Notre Dame Journal of Formal Logic

Computing the Number of Types of Infinite Length

Will Boney

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Abstract

We show that the number of types of sequences of tuples of a fixed length can be calculated from the number of $1$-types and the length of the sequences. Specifically, if $\kappa \leq \lambda$, then

\[\sup_{\Vert M\Vert =\lambda}\vert S^{\kappa}(M)\vert =(\sup_{\Vert M\Vert =\lambda}\vert S^{1}(M)\vert )^{\kappa}.\] We show that this holds for any abstract elementary class with $\lambda$-amalgamation. No such calculation is possible for nonalgebraic types. However, we introduce a subclass of nonalgebraic types for which the same upper bound holds.

Article information

Source
Notre Dame J. Formal Logic Volume 58, Number 1 (2017), 133-154.

Dates
Received: 18 September 2013
Accepted: 24 June 2014
First available in Project Euclid: 25 November 2016

Permanent link to this document
https://projecteuclid.org/euclid.ndjfl/1480042820

Digital Object Identifier
doi:10.1215/00294527-3768177

Mathematical Reviews number (MathSciNet)
MR3595347

Subjects
Primary: 03C48: Abstract elementary classes and related topics [See also 03C45]
Secondary: 03C45: Classification theory, stability and related concepts [See also 03C48]

Keywords
abstract elementary classes types

Citation

Boney, Will. Computing the Number of Types of Infinite Length. Notre Dame J. Formal Logic 58 (2017), no. 1, 133--154. doi:10.1215/00294527-3768177. https://projecteuclid.org/euclid.ndjfl/1480042820.


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