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Mathematics > Logic

arXiv:2008.03214 (math)
[Submitted on 7 Aug 2020 (v1), last revised 16 Jan 2022 (this version, v2)]

Title:Boolean Types in Dependent Theories

Authors:Itay Kaplan, Ori Segel, Saharon Shelah
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Abstract:The notion of a complete type can be generalized in a natural manner to allow assigning a value in an arbitrary Boolean algebra B to each formula. We show some basic results regarding the effect of the properties of B on the behavior of such types, and show they are particularity well behaved in the case of NIP theories. In particular, we generalize the third author's result about counting types, as well as the notion of a smooth type and extending a type to a smooth one. We then show that Keisler measures are tied to certain Boolean types and show that some of the results can thus be transferred to measures - in particular, giving an alternative proof of the fact that every measure in a dependent theory can be extended to a smooth one. We also study the stable case. We consider this paper as an invitation for more research into the topic of Boolean types.
Comments: 32 pages, no figures
Subjects: Logic (math.LO)
MSC classes: 03C95 (Primary) 03C45, 03G05, 28A60 (Secondary)
Cite as: arXiv:2008.03214 [math.LO]
  (or arXiv:2008.03214v2 [math.LO] for this version)
  https://doi.org/10.48550/arXiv.2008.03214

Submission history

From: Ori Segel [view email]
[v1] Fri, 7 Aug 2020 15:06:07 UTC (28 KB)
[v2] Sun, 16 Jan 2022 15:31:15 UTC (29 KB)
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