Forcing-Invariant Elementary Equivalence Models

Definitions and mathematical statement

Let $N\models\mathrm{ZFC}$. For each cardinal $\lambda$ computed in $N$, let $P_\lambda=\operatorname{Col}(\omega,\lambda)^N$ be the intended collapse forcing. Under the usual convention, conditions are finite partial functions from $\omega$ to $\lambda$, ordered by extension, and a generic union makes $\lambda$ countable. The exact collapse convention must be checked in the source. An $N$-generic filter meets every dense subset of $P_\lambda$ belonging to $N$.

The full question asks for a model $N$ such that \[ \forall\lambda\in\operatorname{Card}^N\ \forall H\subseteq P_\lambda \text{ generic over }N,\qquad (N[H],\in)\equiv(N,\in). \] Elementary equivalence here means equality of parameter-free first-order theories in the language of set theory: \[ \forall\varphi\text{ a sentence},\qquad N\models\varphi\ \Longleftrightarrow\ N[H]\models\varphi. \] It does not require $N\prec N[H]$, agreement for formulas with arbitrary ground-model parameters, or an elementary embedding between them.

The restricted question asks the same existence problem with the quantifier over $\lambda$ limited to cardinals that $N$ regards as regular. This is the usual target-regular interpretation of "only when $H$ collapses a regular cardinal." Whether another forcing restriction is intended must be confirmed. The restricted property is weaker than the full one.

Generic filters are understood in suitable outer universes, so the property is not made vacuous by a universe containing no such filter. Equivalently, one can express the desired preservation as an external scheme: for each parameter-free sentence $\varphi$ and each allowed $\lambda$, forcing with $P_\lambda$ preserves its truth value in $N$. This avoids presupposing an internal ZFC truth predicate. The record does not require $N$ to be transitive, countable, or a proper-class model; those choices matter to a precise metamathematical existence or relative-consistency statement.

The supplied assessment dated 2026-08-21 attributes to Golshani--Mitchell equiconsistency results for weaker forms and an upper consistency bound for the full Hamkins--Lowe property from a Woodin cardinal with an inaccessible cardinal above it. In relative-consistency language the reported direction is from consistency of that large-cardinal hypothesis to consistency of the full property, not the converse. The exact weaker forms and the optimal consistency strength are not supplied as settled.

Short English statement

Can a model of set theory have the same parameter-free theory after every collapse to countability, or at least after those targeted at its regular cardinals? The problem concerns elementary equivalence, with a reported large-cardinal consistency upper bound and unresolved exact strength.

Sources

• Local UnsolvedMath registry, permanent ID $30000906$, exact source code catalogue entry, proposed year 2008; no individual proposer is supplied.
• Set Theory (2008), Oberwolfach Reports, DOI 10.4171/owr/2008/02, pp. 79--118. https://doi.org/10.4171/owr/2008/02.
• Supplied reference: Golshani--Mitchell, https://arxiv.org/abs/1609.02633. The supplied assessment calls the full condition the Hamkins--Lowe property.
• The corrected_verified note reports editorial and LaTeX cleanup without added source-dependent content. The partially solved label and assessment checked 2026-08-21 are dated supplied triage requiring independent verification of precise weaker forms, consistency bounds, and current status.