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Properties and consequences of Thorn-independence

  • Alf Onshuus (a1)

We develop a new notion of independence (ϸ-independence, read “thorn”-independence) that arises from a family of ranks suggested by Scanlon (ϸ-ranks). We prove that in a large class of theories (including simple theories and o-minimal theories) this notion has many of the properties needed for an adequate geometric structure.

We prove that ϸ-independence agrees with the usual independence notions in stable, supersimple and o-minimal theories. Furthermore, we give some evidence that the equivalence between forking and ϸ-forking in simple theories might be closely related to one of the main open conjectures in simplicity theory, the stable forking conjecture. In particular, we prove that in any simple theory where the stable forking conjecture holds, ϸ-independence and forking independence agree.

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[1] Steven Buechler , Anand Pillay , and Frank Wagner , Supersimple theories, Journal of the American Mathematical Society, vol. 14 (2001), no. 1, pp. 109124.

[2] Enrique Casanovas and Frank Wagner , The free roots of the complete graph, Proceedings of the American Mathematical Society, vol. 132 (2003), no. 5, pp. 15431548.

[4] Wilfrid Hodges , Model theory, Encyclopedia of Mathematics and its Applications, vol. 42, Cambridge University Press, Cambridge, 1993.

[6] Byunghan Kim and A. Pillay , Around stable forking, Fundamenta Mathematicae, vol. 170 (2001), no. 1–2, pp. 107118, dedicated to the memory of Jerzy Łoś.

[7] Byunghan Kim and Anand Pillay , Simple theories, Annals of Pure and Applied Logic, vol. 88 (1997), no. 2–3, pp. 149164, joint AILA-KGS Model Theory Meeting (Florence, 1995).

[10] Lou van den Dries , Tame topology and o-minimal structures, London Mathematical Society Lecture Note Series, vol. 248, Cambridge University Press, Cambridge, 1998.

[11] Frank O. Wagner , Simple theories, Mathematics and its Applications, vol. 503, Kluwer Academic Publishers, Dordrecht, 2000.

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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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