Cohesive powers of the natural numbers

Dr Paul Shafer, University of Leeds. Part of the proofs, constructions and computations seminar series.

A cohesive power of a computable structure is an effective analogue of an ultrapower of the structure in which a cohesive set plays the role of an ultrafilter. We study the cohesive powers of computable copies of the structure (N, <), i.e., the natural numbers with their usual order. Our findings are as follows:

  • If L is a computable copy of (N, <) with a computable successor function, then every cohesive power of L has order type N + (Q x Z) (which is familiar as the order type of countable non-standard models of Peano arithmetic).
  • There is a computable copy, L, of (N, <) with a non-computable successor function such that every cohesive power of L has order type N + (Q x Z).
  • Most interestingly, there is a computable copy, L, of (N, <) (with a necessarily non-computable successor function) having a cohesive power that is not of order type N + (Q x Z).

This work is joint with Professor Rumen Dimitrov (Western Illinois University), Professor Valentina Harizanov (George Washington University), Professor Andrey Morozov (Sobolev Institute of Mathematics), Dr Alexandra Soskova (Sofia University), and Stefan Vatev (Sofia University).