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An introduction to forking

Published online by Cambridge University Press:  12 March 2014

Daniel Lascar
Affiliation:
Université Paris VII, U.E.R. de Mathématiques, Tour 45–55, 2 Place Jussieu, 75005 Paris, France
Bruno Poizat
Affiliation:
Université Paris VI, 11 Parc d'Ardenay, 91120 Palaiseau, France

Extract

The notion of forking has been introduced by Shelah, and a full treatment of it will appear in his book on stability [S1]. The principal aim of this paper is to show that it is an easy and natural notion.

Consider some well-known examples of ℵ0-stable theories: vector spaces over Q, algebraically closed fields, differentially closed fields of characteristic 0; in each of these cases, we have a natural notion of independence: linear, algebraic and differential independence respectively. Forking gives a generalization of these notions. More precisely, if are subsets of some model and c a point of this model, the fact that the type of c over does not fork over means that there are no more relations of dependence between c and than there already existed between c and . In the case of the vector spaces, this means that c is in the space generated by only if it is already in the space generated by . In the case of differentially closed fields, this means that the minimal differential equations of c with coefficient respectively in and have the same order. Of course, these notions of dependence are essential for the study of the above mentioned structures. Forking is no less important for stable theories. A glance at Shelah's book will convince the reader that this is the case.

What we have to do is the following. Assuming T stable and given and p a type on , we want to distinguish among the extensions of p to some of them that we shall call the nonforking extensions of p.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 1979

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References

[CK]Chang, C. C. and Keisler, H. J., Model theory, North-Holland, Amsterdam, 1973.Google Scholar
[L1]Lascar, Daniel, Rank and definability in superstable theories, Israel Journal of Mathematics, vol. 23 (1976), pp. 5387.CrossRefGoogle Scholar
[L2]Lascar, Daniel, Définissabilité dans les théories stables, Logique et analyse, 19711972, pp. 489507; appearing also in Six Days of Model Theory, Proceedings of a Conference at Louvain-la-Neuve, March 1975 (Paul Henrard, Editor).Google Scholar
[P]Poizat, Bruno, Déviation des types, Doctoral dissertation, Paris, 1977.Google Scholar
[S1]Shelah, Saharon, Classification theory and the number of non-isomorphic models, Studies in logic and the foundations of mathematics, vol. 92, North-Holland, Amsterdam, 1978.Google Scholar
[S2]Shelah, Saharon, Every two elementarily equivalent models have isomorphic ultrapowers, Israel Journal of Mathematics, vol. 10 (1971), pp. 224233.CrossRefGoogle Scholar
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