SACCHARINITY WITH ccc

Abstract Using creature technology, we construct families of Suslin ccc non-sweet forcing notions Q $\mathbb Q$ double struck upper Q such that Z F C $ZFC$ upper Z upper F upper C is equiconsistent with Z F + $ZF+$ upper Z upper F plus “Every set of reals equals a Borel set modulo the ( ≤ ℵ 1 ) $(\leq \aleph _1)$ left parenthesis less than or equals normal first transfinite cardinal 1 right parenthesis -closure of the null ideal associated with Q $\mathbb Q$ double struck upper Q ” + $+$ plus “There is an ω 1 $\omega _1$ 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 Π 2 1 $\Pi ^1_2$ normal upper Pi 2 Superscript 1 singleton over L without relying on L