Boolean Types in Dependent Theories

Journal of Symbolic Logic 87 (4):1349-1373 (2022)
  Copy   BIBTEX

Abstract

The notion of a complete type can be generalized in a natural manner to allow assigning a value in an arbitrary Boolean algebra $\mathcal {B}$ to each formula. We show some basic results regarding the effect of the properties of $\mathcal {B}$ 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.

Links

PhilArchive



    Upload a copy of this work     Papers currently archived: 90,593

External links

Setup an account with your affiliations in order to access resources via your University's proxy server

Through your library

Similar books and articles

Chain conditions in dependent groups.Itay Kaplan & Saharon Shelah - 2013 - Annals of Pure and Applied Logic 164 (12):1322-1337.
Examples in dependent theories.Itay Kaplan & Saharon Shelah - 2014 - Journal of Symbolic Logic 79 (2):585-619.
Small theories of Boolean ordered o-minimal structures.Roman Wencel - 2002 - Journal of Symbolic Logic 67 (4):1385-1390.
On generically stable types in dependent theories.Alexander Usvyatsov - 2009 - Journal of Symbolic Logic 74 (1):216-250.
The number of openly generated Boolean algebras.Stefan Geschke & Saharon Shelah - 2008 - Journal of Symbolic Logic 73 (1):151-164.
More constructions for Boolean algebras.Saharon Shelah - 2002 - Archive for Mathematical Logic 41 (5):401-441.
On uniform definability of types over finite sets.Vincent Guingona - 2012 - Journal of Symbolic Logic 77 (2):499-514.
More on cardinal invariants of Boolean algebras.Andrzej Rosłanowski & Saharon Shelah - 2000 - Annals of Pure and Applied Logic 103 (1-3):1-37.
On ultraproducts of Boolean algebras and irr.Saharon Shelah - 2003 - Archive for Mathematical Logic 42 (6):569-581.
Strict independence.Itay Kaplan & Alexander Usvyatsov - 2014 - Journal of Mathematical Logic 14 (2):1450008.
The number of types in simple theories.Enrique Casanovas - 1999 - Annals of Pure and Applied Logic 98 (1-3):69-86.
Depth of Boolean Algebras.Shimon Garti & Saharon Shelah - 2011 - Notre Dame Journal of Formal Logic 52 (3):307-314.

Analytics

Added to PP
2022-04-08

Downloads
4 (#1,426,245)

6 months
3 (#445,838)

Historical graph of downloads
How can I increase my downloads?

Citations of this work

Add more citations

References found in this work

No references found.

Add more references