MAD Families of Projections on l^2 and Real-Valued Functions on Omega

TL;DR

This paper explores the size of maximal almost disjoint families of projections on l^2 and real-valued functions on omega, demonstrating they can be smaller than the continuum under certain set-theoretic assumptions.

Contribution

It introduces and analyzes analogs of classical almost disjoint families for projections on l^2 and real-valued functions, extending known results to new mathematical objects.

Findings

Maximal almost disjoint families of projections on l^2 can be smaller than continuum.

Maximal almost disjoint families of real-valued functions on omega can be smaller than continuum.

Consistent set-theoretic models show these sizes can vary independently of the continuum.

Abstract

Two sets are said to be almost disjoint if their intersection is finite. Almost disjoint subsets of [omega]^omega and omega^omega have been studied for quite some time. In particular, the cardinal invariants a and a_e, defined to be the minimum cardinality of a maximal infinite almost disjoint family of [omega]^omega and omega^omega respectively, are known to be consistently less than continuum. Here we examine analogs for functions in R^omega and projections on l^2, showing that they too can be consistently less than continuum.