Failure of Higher-Order Truth within Intuitionistic Propositional Logic

Abstract: 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 the free Heyting algebra F₂ on two generators cannot be the lattice of subterminal objects of any elementary topos which using higher-order quantification.
• The proof constructs an upward-closed subset A that is not definable within F₂, demonstrating a higher-order obstruction to its realizability.
• Internal reachability predicates in the logic of an elementary topos contradict the assumption that A is realizable from F₂. This highlights the role of higher-order quantification in creating new definability inaccessible to Heyting algebra terms alone.

Problem and principal result

The paper addresses a categorical-logical realization problem: which Heyting algebras can occur as the lattice of subterminal objects of an elementary topos? For an elementary topos $\mathcal E$, the Heyting algebra of truth values is

$$\operatorname{Sub}_{\mathcal E}(1)\cong \Gamma(\Omega),$$

where $1$ is the terminal object, $\Omega$ is the subobject classifier, and $\Gamma(\Omega)$ denotes its global sections. The question is whether every Heyting algebra can be obtained in this way.

The paper gives a negative answer by proving that the free Heyting algebra $F_2$ on two generators cannot be isomorphic to $\operatorname{Sub}_{\mathcal E}(1)$ for any elementary topos $\mathcal E$ (2608.26874). Consequently, higher-order intuitionistic logic imposes structure on global truth values that is not captured by the abstract Heyting-algebra operations alone. The obstruction is not exhibited through a finite algebraic identity; instead, it arises from a particular infinitary pattern that can be defined internally using higher-order quantification but cannot be represented by any propositional formula of $F_2$.

The proof combines Bellissima’s Kripke representation of finitely generated free Heyting algebras with an internal higher-order construction in an assumed elementary topos. The central strategy is to construct an upward-closed subset $A$ of Bellissima’s universal Kripke model $$\operatorname{Sub}_{\mathcal E}(1)\cong \Gamma(\Omega),$$0 such that $$\operatorname{Sub}_{\mathcal E}(1)\cong \Gamma(\Omega),$$1, and then show that, under the assumption $$\operatorname{Sub}_{\mathcal E}(1)\cong \Gamma(\Omega),$$2, the same subset is represented by a global higher-order proposition $$\operatorname{Sub}_{\mathcal E}(1)\cong \Gamma(\Omega),$$3. This yields the contradiction $$\operatorname{Sub}_{\mathcal E}(1)\cong \Gamma(\Omega),$$4 and $$\operatorname{Sub}_{\mathcal E}(1)\cong \Gamma(\Omega),$$5.

Bellissima’s representation of the free Heyting algebra

The paper realizes $$\operatorname{Sub}_{\mathcal E}(1)\cong \Gamma(\Omega),$$6 as a sub-Heyting algebra of the lattice $$\operatorname{Sub}_{\mathcal E}(1)\cong \Gamma(\Omega),$$7 of upward-closed subsets of a universal image-finite Kripke model $$\operatorname{Sub}_{\mathcal E}(1)\cong \Gamma(\Omega),$$8. Points of $$\operatorname{Sub}_{\mathcal E}(1)\cong \Gamma(\Omega),$$9 encode theories of finite reduced Kripke models of intuitionistic propositional logic with $1$0 propositional generators. The ordering is the Kripke accessibility relation, and each formula $1$1 is interpreted as the upward-closed set

$1$2

Bellissima’s theorem gives an embedding

$1$3

The model is constructed by a rank filtration

$1$4

At each stage, new points $1$5 are added from an atomic valuation $1$6 and an upward-closed subset $1$7 of the preceding stage. The ordering is determined by membership in $1$8, so that $1$9 lies below precisely the points selected by $\Omega$0, subject to the stated reduction conditions. Each $\Omega$1 is finite and has rank $\Omega$2, while $\Omega$3 is image-finite: every principal upset is finite.

This representation is important for two reasons. First, it provides an explicit combinatorial environment in which non-definable upward-closed subsets can be constructed. Second, every principal upset and coprincipal upset generated by a point of $\Omega$4 is definable by a formula of $\Omega$5. If $\Omega$6, the paper denotes formulas defining these sets by $\Omega$7 and $\Omega$8, with

$\Omega$9

according to the paper’s ordering conventions. This definability later allows finite portions of the Kripke model to be referenced inside the internal language of $\Gamma(\Omega)$0.

The paper also places Bellissima’s model in the context of profinite Heyting algebras. The image-finite poset $\Gamma(\Omega)$1 can be viewed as the dual object corresponding to the profinite completion of $\Gamma(\Omega)$2, and the embedding of $\Gamma(\Omega)$3 into $\Gamma(\Omega)$4 is the canonical embedding into that completion. This perspective clarifies why $\Gamma(\Omega)$5 contains more elements than the free Heyting algebra itself: it is a completion-like object, whereas $\Gamma(\Omega)$6 consists only of the finitely generated propositional truth values.

The explicit non-propositional obstruction

The obstruction is constructed inside $\Gamma(\Omega)$7 using three recursively defined families of points:

$\Gamma(\Omega)$8

and

$\Gamma(\Omega)$9

The three points at each level are pairwise incomparable. The paper then defines

$F_2$0

and proves that the sequence $F_2$1 is an antichain. The desired upward-closed subset is

$F_2$2

The antichain structure is essential. Each $F_2$3 generates one component of $F_2$4, but no point of the sequence lies below another. Thus $F_2$5 contains infinitely many mutually incompatible distinguished points.

The paper proves that $F_2$6. The argument uses the representation of elements of a free Heyting algebra as finite joins of join-irreducible formulas. If $F_2$7 were in $F_2$8, then it could be written as a finite join

$F_2$9

of join-irreducible elements. The cited structural results imply that each $\operatorname{Sub}_{\mathcal E}(1)$0 is downward filtered: whenever it contains two points, it contains a common predecessor. Since the finite family $\operatorname{Sub}_{\mathcal E}(1)$1 covers the infinitely many points $\operatorname{Sub}_{\mathcal E}(1)$2, one $\operatorname{Sub}_{\mathcal E}(1)$3 must contain two distinct members $\operatorname{Sub}_{\mathcal E}(1)$4 and $\operatorname{Sub}_{\mathcal E}(1)$5. Downward filteredness then gives a point below both. The recursive construction forces some $\operatorname{Sub}_{\mathcal E}(1)$6 below that common predecessor, contradicting the antichain property.

This establishes a precise separation:

• $\operatorname{Sub}_{\mathcal E}(1)$7 is an upward-closed subset of $\operatorname{Sub}_{\mathcal E}(1)$8;
• $\operatorname{Sub}_{\mathcal E}(1)$9 is not the interpretation of any formula in the free Heyting algebra $\mathcal E$0;
• nevertheless, $\mathcal E$1 has a regular recursive description that can be expressed using higher-order quantification.

The implication is that the failure is not merely a consequence of the size or incompleteness of $\mathcal E$2. The subset $\mathcal E$3 is generated by a uniform finite-state recurrence, but its infinite reachability closure cannot be captured by finite propositional syntax.

Internal higher-order realization

Assume, for contradiction, that there exists an elementary topos $\mathcal E$4 with

$\mathcal E$5

The paper identifies the propositional generators of $\mathcal E$6 with global truth values in $\mathcal E$7. Since each point $\mathcal E$8 has definable principal and coprincipal sets, the corresponding formulas become global sections of $\mathcal E$9. Write

$F_2$0

and similarly for $F_2$1.

The recursive definition of the Kripke points induces uniform intuitionistic relations among these six truth values. For example, the successor relations are represented by formulas of the form

$F_2$2

with analogous equations for $F_2$3 and $F_2$4. The coprincipal terms satisfy corresponding relations such as

$F_2$5

The six components are packaged into an object

$F_2$6

with global points

$F_2$7

The recursive equations define an internal endomorphism-like predicate $F_2$8 satisfying $F_2$9. The paper then defines an internal reachability predicate by quantifying over all predicates $A$0 that contain $A$1 and are closed under $A$2:

$A$3

This is a constructive, impredicative characterization of membership in the least $A$4-invariant predicate containing $A$5. It does not assert the existence of a natural-number witness or rely on an externally supplied induction principle. Instead, it defines reachability as intersection of all internally inductive predicates.

The terms associated with the points $A$6 define another predicate $A$7, with

$A$8

The crucial global truth value is then

$A$9

Externally, each $$\operatorname{Sub}_{\mathcal E}(1)\cong \Gamma(\Omega),$$00 is reachable from $$\operatorname{Sub}_{\mathcal E}(1)\cong \Gamma(\Omega),$$01, so every $$\operatorname{Sub}_{\mathcal E}(1)\cong \Gamma(\Omega),$$02 implies $$\operatorname{Sub}_{\mathcal E}(1)\cong \Gamma(\Omega),$$03. Therefore,

$$\operatorname{Sub}_{\mathcal E}(1)\cong \Gamma(\Omega),$$04

This is the easy inclusion. Its significance is that higher-order quantification has formed the closure of the recursively generated sequence, even though the resulting union is not a member of the original propositional algebra.

Finite approximations and the reverse inclusion

The difficult direction is $$\operatorname{Sub}_{\mathcal E}(1)\cong \Gamma(\Omega),$$05. The paper obtains it by exploiting the finite filtration of $$\operatorname{Sub}_{\mathcal E}(1)\cong \Gamma(\Omega),$$06.

For each rank $$\operatorname{Sub}_{\mathcal E}(1)\cong \Gamma(\Omega),$$07, the finite submodel $$\operatorname{Sub}_{\mathcal E}(1)\cong \Gamma(\Omega),$$08 is definable by some formula $$\operatorname{Sub}_{\mathcal E}(1)\cong \Gamma(\Omega),$$09. This formula is used internally to define an equivalence relation on $$\operatorname{Sub}_{\mathcal E}(1)\cong \Gamma(\Omega),$$10:

$$\operatorname{Sub}_{\mathcal E}(1)\cong \Gamma(\Omega),$$11

Intuitively, two elements are equivalent at level $$\operatorname{Sub}_{\mathcal E}(1)\cong \Gamma(\Omega),$$12 if they agree when restricted to the finite approximation represented by $$\operatorname{Sub}_{\mathcal E}(1)\cong \Gamma(\Omega),$$13. Since $$\operatorname{Sub}_{\mathcal E}(1)\cong \Gamma(\Omega),$$14 is finite, the sequence $$\operatorname{Sub}_{\mathcal E}(1)\cong \Gamma(\Omega),$$15 has only finitely many $$\operatorname{Sub}_{\mathcal E}(1)\cong \Gamma(\Omega),$$16-classes. Choosing representatives yields an internally definable predicate $$\operatorname{Sub}_{\mathcal E}(1)\cong \Gamma(\Omega),$$17 asserting that $$\operatorname{Sub}_{\mathcal E}(1)\cong \Gamma(\Omega),$$18 agrees, under $$\operatorname{Sub}_{\mathcal E}(1)\cong \Gamma(\Omega),$$19, with one of these finitely many representatives.

The paper proves that $$\operatorname{Sub}_{\mathcal E}(1)\cong \Gamma(\Omega),$$20 contains $$\operatorname{Sub}_{\mathcal E}(1)\cong \Gamma(\Omega),$$21 and is closed under $$\operatorname{Sub}_{\mathcal E}(1)\cong \Gamma(\Omega),$$22:

$$\operatorname{Sub}_{\mathcal E}(1)\cong \Gamma(\Omega),$$23

By the impredicative definition of reachability, this implies

$$\operatorname{Sub}_{\mathcal E}(1)\cong \Gamma(\Omega),$$24

This step is where the finiteness of the rank approximation enters constructively. No claim is made that the entire infinite sequence is periodic. Rather, only its behavior modulo the finite observational quotient induced by $$\operatorname{Sub}_{\mathcal E}(1)\cong \Gamma(\Omega),$$25 is represented by finitely many states.

Now fix any point $$\operatorname{Sub}_{\mathcal E}(1)\cong \Gamma(\Omega),$$26. Since $$\operatorname{Sub}_{\mathcal E}(1)\cong \Gamma(\Omega),$$27 is not the total subset of $$\operatorname{Sub}_{\mathcal E}(1)\cong \Gamma(\Omega),$$28, such points exist. Choose $$\operatorname{Sub}_{\mathcal E}(1)\cong \Gamma(\Omega),$$29 large enough that $$\operatorname{Sub}_{\mathcal E}(1)\cong \Gamma(\Omega),$$30. The relation $$\operatorname{Sub}_{\mathcal E}(1)\cong \Gamma(\Omega),$$31 ensures that equality under $$\operatorname{Sub}_{\mathcal E}(1)\cong \Gamma(\Omega),$$32 implies equality under $$\operatorname{Sub}_{\mathcal E}(1)\cong \Gamma(\Omega),$$33. Because no $$\operatorname{Sub}_{\mathcal E}(1)\cong \Gamma(\Omega),$$34 lies below $$\operatorname{Sub}_{\mathcal E}(1)\cong \Gamma(\Omega),$$35, every $$\operatorname{Sub}_{\mathcal E}(1)\cong \Gamma(\Omega),$$36 is contained in the coprincipal definable set $$\operatorname{Sub}_{\mathcal E}(1)\cong \Gamma(\Omega),$$37.

Suppose internally that $$\operatorname{Sub}_{\mathcal E}(1)\cong \Gamma(\Omega),$$38 holds. By the definition of $$\operatorname{Sub}_{\mathcal E}(1)\cong \Gamma(\Omega),$$39, $$\operatorname{Sub}_{\mathcal E}(1)\cong \Gamma(\Omega),$$40 agrees with one of the representative points $$\operatorname{Sub}_{\mathcal E}(1)\cong \Gamma(\Omega),$$41 under $$\operatorname{Sub}_{\mathcal E}(1)\cong \Gamma(\Omega),$$42, and hence under $$\operatorname{Sub}_{\mathcal E}(1)\cong \Gamma(\Omega),$$43. The corresponding $$\operatorname{Sub}_{\mathcal E}(1)\cong \Gamma(\Omega),$$44 implies $$\operatorname{Sub}_{\mathcal E}(1)\cong \Gamma(\Omega),$$45. Since $$\operatorname{Sub}_{\mathcal E}(1)\cong \Gamma(\Omega),$$46 itself has the form of an implication out of $$\operatorname{Sub}_{\mathcal E}(1)\cong \Gamma(\Omega),$$47, the paper derives that $$\operatorname{Sub}_{\mathcal E}(1)\cong \Gamma(\Omega),$$48 implies $$\operatorname{Sub}_{\mathcal E}(1)\cong \Gamma(\Omega),$$49 under the relevant finite approximation. Consequently,

$$\operatorname{Sub}_{\mathcal E}(1)\cong \Gamma(\Omega),$$50

for every $$\operatorname{Sub}_{\mathcal E}(1)\cong \Gamma(\Omega),$$51.

Finally, because $$\operatorname{Sub}_{\mathcal E}(1)\cong \Gamma(\Omega),$$52 is upward-closed,

$$\operatorname{Sub}_{\mathcal E}(1)\cong \Gamma(\Omega),$$53

Each coprincipal set $$\operatorname{Sub}_{\mathcal E}(1)\cong \Gamma(\Omega),$$54 is interpreted by $$\operatorname{Sub}_{\mathcal E}(1)\cong \Gamma(\Omega),$$55, so the preceding argument gives $$\operatorname{Sub}_{\mathcal E}(1)\cong \Gamma(\Omega),$$56. Together with the previously established $$\operatorname{Sub}_{\mathcal E}(1)\cong \Gamma(\Omega),$$57, this proves

$$\operatorname{Sub}_{\mathcal E}(1)\cong \Gamma(\Omega),$$58

But $$\operatorname{Sub}_{\mathcal E}(1)\cong \Gamma(\Omega),$$59 is a global proposition of $$\operatorname{Sub}_{\mathcal E}(1)\cong \Gamma(\Omega),$$60, hence belongs to $$\operatorname{Sub}_{\mathcal E}(1)\cong \Gamma(\Omega),$$61, whereas Proposition 3.2 establishes $$\operatorname{Sub}_{\mathcal E}(1)\cong \Gamma(\Omega),$$62. The contradiction proves the main theorem.

Consequences for higher-order truth

The result shows that the lattice of subterminal objects of an elementary topos cannot, in general, be treated as an arbitrary Heyting algebra equipped only with finite intuitionistic operations. The internal higher-order language can define predicates obtained through impredicative closure and quantification over predicates. In the constructed case, this produces the truth value $$\operatorname{Sub}_{\mathcal E}(1)\cong \Gamma(\Omega),$$63, whose external interpretation is the non-propositional upset $$\operatorname{Sub}_{\mathcal E}(1)\cong \Gamma(\Omega),$$64.

The paper therefore establishes a concrete form of failure of “higher-order truth” within intuitionistic propositional logic: propositional truth values may form $$\operatorname{Sub}_{\mathcal E}(1)\cong \Gamma(\Omega),$$65 algebraically, while higher-order quantification necessarily generates an additional global truth value. The contradiction does not depend on classical principles, excluded middle, or an external natural-number object. The reachability construction is formulated in the internal intuitionistic logic of an arbitrary elementary topos satisfying the assumed isomorphism.

A further consequence is the stated quotient obstruction. If a Heyting algebra $$\operatorname{Sub}_{\mathcal E}(1)\cong \Gamma(\Omega),$$66 admits a surjective homomorphism

$$\operatorname{Sub}_{\mathcal E}(1)\cong \Gamma(\Omega),$$67

then $$\operatorname{Sub}_{\mathcal E}(1)\cong \Gamma(\Omega),$$68 cannot be the lattice of subterminal objects of an elementary topos. Indeed, the class of topos-realizable Heyting algebras is closed under neither arbitrary surjective presentation nor the presence of $$\operatorname{Sub}_{\mathcal E}(1)\cong \Gamma(\Omega),$$69 as a homomorphic quotient. This extends the negative result beyond $$\operatorname{Sub}_{\mathcal E}(1)\cong \Gamma(\Omega),$$70 itself.

The result contrasts with positive realization theorems for important subclasses, including complete Heyting algebras and Boolean algebras, as well as the class of étale-finite Heyting algebras studied in related work (Abbadini et al., 2 Jun 2026). The paper thus isolates a specific obstruction within the general realization problem rather than classifying all realizable Heyting algebras.

Limitations and open questions

The argument treats only the free Heyting algebra on two generators. It does not establish whether $$\operatorname{Sub}_{\mathcal E}(1)\cong \Gamma(\Omega),$$71 fails to be realizable for every $$\operatorname{Sub}_{\mathcal E}(1)\cong \Gamma(\Omega),$$72, although the construction strongly depends on the availability of two independent propositional generators. Nor does it characterize the exact algebraic or model-theoretic property distinguishing realizable Heyting algebras from non-realizable ones.

The proof also relies substantially on structural results about join-irreducibles and Bellissima’s representation, including the downward-filteredness of join-irreducible elements in the chosen Kripke model. The separation between $$\operatorname{Sub}_{\mathcal E}(1)\cong \Gamma(\Omega),$$73 and $$\operatorname{Sub}_{\mathcal E}(1)\cong \Gamma(\Omega),$$74 is therefore representation-sensitive in its execution, even though the final non-realizability statement is categorical.

The internal reachability predicate uses quantification over the exponential $$\operatorname{Sub}_{\mathcal E}(1)\cong \Gamma(\Omega),$$75. The construction consequently depends on the full higher-order structure available in an elementary topos. An open technical question is whether analogous obstructions can be formulated using weaker fragments of higher-order intuitionistic logic, or whether the impredicative quantification over predicates is essential to the failure.

Finally, the paper does not determine whether the non-realizability phenomenon can be detected by a purely algebraic condition on a Heyting algebra. Its proof constructs a particular higher-order definability obstruction rather than extracting a finite equational or quasi-equational criterion.

Conclusion

The paper proves that the free Heyting algebra $$\operatorname{Sub}_{\mathcal E}(1)\cong \Gamma(\Omega),$$76 on two generators is not the lattice of subterminal objects of any elementary topos (2608.26874). The proof constructs an antichain-generated upset $$\operatorname{Sub}_{\mathcal E}(1)\cong \Gamma(\Omega),$$77 outside $$\operatorname{Sub}_{\mathcal E}(1)\cong \Gamma(\Omega),$$78, then defines an internal higher-order proposition $$\operatorname{Sub}_{\mathcal E}(1)\cong \Gamma(\Omega),$$79 whose reachability semantics forces $$\operatorname{Sub}_{\mathcal E}(1)\cong \Gamma(\Omega),$$80. This contradiction demonstrates that higher-order intuitionistic truth carries definability structure beyond the underlying Heyting algebra and supplies a concrete negative instance for the general topos-realization problem.