A model with a precipitous ideal, but no normal precipitous ideal (Q2853978)

\textit{T. Jech} and \textit{K. Prikry} [Bull. Am. Math. Soc. 82, 593--595 (1976; Zbl 0339.02060)] introduced the notion of precipitous ideal as a way to construct generic elementary embeddings of \( \mathrm{V} \) into a transitive class \( M \). More precisely, if \( I \) is a precipitous ideal on \( \kappa \) and \( G \subseteq \mathscr{P} ( \kappa ) \), then in \( \mathrm{V} [ G ] \) there is an elementary embedding \( j \colon \mathrm{V} \to M \) with critical point \( \kappa \). The existence of a precipitous ideal is equiconsistent with the existence of a measurable cardinal, but since \( j \) lives in \( \mathrm{V} [ G ] \) but not in \(\mathrm{V} \), the cardinal \( \kappa \) need not be large at all. In fact, it can be quite small, for example \( \omega_1 \). In the following four decades, precipitous ideals (and their stronger siblings: saturated ideals) became a standard technique in the set-theorist's toolbox. Despite being studied for such a long time, a basic question about precipitousness remained open: does the existence of a precipitous ideal imply the existence of a normal one? The present paper is the culmination of a series of papers (some of which by the author) and shows that the answer is negative: assuming the existence of a measurable cardinal \( \kappa \) with Mitchell order \( \kappa ^{++} \) there is a model where \( \omega_1 \) carries a precipitous ideal, but there are no normal precipitous ideals.NEWLINENEWLINE The paper is clearly written, with a clear notation, despite the proofs being very intricate and difficult.