A parallel to the null ideal for inaccessible lambda. Part I

TL;DR

This paper explores a generalization of the null ideal for inaccessible lambda, establishing the existence of a corresponding forcing notion and Boolean algebra, and deriving related consistency results for cardinal invariants.

Contribution

It introduces a new forcing-like construction and Boolean algebra for inaccessible lambda, extending properties of the null ideal and random real forcing to larger cardinals.

Findings

Existence of a forcing notion similar to random real forcing for inaccessible lambda.

Boolean algebra of lambda-Borel sets modulo the ideal exists for weakly compact cardinals.

Derived consistency results for cardinal invariants at these large cardinals.

Abstract

It is well known to generalize the meagre ideal replacing aleph_0 by a (regular) cardinal lambda > aleph_0 and requiring the ideal to be lambda^+-complete. But can we generalize the null ideal? In terms of forcing, this means finding a forcing notion similar to the random real forcing, replacing aleph_0 by lambda, so requiring it to be (<lambda)-complete. Of course, we would welcome additional properties generalizing the ones of the random real forcing. Returning to the ideal (instead forcing) we may look at the Boolean Algebra of lambda-Borel sets modulo the ideal. Surprisingly we get an positive = existence answer for lambda a "mild" large cardinals: the weakly compact ones. We apply this to get consistency results on cardinal invariants for such lambda's. We shall deal with other cardinals more properties related forcing notions in a continuation.