka ◽  
Ali El‐Siblani
Keyword(s):  
Real Closed Fields ◽  
Closed Fields

On Groups and Rings Definable In O-Minimal Expansions of Real Closed Fields

1996 ◽  
Vol 28 (1) ◽  
pp. 7-14 ◽  
Author(s):  
Margarita Otero ◽  
Ya'acov Peterzil ◽  
Anand Pillay
Keyword(s):  
Real Closed Fields ◽  
Closed Fields

scholarly journals A question of van den Dries and a theorem of Lipshitz and Robinson; Not everything is standard

2007 ◽  
Vol 72 (1) ◽  
pp. 119-122 ◽  
Author(s):  
Ehud Hrushovski ◽  
Ya'acov Peterzil
Keyword(s):  
Real Numbers ◽  
The Real ◽  
First Order ◽  
Real Closed Fields ◽  
Minimal Structure ◽  
Closed Fields ◽  
New Construction ◽  
The Relationship

AbstractWe use a new construction of an o-minimal structure, due to Lipshitz and Robinson, to answer a question of van den Dries regarding the relationship between arbitrary o-minimal expansions of real closed fields and structures over the real numbers. We write a first order sentence which is true in the Lipshitz-Robinson structure but fails in any possible interpretation over the field of real numbers.

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, <).

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