Combinatorial and number-theoretic properties of generic reals

TL;DR

This paper explores the properties of Cohen and random reals, showing their prevalence in large families, their classification in Mahler's hierarchy, and their complexity in o-minimal structures.

Contribution

It provides new insights into the combinatorial, number-theoretic, and definability properties of generic reals, linking them to various areas in logic and number theory.

Findings

Cohen and random reals belong to all definable partition regular families.

They are classified within Mahler's hierarchy, providing information about Liouville numbers.

They are 'wild' in o-minimality, capable of defining the set of integers.

Abstract

We discuss some properties of Cohen and random reals. We show that they belong to any definable partition regular family, and hence they satisfy most "largeness" properties studied in Ramsey theory. We determine their position in the Mahler's classification of the reals and using it, we get some information about Liouville numbers. We also show that they are wild in the sense of o-minimality, i.e., they define the set of integers.