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TL;DR

This paper proves the existence of a specific sigma-ideal on the reals with certain properties that cannot be decomposed into an increasing union of smaller sigma-subideals, answering a previously open question.

Contribution

It establishes the consistency of a sigma-ideal with additivity omega_1 that defies representation as an increasing union of sigma-subideals, resolving an open problem.

Findings

Existence of a sigma-ideal with additivity omega_1

Such an ideal cannot be expressed as an increasing union of sigma-subideals

Answers a question posed by Borodulin-Nadzieja and Glab

Abstract

We show the consistency of "there is a nice sigma --ideal I on the reals with add(I)= omega_1 which cannot be represented as the union of a strictly increasing sequence of length omega_1 of sigma-subideals". This answers a question by Borodulin-Nadzieja and Glab.