Model-theoretic applications of cofinality spectrum problems
- 58 Downloads
We apply the recently developed technology of cofinality spectrum problems to prove a range of theorems in model theory. First, we prove that any model of Peano arithmetic is λ-saturated iff it has cofinality ≥ λ and the underlying order has no (κ, κ)-gaps for regular κ < λ. We also answer a question about balanced pairs of models of PA. Second, assuming instances of GCH, we prove that SOP2 characterizes maximality in the interpretability order ∇*, settling a prior conjecture and proving that SOP2 is a real dividing line. Third, we establish the beginnings of a structure theory for NSOP2, proving that NSOP2 can be characterized by the existence of few so-called higher formulas. In the course of the paper, we show that ps = ts in any weak cofinality spectrum problem closed under exponentiation (naturally defined). We also prove that the local versions of these cardinals need not coincide, even in cofinality spectrum problems arising from Peano arithmetic.
Unable to display preview. Download preview PDF.
- H. Gaifman and C. Dimitracopoulos, Fragments of Peano’s arithmetic and the MRDP theorem, in Logic and Algorithmic (Zurich, 1980), Monographies de L’Enseignement Mathématique, Vol. 30, Université de Genève, Geneva, 1982, pp. 187–206.Google Scholar
- I. Kaplan, S. Shelah and P. Simon, Exact saturation in simple and NIP theories, Journal of Mathematical Logic, accepted.Google Scholar
- R. Kossak, private communication.Google Scholar
- M. Malliaris and S. Shelah, Keisler’s order has infinitely many classes, Israel Journal of Mathematics, accepted.Google Scholar
- M. Malliaris and S. Shelah, Open problems on ultrafilters and some connections to the continuum, in Foundations of Mathematics, Contemporary Mathematics, Vol. 690, American Mathematical Society, Providence, RI, to appear.Google Scholar
- R. MacDowell and E. Specker, Modelle der arithmetic, in Infinitistic Methods (Proceedings of the Symposium on Foundations of Mathematics, Warsaw, 1959), Pergamon Press, Oxford; Państwowe Wydawnictwo Naukowe, Warsaw, 1961, pp. 257–263.Google Scholar
- S. Shelah, Classification Theory and the Number of Nonisomorphic Models, Studies in Logic and the Foundations of Mathematics, Vol. 92, North-Holland, Amsterdam–New York, 1978.Google Scholar
- S. Shelah, General non-structure theory, Paper E59, available at http://shelah.logic.at/Google Scholar
- S. Shelah, Dependent theories and the generic pair conjecture, Communications in Contemporary Mathematics 17 (2015).Google Scholar
- S. Shelah, Atomic saturation of reduced powers, arXiv:1601.04824.Google Scholar
- S. Shelah, Dependent dreams: recounting types, arXiv: 1202.5795.Google Scholar