Richard Mansfield. Perfect subsets of definable sets of real numbers. Pacific journal of mathematics, vol. 35 (1970), pp. 451–457.

1975 ◽  
Vol 40 (3) ◽  
pp. 462-462
Author(s):  
Yiannis N. Moschovakis
Keyword(s):  
Real Numbers ◽  
Definable Sets

sets of reals

1997 ◽  
Vol 62 (4) ◽  
pp. 1379-1428 ◽  
Author(s):  
Joan Bagaria ◽  
W. Hugh Woodin
Keyword(s):  
Set Theory ◽  
Real Numbers ◽  
Borel Sets ◽  
Definable Sets ◽  
Analytic Sets ◽  
Basic Facts ◽  
Continuous Images

Some of the most striking results in modern set theory have emerged from the study of simply-definable sets of real numbers. Indeed, simple questions like: what are the posible cardinalities?, are they measurable?, do they have the property of Baire?, etc., cannot be answered in ZFC.When one restricts the attention to the analytic sets, i.e., the continuous images of Borel sets, then ZFC does provide an answer to these questions. But this is no longer true for the projective sets, i.e., all the sets of reals that can be obtained from the Borel sets by taking continuous images and complements. In this paper we shall concentrate on particular projective classes, the , and using forcing constructions we will produce models of ZFC where, for some n, all , sets have some specified property. For the definition and basic facts about the projective classes , and , as well as the Kleene (or lightface) classes , and , we refer the reader to Moschovakis [19].The first part of the paper is about measure and category. Early in this century, Luzin [16] and Luzin-Sierpiński [17] showed that all analytic (i.e., ) sets of reals are Lebesgue measurable and have the property of Baire.

Reviews - J. H. Woodger. Translator's preface. Logic, semantics, metamathematics, papers from 1923 to 1938.Oxford at the Clarendon Press, London1956, pp. vii–ix. - Alfred Tarski. Author's acknowledgments.Logic, semantics, metamathematics, papers from 1923 to 1938.Oxford at the Clarendon Press, London1956, pp. xi–xii. - Alfred Tarski. On the primitive term of logistic. Modified English translation based on 2852–4. Logic, semantics, metamathematics, papers from 1923 to 1938.Oxford at the Clarendon Press, London1956, pp. 1–23. - Alfred Tarski. Foundations of the geometry of solids.Logic, semantics, metamathematics, papers from 1923 to 1938.Oxford at the Clarendon Press, London1956, pp. 24–29. (Translated, with additions, from Księga Pamiątkowa Pierwszego Polskiego Zjazdu Matematycznego, supplement to Annales de la Société Polonaise de Mathématique, Cracow 1929, pp. 29-33.) - Alfred Tarski. On some fundamental concepts of metamathematics. English translation of 2857. Logic, semantics, metamathematics, papers from 1923 to 1938.Oxford at the Clarendon Press, London1956, 30–37. - Jan Łukasiewicz and Alfred Tarski. Investigations into the sentential calculus. English translation of 4077, with added footnotes. Logic, semantics, metamathematics, papers from 1923 to 1938.Oxford at the Clarendon Press, London1956, pp. 38–59. - Alfred Tarski. Fundamental concepts of the methodology of the deductive sciences. English translation of 2858, with added footnotes. Logic, semantics, metamathematics, papers from 1923 to 1938.Oxford at the Clarendon Press, London1956, pp. 60–109. - Alfred Tarski. On definable sets of real numbers. English translation of 28510, with additions in the text by the author as well as added footnotes. Logic, semantics, metamathematics, papers from 1923 to 1938.Oxford at the Clarendon Press, London1956, pp. 110–142. - Kazimierz Kuratowski and Alfred Tarski. Logical operations and projective sets. English translation of 4321, with added footnotes. Logic, semantics, metamathematics, papers from 1923 to 1938.Oxford at the Clarendon Press, London1956, pp. 143–151. - Alfred Tarski. The concept of truth in formalized languages. English translation of 28516, with added footnotes. Logic, semantics, metamathematics, papers from 1923 to 1938.Oxford at the Clarendon Press, London1956, pp. 152–278.

1969 ◽  
Vol 34 (1) ◽  
pp. 99-106 ◽  
Author(s):  
W. A. Pogorzelski ◽  
S. J. Surma
Keyword(s):  
English Translation ◽  
Real Numbers ◽  
Sentential Calculus ◽  
Alfred Tarski ◽  
Primitive Term ◽  
Logical Operations ◽  
Definable Sets ◽  
Concept Of Truth

scholarly journals Perfect subsets of definable sets of real numbers

1970 ◽  
Vol 35 (2) ◽  
pp. 451-457 ◽  
Author(s):  
Richard Mansfield
Keyword(s):  
Real Numbers ◽  
Definable Sets

First order topological structures and theories

1987 ◽  
Vol 52 (3) ◽  
pp. 763-778 ◽  
Author(s):  
Anand Pillay
Keyword(s):  
Topological Structure ◽  
Complex Numbers ◽  
Restricted Class ◽  
Real Numbers ◽  
Boolean Combination ◽  
Topological Structures ◽  
First Order ◽  
Definable Set ◽  
Material Basis ◽  
Definable Sets

In this paper we introduce the notion of a first order topological structure, and consider various possible conditions on the complexity of the definable sets in such a structure, drawing several consequences thereof.Our aim is to develop, for a restricted class of unstable theories, results analogous to those for stable theories. The “material basis” for such an endeavor is the analogy between the field of real numbers and the field of complex numbers, the former being a “nicely behaved” unstable structure and the latter the archetypal stable structure. In this sense we try here to situate our work on o-minimal structures [PS] in a general topological context. Note, however, that the p-adic numbers, and structures definable therein, will also fit into our analysis.In the remainder of this section we discuss several ways of studying topological structures model-theoretically. Eventually we fix on the notion of a structure in which the topology is “explicitly definable” in the sense of Flum and Ziegler [FZ]. In §2 we introduce the hypothesis that every definable set is a Boolean combination of definable open sets. In §3 we introduce a “dimension rank” on (closed) definable sets. In §4 we consider structures on which this rank is defined, and for which also every definable set has a finite number of definably connected definable components. We show that prime models over sets exist under such conditions.

A version of p-adic minimality

2012 ◽  
Vol 77 (2) ◽  
pp. 621-630 ◽  
Author(s):  
Raf Cluckers ◽  
Eva Leenknegt
Keyword(s):  
Positive Characteristic ◽  
Quantifier Elimination ◽  
Cell Decomposition ◽  
Real Numbers ◽  
Valued Fields ◽  
The Real ◽  
Definable Sets ◽  
A Cell ◽  
Definable Functions

AbstractWe introduce a very weak language on p-adic fields K, which is just rich enough to have exactly the same definable subsets of the line K that one has using the ring language. (In our context, definable always means definable with parameters.) We prove that the only definable functions in the language are trivial functions. We also give a definitional expansion of in which K has quantifier elimination, and we obtain a cell decomposition result for -definable sets.Our language can serve as a p-adic analogue of the very weak language (<) on the real numbers, to define a notion of minimality on the field of p-adic numbers and on related valued fields. These fields are not necessarily Henselian and may have positive characteristic.

Preliminaries

2017 ◽  
Author(s):  
Ehud Hrushovski ◽  
François Loeser
Keyword(s):  
Valued Fields ◽  
Definable Set ◽  
Valued Field ◽  
Galois Coverings ◽  
Definable Type ◽  
Definable Sets ◽  
Background Material

This chapter provides some background material on definable sets, definable types, orthogonality to a definable set, and stable domination, especially in the valued field context. It considers more specifically these concepts in the framework of the theory ACVF of algebraically closed valued fields and describes the definable types concentrating on a stable definable V as an ind-definable set. It also proves a key result that demonstrates definable types as integrals of stably dominated types along some definable type on the value group sort. Finally, it discusses the notion of pseudo-Galois coverings. Every nonempty definable set over an algebraically closed substructure of a model of ACVF extends to a definable type.

On existence result of a class of nonlinear integral equation

Filomat ◽  
2017 ◽  
Vol 31 (11) ◽  
pp. 3593-3597
Author(s):  
Ravindra Bisht
Keyword(s):  
Integral Equation ◽  
Fixed Point ◽  
Existence And Uniqueness ◽  
Existence Result ◽  
Nonlinear Integral Equation ◽  
Continuous Functions ◽  
Concave Functions ◽  
Real Numbers ◽  
Fixed Point Techniques ◽  
Uniqueness Of A Solution

Combining the approaches of functionals associated with h-concave functions and fixed point techniques, we study the existence and uniqueness of a solution for a class of nonlinear integral equation: x(t) = g1(t)-g2(t) + ? ?t,0 V1(t,s)h1(s,x(s))ds + ? ?T,0 V2(t,s)h2(s,x(s))ds; where C([0,T];R) denotes the space of all continuous functions on [0,T] equipped with the uniform metric and t?[0,T], ?,? are real numbers, g1, g2 ? C([0, T],R) and V1(t,s), V2(t,s), h1(t,s), h2(t,s) are continuous real-valued functions in [0,T]xR.

Dynamics of a family of orbitally continuous mappings

Filomat ◽  
2017 ◽  
Vol 31 (11) ◽  
pp. 3507-3517
Author(s):  
Abhijit Pant ◽  
R.P. Pant ◽  
Kuldeep Prakash
Keyword(s):  
Fixed Points ◽  
Julia Sets ◽  
Real Numbers ◽  
Continuous Mappings ◽  
Non Linear

The aim of the present paper is to study the dynamics of a class of orbitally continuous non-linear mappings defined on the set of real numbers and to apply the results on dynamics of functions to obtain tests of divisibility. We show that this class of mappings contains chaotic mappings. We also draw Julia sets of certain iterations related to multiple lowering mappings and employ the variations in the complexity of Julia sets to illustrate the results on the quotient and remainder. The notion of orbital continuity was introduced by Lj. B. Ciric and is an important tool in establishing existence of fixed points.

Sign in / Sign up

Export Citation Format

Share Document