# Ranks for families of theories and their spectra

## TL;DR

This paper introduces ranks and degrees for families of theories, extending classical notions like Morley rank, and explores their properties, bounds, and criteria for total transcendentality within model theory.

## Contribution

It defines new rank and degree notions for families of theories, analyzes their preservation under E-closures, and establishes criteria for total transcendentality in countable languages.

## Key findings

- Bounds for e-spectra with respect to ranks and degrees
- Ranks and degrees are preserved under E-closures
- Criteria for total transcendental families based on cardinality

## Abstract

We define ranks and degrees for families of theories, similar to Morley rank and degree, as well as Cantor-Bendixson rank and degree, and the notion of totally transcendental family of theories. Bounds for $e$-spectra with respect to ranks and degrees are found. It is shown that the ranks and the degrees are preserved under $E$-closures and values for the ranks and the degrees are characterized. Criteria for totally transcendental families in terms of cardinality of $E$-closure and of the $e$-spectrum value, for a countable language, are proved.