On pairs of definable orthogonal families

TL;DR

This paper introduces the concept of M-families of infinite subsets of natural numbers, exploring the structure of orthogonal hereditary families with specific measurability and definability properties.

Contribution

It defines M-families within the context of orthogonal hereditary families and analyzes their structural properties, extending Mathias's implicit work.

Findings

Characterization of M-families in the context of orthogonal families

Structural insights into analytic and C-measurable hereditary families

Extension of Mathias's concepts to new classes of families

Abstract

We introduce the notion of an M-family of infinite subsets of $\nn$ which is implicitly contained in the work of A. R. D. Mathias. We study the structure of a pair of orthogonal hereditary families $\aaa$ and $\bbb$, where $\aaa$ is analytic and $\bbb$ is $C$-measurable and an M-family.