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EXAMPLES IN DEPENDENT THEORIES

Published online by Cambridge University Press:  25 June 2014

ITAY KAPLAN  and
SAHARON SHELAH
ITAY KAPLAN*
Affiliation:
THE HEBREW UNIVERSITY OF JERUSALEM, EINSTEIN INSTITUTE OF MATHEMATICS, EDMOND J. SAFRA CAMPUS, GIVAT RAM, JERUSALEM 91904, ISRAEL
SAHARON SHELAH
Affiliation:
DEPARTMENT OF MATHEMATICS, HILL CENTER-BUSCH CAMPUS, RUTGERS, THE STATE UNIVERSITY OF NEW JERSEY, 110 FRELINGHUYSEN ROAD, PISCATAWAY, NJ 08854-8019 USAE-mail: [email protected]
*
*THE HEBREW UNIVERSITY OF JERUSALEM, EINSTEIN INSTITUTE OF MATHEMATICS, EDMOND J. SAFRA CAMPUS, GIVAT RAM, JERUSALEM 91904, ISRAEL E-mail: [email protected]
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Abstract

In the first part we show a counterexample to a conjecture by Shelah regarding the existence of indiscernible sequences in dependent theories (up to the first inaccessible cardinal). In the second part we discuss generic pairs, and give an example where the pair is not dependent. Then we define the notion of directionality which deals with counting the number of coheirs of a type and we give examples of the different possibilities. Then we discuss nonsplintering, an interesting notion that appears in the work of Rami Grossberg, Andrés Villaveces and Monica VanDieren, and we show that it is not trivial (in the sense that it can be different than splitting) whenever the directionality of the theory is not small. In the appendix we study dense types in RCF.

Keywords

03C4503C9503C64Model theorydependent theoriesNIPindiscernible sequencesgeneric pairdirectionalitysplinteringdense types
Type
Articles
Information
The Journal of Symbolic Logic , Volume 79 , Issue 2 , June 2014 , pp. 585 - 619
Copyright
Copyright © The Association for Symbolic Logic 2014 

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