Turing Computable Embeddings and Coding Families of Sets

Abstract

In [7] the notion of Turing computable embeddings is introduced as an effective counterpart for Borel embeddings. The former allows for the study of classes of structures with universe a subset of ω. It also allows for finer distinctions, in particular, among classes with $\aleph_0$ isomorphism types. The hierarchy of effective cardinalities that arises from TC embeddings has been studied, among other places, in [7] and [2]. In this work, we prove that the special class of ‘daisy graphs', a subclass of undirected graphs used to code families of sets, has the same effective cardinality as the class of archimedian real closed fields. As a consequence, the class of abelian p-groups and the class of archimedian real closed fields are TC incomparable.