## Notre Dame Journal of Formal Logic

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

### Computing the Number of Types of Infinite Length

#### 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 \le \lambda $, then

$${sup\hspace{0.17em}}_{\Vert M\Vert =\lambda}\left|{S}^{\kappa}\right(M\left)\right|=\left({sup\hspace{0.17em}}_{\Vert M\Vert =\lambda}\right|{S}^{1}\left(M\right)|{)}^{\kappa}.$$ 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

**Zentralblatt MATH identifier**

06686423

**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