Saccharinity with ccc

TL;DR

Using creature technology, families of Suslin ccc non-sweet forcing notions are constructed such that $ZFC$ is equiconsistent with $ZF+$ and every set of reals equals a Borel set modulo the null ideal associated with Q.

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 $(\leq \aleph_1)$-closure of the null ideal associated with $\mathbb Q$"+"there is an $ω_1$-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 $Π^1_2$ singleton over $L$ without relying on $L$-combinatorics.