Dependent first order theories, continued. (English) Zbl 1195.03040
This paper continues the investigation, started in [S. Shelah, Sci.Math.Jpn.59, No.2, 265–316 (2004; Zbl 1081.03031)], of first-order theories without the dependence property (i.e., dependent theories). It is written in the author’s characteristic style.
In the first section, it is shown that if we expand a model of a dependent theory with quantifier elimination by all traces of sets definable in a (sufficiently saturated) elementary extension, the resulting structure still allows quantifier elimination. In particular it is still dependent. A far more readable account of this result is given by A. Pillay [in: B. Löwe (ed.), Algebra, logic, set theory. Festschrift für Ulrich Felgner zum 65.Geburtstag. London: King’s College Publications. Studies in Logic (London) 4, 175–181 (2007; Zbl 1135.03337)], and a stronger result was shown by A. Chernikov and P. Simon [arXiv:1007.4468v1].
In the second section, the author gives examples to show that the cofinality restrictions in Section 5 of his paper above are necessary.
In the third section, the author studies strongly dependent theories: \(T\) is strongly dependent if for some infinite \(\kappa\) in no model there are formulas \((\varphi_n(\bar x,\bar y):n<\omega)\) and tuples \((\bar a^n_\alpha:n<\omega,\alpha<\kappa)\) such that for all \(\eta\in{}^\omega\kappa\) the set
\[ p_\eta(\bar x)=\{\varphi_n(\bar x,\bar a^n_{\eta(\alpha)}):n<\omega\}\cup\{\neg\varphi_n(\bar x,\bar a^n_\alpha):n<\omega,\;\alpha\not=\eta(n)\} \]
is consistent. In this case, \(\kappa_{\text{ict}}(T)\) is the minimal \(\kappa\) with this property. If in addition we require all the \(p_\eta\) to be consistent with some type \(q\), this is called the burden of \(q\) by H. Adler [Strong theories, burden, and weight. Preprint (2007)] and generalizes the notion of weight for a type in a stable theory. The author shows that in a strongly dependent theory there are no non-algebraic types \(p\) and \(q\) such that for any countable set \(A\) of realizations of \(p\) there is \(b\) realizing \(q\) algebraising \(B\) (Shelah calls \((p,q)\) a \((1=\aleph_0)\)-pair, a terminology which I feel should be avoided). The study of strongly dependent theories will be continued in [Strongly dependent theories. Preprint].
In the fourth section, is is shown that a dependent group with an infinite abelian subgroup contains a definable abelian subgroup. This was generalized by R. de Aldama in his thesis [Chaînes et dépendance. Lyon (2009)] to the nilpotent case; moreover he shows that the definable group actually contains the given one.
In the fifth section, the author studies forking in the dependent context. He defines explicit dividing and forking, strong splitting, exact forking, strict non-dividing (negated as strict dividing), strict forking, strict non-forking (which is not the negation of strict forking) and shows various properties.
In the first section, it is shown that if we expand a model of a dependent theory with quantifier elimination by all traces of sets definable in a (sufficiently saturated) elementary extension, the resulting structure still allows quantifier elimination. In particular it is still dependent. A far more readable account of this result is given by A. Pillay [in: B. Löwe (ed.), Algebra, logic, set theory. Festschrift für Ulrich Felgner zum 65.Geburtstag. London: King’s College Publications. Studies in Logic (London) 4, 175–181 (2007; Zbl 1135.03337)], and a stronger result was shown by A. Chernikov and P. Simon [arXiv:1007.4468v1].
In the second section, the author gives examples to show that the cofinality restrictions in Section 5 of his paper above are necessary.
In the third section, the author studies strongly dependent theories: \(T\) is strongly dependent if for some infinite \(\kappa\) in no model there are formulas \((\varphi_n(\bar x,\bar y):n<\omega)\) and tuples \((\bar a^n_\alpha:n<\omega,\alpha<\kappa)\) such that for all \(\eta\in{}^\omega\kappa\) the set
\[ p_\eta(\bar x)=\{\varphi_n(\bar x,\bar a^n_{\eta(\alpha)}):n<\omega\}\cup\{\neg\varphi_n(\bar x,\bar a^n_\alpha):n<\omega,\;\alpha\not=\eta(n)\} \]
is consistent. In this case, \(\kappa_{\text{ict}}(T)\) is the minimal \(\kappa\) with this property. If in addition we require all the \(p_\eta\) to be consistent with some type \(q\), this is called the burden of \(q\) by H. Adler [Strong theories, burden, and weight. Preprint (2007)] and generalizes the notion of weight for a type in a stable theory. The author shows that in a strongly dependent theory there are no non-algebraic types \(p\) and \(q\) such that for any countable set \(A\) of realizations of \(p\) there is \(b\) realizing \(q\) algebraising \(B\) (Shelah calls \((p,q)\) a \((1=\aleph_0)\)-pair, a terminology which I feel should be avoided). The study of strongly dependent theories will be continued in [Strongly dependent theories. Preprint].
In the fourth section, is is shown that a dependent group with an infinite abelian subgroup contains a definable abelian subgroup. This was generalized by R. de Aldama in his thesis [Chaînes et dépendance. Lyon (2009)] to the nilpotent case; moreover he shows that the definable group actually contains the given one.
In the fifth section, the author studies forking in the dependent context. He defines explicit dividing and forking, strong splitting, exact forking, strict non-dividing (negated as strict dividing), strict forking, strict non-forking (which is not the negation of strict forking) and shows various properties.
Reviewer: Frank Wagner (Villeurbanne)
MSC:
| 03C45 | Classification theory, stability, and related concepts in model theory |
Keywords:
dependent theory; independence property; externally definable sets; cofinality; strong dependence; weight; burden; abelian subgroup; non-forkingReferences:
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