Failure of Higher-Order Truth within Intuitionistic Propositional Logic

We answer the question whether all Heyting algebras can appear as the lattice of subterminal objects of an elementary topos in the negative. Concretely, we have shown that the free Heyting algebra on two generators cannot be such a Heyting algebra. The mathematical results in this document were obtained with the help of ChatGPT 5.6 Sol, although the document itself was written entirely by us and we take full responsibility for its contents.

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.