Arithmetic separation and Banach–Saks sets

On bD-Sets and Associated Separation Axioms

on bd-sets and associated separation axioms

Models of Arithmetic and Subuniform Bounds for the Arithmetic Sets

It has been known for more than thirty years that the degree of a non-standard model of true arithmetic is a subuniform upper bound for the arithmetic sets (suub). Here a notion of generic enumeration is presented with the property that the degree of such an enumeration is an suub but not the degree of a non-standard model of true arithmetic. This anwers a question posed in the literature.

Models of Arithmetic and Upper Bounds for Arithmetic Sets

We settle a question in the literature about degrees of models of true arithmetic and upper bounds for the arithmetic sets. We prove that there is a model of true arithmetic whose degree is not a uniform upper bound for the arithmetic sets. The proof involves two forcing constructions.

Automata for Arithmetic Meyer Sets

The set Zβ of β-integers is a Meyer set when β is a Pisot number, and thus there exists a finite set F such that Zβ −Zβ ⊂ Zβ +F . We give finite automata describing the expansions of the elements of Zβ and of Zβ − Zβ . We present a construction of such a finite set F , and a method to minimize the size of F . We obtain in this way a finite transducer that performs the decomposition of the eleme...