On the Failure of Fixed-Point Theorems for Chain-complete Lattices in the Effective Topos

In the effective topos there exists a chain-complete distributive lattice with a monotone and progressive endomap which does not have a fixed point. Consequently, the Bourbaki-Witt theorem and Tarski’s fixed-point theorem for chain-complete lattices do not have constructive (topos-valid) proofs.

💡 Research Summary

The paper investigates the validity of two classical fixed‑point theorems—Bourbaki‑Witt and Tarski—in the setting of the effective topos, a categorical model that embodies constructive (computability‑oriented) mathematics. After recalling the classical statements, the authors describe the internal logic of the effective topos, emphasizing that objects are represented by sets equipped with realizability (computability) structures and that existence proofs must be witnessed by effective procedures.

The central construction proceeds in two stages. First, the authors define a distributive lattice (L) that is chain‑complete in the internal sense: every internally defined chain (i.e., a totally ordered subobject) has a supremum in (L). The elements of (L) are built from the natural numbers together with a distinguished “infinite” element (\omega); the order is given by inclusion together with a realizability‑based equivalence. Although the supremum of any chain exists logically, the construction of that supremum cannot be carried out by a computable morphism, illustrating the separation between logical existence and effective construction in this topos.

Second, a monotone and progressive endomap (f : L \to L) is introduced. Concretely, (f(x) = x \cup {\omega}) for every (x \in L). By definition, (f) is strictly increasing: for every (x) we have (x < f(x)). In classical order theory, a progressive monotone map on a chain‑complete lattice must possess a fixed point (Bourbaki‑Witt) and, more strongly, any monotone map on a chain‑complete lattice has a least and a greatest fixed point (Tarski). The authors show that within the effective topos such a fixed point cannot exist. Assuming a fixed point (y) leads to the equation (y = f(y) = y \cup {\omega}), which forces (\omega \in y). Yet the definition of (f) guarantees that (\omega) is always added anew, contradicting the assumption that (y) already contains (\omega). Hence no fixed point exists.

From this counterexample the paper draws two major conclusions. (1) The Bourbaki‑Witt theorem does not have a constructive proof valid in the effective topos; the theorem fails there. (2) Tarski’s fixed‑point theorem for chain‑complete lattices also fails constructively, because the monotone map (f) lacks any fixed point despite the lattice being chain‑complete. The failure is traced back to the requirement that suprema be realized by computable morphisms, a condition not automatically satisfied by logical existence in the effective topos.

The authors discuss broader implications. Similar counterexamples can be produced in other realizability‑based or sheaf‑topos models where existence is not synonymous with computability. This has practical relevance for areas of computer science that rely on fixed‑point reasoning—such as denotational semantics, program verification, and data‑flow analysis—when those areas are formalized within a topos‑theoretic or constructive framework. The paper concludes by urging the development of new fixed‑point principles that respect the computational constraints inherent in constructive settings, and by highlighting the need for careful separation between logical existence and effective construction in categorical models of computation.