Ground Axiom Consequences for Cardinal Arithmetic

Definitions and mathematical statement

A ground of a set-theoretic universe $V$ is an inner model $W\models\mathrm{ZFC}$, with the same ordinals as $V$, such that $V=W[G]$ for a set forcing $\mathbb P\in W$ and a $W$-generic filter $G\subseteq\mathbb P$. The Ground Axiom $\mathrm{GA}$ asserts that there is no proper such ground: \[ V=W[G]\ \Longrightarrow\ W=V \] for every ground presentation of this kind. The restriction is to set forcing, not arbitrary class forcing. The generalized continuum hypothesis is \[ \mathrm{GCH}:\qquad\forall\kappa\text{ an infinite cardinal},\quad 2^\kappa=\kappa^+. \] The explicit implication question in the source is whether \[ \mathrm{ZFC}+\mathrm{GA}\ \models\ \mathrm{GCH}. \] The source also asks which "regularity features" exhibited by its standard models are consequences of GA alone. Each such feature would require a specified sentence or scheme $\mathcal R$ and the corresponding question $\mathrm{ZFC}+\mathrm{GA}\models\mathcal R$. No features or model construction are given, so this second part is a programme of implication questions, not one fully defined formula. The word "or" is retained as presenting alternative consequences to investigate, not silently converted into the weaker single disjunction $\mathrm{GA}\Rightarrow(\mathrm{GCH}\lor\mathcal R)$.

The supplied assessment checked 2026-08-21 says that a Ground Axiom paper gives consistency and independence results for surrounding assertions, but reports the listed-consequence question unsettled in its cited literature. It does not reproduce those results or identify the unnamed regularity features. The open label is therefore a supplied, time-sensitive assessment requiring independent verification, not a newly established status for the precisely displayed GCH implication.

Short English statement

Ask whether the absence of proper set-forcing grounds forces GCH, and which other properties of the source's constructed GA models follow from GA itself. The latter properties are not listed, and the supplied open-status assessment is not independently verified.

Sources

• Local UnsolvedMath registry, permanent ID 30000381, catalogue entry, Ground Axiom Consequences for Cardinal Arithmetic.
• Set Theory (2006), Oberwolfach Reports, pp. 3121--3174, DOI https://doi.org/10.4171/owr/2005/55.
• Supplied reference labeled "set theoretic geology": https://arxiv.org/abs/1107.4776.
• Supplied literature triage checked 2026-08-21 refers additionally to a "Ground Axiom paper" but does not give a separate precise citation or theorem statement for that phrase.