Generalizing random real forcing for inaccessible cardinals

TL;DR

This paper develops a new forcing notion for inaccessible cardinals that generalizes random real forcing, preserving cardinals and cofinalities without adding undominated reals, extending the analogy between measure and forcing concepts.

Contribution

It introduces a forcing for inaccessible cardinals that generalizes random real forcing, filling a gap in the theory for uncountable cardinals beyond weakly compact ones.

Findings

The forcing preserves cardinals and cofinalities.

It does not add undominated reals.

It extends the analogy between measure and forcing for uncountable cardinals.

Abstract

The two parallel concepts of "small" sets of the real line are meagre sets and null sets. Those are equivalent to Cohen forcing and Random real forcing for $\aleph^{\aleph_0}_0$; in spite of this similarity, the Cohen forcing and Random Real forcing have very different shapes. One of these differences is in the fact that the Cohen forcing has an easy natural generalization for $\lambda^{\lambda}$ while $\lambda > \aleph_0$, corresponding to an extension for the meagre sets, while the Random real forcing didn't see to have a natural generalization, as Lebesgue measure doesn't have a generalization for space $\lambda^\lambda$ while $\lambda > \aleph_0$. Shelah found a forcing resembling the properties of Random Real Forcing for $\lambda^\lambda$ while $\lambda$ is a weakly compact cardinal. Here we describe, with additional assumptions, such a forcing for $\lambda^\lambda$ while $\lambda$ is an inaccessible cardinal; this forcing preserves cardinals and cofinalities, however unlike Cohen forcing, does not add undominated reals.