Failure of Higher-Order Truth within Intuitionistic Propositional Logic

The paper proves that not every Heyting algebra can occur as the lattice of subterminal objects of an elementary topos. Specifically, the free Heyting algebra $F_2$ on two generators cannot occur. Using Bellissima’s representation $F_2\hookrightarrow\mathcal O_\uparrow(K_2)$ , the authors construct an upward-closed subset $A\subseteq K_2$ with $A\notin F_2$ . They show that if some elementary topos $\mathcal E$ satisfied $\operatorname{Sub}_{\mathcal E}(1)\cong F_2$ , then higher-order internal logic would make $A$ definable as a global proposition, forcing $A$ to correspond to an element of $F_2$ , a contradiction. Thus no elementary topos has subterminal lattice isomorphic to $F_2$ , disproving the claim that every Heyting algebra can arise this way. The paper does not classify which Heyting algebras are realizable.

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Occurred: Oct 29, 2026

Delta type: REGISTRY REVISION

Assumptions: VibeMathed verification: unreviewed. Publication: preprint. AI contribution: ai-assisted. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: Failure of Higher-Order Truth within Intuitionistic Propositional Logic · Higher-order truth in IPL

Confidence: Not scored

Registry verification: unreviewed · preprint · resolved

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VibeMathed
registry · event recorded by

ChatGPT 5.6 Sol
model · ai model contributor · OpenAI

Lingyuan Ye
human · human collaborator

Yiqi Xu
human · human collaborator

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VibeMathed record: Failure of Higher-Order Truth within Intuitionistic Propositional Logic

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Unreviewed: an arXiv preprint one day old (v1, 27 August 2026, math.CT), unrefereed, with no formalization, and the argument runs through Bellissima's representation and higher-order internal logic, none of which was checked here. Verified on 28 August 2026: the paper exists at arXiv:2608.26874 with this title and both authors; the statement and the negative answer are its abstract; the target is described in its introduction as "a long-standing problem in categorical logic" with Pitts cited for a recent summary and a related positive result of Awodey et al. cited alongside; and the proof strategy is as the entry describes it, constructing $A\in\mathcal O_\uparrow(K_2)\setminus F_2$ and deriving a contradiction from its definability as a global proposition. The single AI sentence in the abstract is the paper's only mention of a model.

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Failure of Higher-Order Truth within Intuitionistic Propositional Logic

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