Sequences in countable nonstandard models of the natural numbers
The author studies countable models of the set of natural numbers. Let \({\mathcal M}\) be such a model, \({\mathcal M}=<M,+,\cdot,0,1,A^ m>,\) where A is a unary predicate symbol whose interpretation in \({\mathbb{N}}\) is the standard sequence A. There is an order-preserving mapping q: \(M\to {\mathbb{Q}}\cap [0,1]\). Now the addition within the model induces a new ``addition'' on \({\mathbb{Q}}\cap [0,1]\). This addition function is continuous but it does not extend continuously to [0,1]. A sequence \(A\subseteq {\mathbb{N}}\) is associated with a subset q(A) of \({\mathbb{Q}}\cap [0,1]\). It is proved that the topological properties of the subset q(A) are independent of the way in which q is chosen. Relationships between A and q(A) are studied. The following result is typical: q(A) has at least one isolated point iff there is an \(n\in {\mathbb{N}}\) such that for all \(m\in {\mathbb{N}}\) there exists an interval of length m which contains exactly n elements of A. A mapping e: \(M\to [0,1]\) is introduced so that \(e(m)=e(n)\) iff m/n is in the monad of 1. The mapping e induces another addition \([+]\) on [0,1]\(\cap {\mathbb{Q}}\). The function \([+]\) can be extended continuously to [0,1].
scientific article; zbMATH DE number 4104954