Saccharinity with ccc

Abstract

Using creature technology, we construct families of Suslin ccc non-sweet forcing notions $\mathbb{Q}$ such that $ZFC$ is equiconsistent with $ZF+$"every set of reals equals a Borel set modulo the $(\le {\aleph }_1)$-closure of the null ideal associated with $\mathbb{Q}$"+"there is an ${\omega }_1$-sequence of distinct reals". This answers a question of the second author and Kellner. As an application of independent interest, we also show how our forcing adds a new ${\Pi }_2^1$ singleton over $L$ without relying on $L$-combinatorics.

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