Failure of Higher-Order Truth within Intuitionistic Propositional Logic

Yiqi Xu, Lingyuan Ye

The free Heyting algebra F₂ on two generators cannot be the lattice of subterminal objects of an elementary topos. The obstruction extends to any Heyting algebra surjecting onto F₂, including free F_N for N≥2.

Sources for this record

What was already known

Whether every Heyting algebra could arise as a topos subterminal lattice was open. Earlier positive results addressed a narrower logical setting.

AI involvement AI-centered

Ye proposed the F₂ target and strategy; Xu directed and simplified a parallel search. ChatGPT supplied the central reachability construction within that human direction.

The model supplied the central reachability-predicate and Bellissima argument, while humans proposed the target and strategy, steered the search and completed the final proof.

Importance Notable

Provides a counterexample to a longstanding categorical-logic representation question.

Limitations

Xu scrutinized model-generated steps and the authors rewrote the argument, but a completed rigorous check of the full final proof is not documented. The AI transcripts are unavailable.

Technical verification Partial

Xu reports adversarial scrutiny of the AI’s steps and the authors reviewed/edited the final content, but no completed rigorous check of the entire final proof is documented.

Independent checking

Xu reports adversarial scrutiny of the AI’s steps and the authors reviewed/edited the final content, but no completed rigorous check of the entire final proof is documented.

Checker
Yiqi Xu
Scope
model-generated steps toward the F₂ Bellissima–reachability obstruction
Basis
Reported by the linked checker
Evidence source

Human understanding Full

Xu explicitly disentangled and reorganized the reasoning and simplified its central Kripke-model and separation mechanism; the paper explains the obstruction.

Correction and publication history