Discoveries / logic-foundations

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.