Open Problems in Mathematical Logic — Problem 18

Definitions and mathematical statement

Let $M$ be a model and let $\mathbb P\in M$ be a forcing notion, as stipulated in the supplied record. The question is the incomplete implication schema \[ \text{Condition }(*)\text{ holds} \quad\stackrel{?}{\Longrightarrow}\quad M\models\text{``there exists a large cardinal''}. \] Here $M\models\varphi$ means that the sentence $\varphi$ holds in $M$. However, the supplied record does not state condition $(*)$, specify the theory or structural requirements for $M$, or identify the large-cardinal assertion intended in the conclusion. The quoted conclusion therefore does not specify a fixed sentence. This schema preserves the supplied question, but these missing qualifications prevent a complete mathematical formulation.

Short English statement

For a model $M$ containing a forcing notion $\mathbb P$, does condition $(*)$ imply that $M$ satisfies a large-cardinal existence assertion? The condition, requirements on the model, and intended large-cardinal assertion are missing from the supplied record.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $801006$, source problem code catalogue entry, title Open Problems in Mathematical Logic --- Problem 18. Supplied upstream revision: d371ee917ac9647fbb06cd6fc529deec6f3dda82.
• Supplied source-list attribution: Barmpalias, Gao et al., Open Problems in Mathematical Logic (2026), Problem 18, https://arxiv.org/abs/2608.26628. The registry also gives Barmpalias, Gao et al. as the proposers and 2026 as the proposed year; this attribution has not been independently verified.
• Supplied provenance: accessed 2026-10-16; extraction method arxiv_latex. These are supplied provenance statements, not a claim of access or verification for this draft.