---
title: Strictly Incompletable Sets
url: https://www.emergentmind.com/topics/strictly-incompletable-sets
type: topic
---

# Strictly Incompletable Sets

Searching arXiv for the cited works and adjacent material on “strictly incompletable sets” across logic, computability, combinatorics on words, and quantum information.
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Searching arXiv for exact matches to the primary sources.
{"query":"ti:\"Incompleteness for stably computable formal systems\"","max_results":5,"sort_by":"relevance","sort_order":"descending"}
“Strictly incompletable sets” is not a single standardized term across the arXiv literature. It instead names a family of completion-obstruction phenomena that arise in several technically distinct settings. In arithmetic and computability, the relevant objects are stably computably enumerable theories that extend \(PA\) yet, under 1-consistency or 2-consistency, necessarily omit a constructible true sentence and cannot prove their own 1-consistency [2208.04752]. In combinatorics on words, the phrase is not introduced as a formal notion, but it is naturally read through non-complete finite sets \(S\subseteq \Sigma^*\) whose incompleteness is witnessed only by long uncompletable words outside \(\operatorname{Fact}(S^*)\) [1002.1928, 1104.0388, 1612.07881]. In quantum information, the term is used more specifically for orthogonal product-state sets of UPB type that cannot be completed in the relevant sense and are locally indistinguishable [2607.00797]. In the theory of c.e. sets and in four-valued set theory, closely related notions appear as definable classes of necessarily incomplete c.e. sets and as sets with intrinsically gappy membership [1101.0228, 2210.00057]. This suggests a unifying theme: a “strictly incompletable” object is one whose admissible completion is blocked by the ambient notion of computability, factor coverage, locality, or classicality.

## 1. Stable computability and strict incompletability of arithmetic theories

The paper “Incompleteness for stably computable formal systems” defines a set \(S\subset B\) to be **stably computably enumerable** if there is a Turing machine \(M:\mathbb N\to B\times\{\pm\}\) such that \(S=M^s\), where \(M^s\) is the set of all \(M\)-stable elements, and \(b\in B\) is \(M\)-stable if there is an \(m\) with \(M(m)=(b,+)\) and there is no \(n>m\) with \(M(n)=(b,-)\). A typical stably c.e. set is the set of Diophantine polynomials with no integer roots; similarly, one can stably compute the set of non-halting pairs \((T,n)\). These sets are generally not c.e., but they are stably c.e. in the limit sense captured by stabilization [2208.04752].

Within this framework, a theory \(F\subseteq\mathcal A\) is stably c.e. if \(F=T^s\) for some Turing machine \(T:\mathbb N\to\mathcal A\times\{\pm\}\). The paper proves direct analogues of Gödel’s first and second incompleteness theorems for such \(F\). If \(F\) is stably c.e. and \(F\vdash PA\), there is a constructible sentence \(\alpha(F)\) such that \(F\nvdash \alpha(F)\) if \(F\) is 1-consistent, and \(F\nvdash\neg\alpha(F)\) if \(F\) is 2-consistent. Moreover, \(PA\) proves \((\text{$F$ is 1-consistent})\implies \alpha(F)\), so \(F\) cannot prove its own 1-consistency.

The significance is that the second incompleteness barrier now applies beyond classically c.e. theories. The paper explicitly treats this as a class of “necessarily incomplete” theories broader than the classical Gödel setting, including non-c.e. but stably c.e. theories arising from limit processes. It also formulates a sharper Gödel disjunction for the idealized mathematical output of humanity, \(\mathcal H\): if \(\mathcal H\vdash PA\), then either \(\mathcal H\) is not stably computably enumerable, or \(\mathcal H\) is not 1-consistent, or \(\mathcal H\) cannot prove a certain true arithmetic statement, and cannot refute it if it is 2-consistent. In this sense, “strict incompletability” names a Gödel-2-type obstruction for stably computable but non-c.e. formal systems.

## 2. Definable incompleteness among c.e. sets: \(n\)-tardy hierarchies

In computability theory, a related use of “strict incompletability” arises not through completion of formal theories, but through definable classes of c.e. sets all of whose members are incomplete. The paper “On n-Tardy Sets” studies **almost prompt**, **very tardy**, and **\(n\)-tardy** sets. A c.e. set \(A\) is \(n\)-tardy if for every nondecreasing computable function \(p\) there exists an \(n\)-c.e. approximation \(X_e^n=\overline A\) such that whenever \(x\in X_{e,s}^n\), one has \(x\notin A_{p(s)}\). A set is **properly \(n\)-tardy** if it is \(n\)-tardy but not \((n-1)\)-tardy [1101.0228].

Harrington and Soare had shown that there is a nonempty \(\mathcal E\)-property \(Q(A)\) such that \(Q(A)\) implies \(A\) is 2-tardy, and that no 2-tardy set is complete. The paper extends this by constructing nonempty \(\mathcal E\)-properties \(Q_n(A)\) such that \(Q_n(A)\) implies \(A\) is properly \(n\)-tardy. It also proves that there is a 3-tardy set \(A\) that is not computed by any 2-tardy set \(B\).

The orbit-theoretic importance is explicit. Since the \(Q_n\) are first-order definable in \(\mathcal E\), they determine automorphism-invariant classes of c.e. sets, and every set in such a class is incomplete. This yields a hierarchy of definable, disjoint orbits of incomplete c.e. sets. In the sense relevant here, these are “strictly incompletable” not because they fail factor completion or basis completion, but because incompleteness is forced by a robust structural slowness property and preserved across the relevant automorphism classes.

## 3. Non-complete sets of words and minimal uncompletable words

In combinatorics on words, the formal objects are **complete** and **incomplete** finite sets of words. For a finite alphabet \(\Sigma\), a finite set \(S\subseteq\Sigma^*\) is complete if \(\operatorname{Fact}(S^*)=\Sigma^*\), and incomplete otherwise. A word in \(\Sigma^*\setminus \operatorname{Fact}(S^*)\) is an **uncompletable word** for \(S\). The length of a shortest such word is
\[
uwl(S)=\min\{\,|x|:x\in \Sigma^*\setminus \operatorname{Fact}(S^*)\,\},
\]
and the extremal function
\[
UWL(k,\sigma)=\max\{\,uwl(S):S\subseteq \Sigma^{\le k},\ |\Sigma|=\sigma\,\}
\]
measures how long incompleteness can remain hidden when all words in \(S\) have length at most \(k\) [1002.1928].

These papers do not formalize the phrase “strictly incompletable set” as a separate definition. Their natural quantitative proxy is large \(uwl(S)\). The baseline construction \(S=\Sigma^k\setminus\{u\}\), with \(u\) unbordered, yields a shortest uncompletable word of length \(k^2+k-1\), so \(UWL(k)\ge k^2+k-1\). The same paper gives stronger lower-bound families: for \(5\le k\le 12\), \(UWL(k)\ge 2k^2-2k+1\), and for \(7\le k\le 12\), \(UWL(k)\ge 3k^2-9k+1\), thereby refuting the conjectured upper bound \(2k^2\) in the tested range [1002.1928].

The later paper “On Non-Complete Sets and Restivo’s Conjecture” strengthens this dramatically. It constructs an infinite family
\[
S_k=\big(E^k\setminus\{ba^{k-1},\,b^{k-1}a\}\big)\cup\big(E^{k-1}\setminus\{a^{k-1},\,b^{k-1}\}\big)
\]
over \(E=\{a,b\}\), and proves that
\[
\mathrm{uwl}(S_k)=5k^2-17k+13
\]
for \(k\ge 4\). This yields an infinite series of counterexamples to Restivo’s conjecture that every non-complete set has an uncompletable word of length at most \(2k^2\) [1104.0388].

Structural results accompany the extremal bounds. For \(S=\Sigma^k\setminus\{u\}\) with \(u\) unbordered, any minimal uncompletable word has the form
\[
w=u v_1 u v_2 u\cdots v_m u,
\]
and \(u\) must be both a prefix and a suffix. The papers also show that the lengths of the intermediate \(v_i\) need not be bounded by \(k\), so even the detailed structural part of Restivo’s original conjecture fails [1002.1928]. A broader survey perspective appears in “On incomplete and synchronizing finite sets,” which defines universal parameters \(R_{\mathcal C}(n,d)\) for minimal incompletable-word length and relates them to synchronizing-pair bounds for complete finite codes. In that setting, the stronger bound \(R_{\mathcal F}(n)\le 2n^2\) had already been disproved for general finite languages, while the weaker asymptotic conjecture \(R_{\mathcal F}(n)=O(n^2)\) remained the operative form [1612.07881].

## 4. Quantum strictly incompletable sets and activation phenomena

In quantum information, the phrase becomes more specialized. The paper “Hierarchy of hidden nonlocality: A genuine activation of Incompletability” works with finite sets of mutually orthogonal pure states, mostly product states, in multipartite Hilbert spaces. It distinguishes **incompletable** sets of product states, which cannot be completed to a full orthogonal product basis of the Hilbert space, from **strongly incompletable** sets, which admit no such completion even in any enlarged Hilbert space. Within the paper’s own usage, “strictly incompletable” refers to UPB-type sets such as the standard \(3\times 3\) unextendible product basis, which are incompletable and locally indistinguishable [2607.00797].

The central new notion is **activation of incompletability**. A locally distinguishable set \(\mathcal S\) is incompletability activable if it can be transformed by orthogonality-preserving local measurements into a set of locally incompletable orthogonal states. The paper constructs an explicit 10-state product set \(\mathcal S_2\subset \mathbb C^6\otimes\mathbb C^6\) that is free from local redundancy, perfectly distinguishable by LOCC, and completable to a full orthonormal product basis. A deterministic LOCC protocol based on binary orthogonality-preserving projectors maps every branch of \(\mathcal S_2\) to the standard five-state \(3\times 3\) UPB “Tiles”, hence to a strictly incompletable set.

This establishes a hierarchy. The paper proves that activation of incompletability necessarily implies activation of nonlocality, because a strictly incompletable post-measurement set cannot be LOCC-distinguishable. The converse fails. A second explicit set, \(\mathcal S_3\subset \mathbb C^6\otimes\mathbb C^6\), is LOCC-distinguishable and locally irredundant, and can be transformed into locally indistinguishable five-state sets, but those post-measurement sets remain completable. Thus activation of nonlocality is strictly weaker than activation of incompletability.

A further refinement appears in the LICC setting. If the incompletability of a product-state set can be activated by local incoherent operations and classical communication, then the set can still be extended to an orthonormal basis of the Hilbert space, but the completed basis is no longer perfectly distinguishable by LOCC. The paper treats this as an interaction among incompletability, nonlocality, and coherence.

## 5. Non-classical set theory and intrinsically incomplete membership

A different notion of strict incompletability appears in four-valued set theory. “Paraconsistent and Paracomplete Zermelo-Fraenkel Set Theory” introduces \(BZFC\), a version of set theory based on the four-valued logic \(BS4\), where a formula may be true only, false only, both true and false, or neither true nor false. For any set \(x\), the theory defines two associated classical sets,
\[
x^!:=\{y: !(y\in x)\},\qquad x^?:=\{y: ?(y\in x)\},
\]
which record, respectively, true membership and the absence of falsity of membership. A set is **complete** iff \(\forall y\,(y\in x^?\to y\in x^!)\), **consistent** iff \(\forall y\,(y\in x^!\to y\in x^?)\), and **classical** iff \(x^!=x^?\) [2210.00057].

The anti-classicality axiom postulates the existence of non-classical sets. The paper then proves a strong representation theorem: if there is an inconsistent set and an incomplete set, then for any classical sets \(u,v\) there is an \(x\) such that
\[
x^!=u,\qquad x^?=v.
\]
Thus any prescribed pattern of true membership and “not false” membership can be realized by a set. In particular, whenever \(u\subsetneq v\), the elements of \(v\setminus u\) are precisely those at which membership in \(x\) is gappy.

The paper does not use the term “strictly incompletable sets,” but it provides the exact machinery needed to define them. A natural reading is that a set is strictly incompletable when its incompleteness is encoded globally in the gap between \(x^!\) and \(x^?\), rather than arising from a single accidental failure of excluded middle. This reading fits the paper’s broader aim: to reason formally about incomplete and inconsistent phenomena while remaining sufficiently close to ZFC. The same work proves that \(BZFC\) is naturally bi-interpretable with ZFC, so these non-classical sets form an ontological extension rather than a replacement of the classical universe.

## 6. Comparative perspective

Across these literatures, “strictly incompletable sets” always concern a failure of completion, but the operative notion of completion changes with the ambient theory.

In arithmetic and computability, the blocked completion is deductive: a stably c.e. theory extending \(PA\) cannot be made complete while retaining the relevant consistency properties. In c.e.-set theory, the obstruction is degree-theoretic and orbit-theoretic: definable classes such as the \(Q_n\)-classes contain only incomplete sets. In combinatorics on words, completion means factor universality of \(S^*\), and the central quantity is how large the shortest missing factor must be. In quantum information, completion means extension to a full orthogonal product basis compatible with the locality structure, and strict incompletability marks a stronger level of hidden nonlocality than ordinary local indistinguishability. In four-valued set theory, completion means restoration of classical two-valued membership, blocked by the intrinsic separation between \(x^!\) and \(x^?\).

The term is therefore best understood as a domain-relative umbrella rather than a single theorematic concept. What unifies these usages is not a common formal definition, but a common structural pattern: admissible completion is impossible for reasons internal to the representation itself. Whether the relevant admissibility condition is stable computability, c.e. definability, factor coverage, LOCC implementability, or classicality of membership, the object retains a non-eliminable deficit with respect to completion. That is the sense in which the modern arXiv literature repeatedly arrives at “strict incompletability.”

Source: https://www.emergentmind.com/topics/strictly-incompletable-sets