The Axiom of Double Complement and its opposites

Abstract: Powell introduced the Axiom of Double Complement ($\mathsf{DCom}$) to give his double-negation interpretation of $\mathsf{ZF}$ into $\mathsf{IZF_{Rep}}$. However, the consistency, strength, and compatibility of $\mathsf{DCom}$ remain open problems. This article aims to survey the compatibility and consistency strength of $\mathsf{DCom}$, its consequence and opposites, which will be named $\mathsf{NDCom}$ and $\mathsf{ADCom}$. We will also develop Lubarsky's Kripke models over $\mathsf{CZF}$ to derive these results. We will show that $\mathsf{DCom}$ proves the Powerset axiom over $\mathsf{CZF}$ and is independent of $\mathsf{IZF}$. We will also show that $\mathsf{ADCom}$ does not add consistency strength over $\mathsf{CZF}$, by modifying the construction of Lubarsky's model for $\mathsf{CZF+\lnot Pow}$. We will also show that $\mathsf{DCom}$, $\mathsf{ADCom}$, and $\mathsf{NDCom}$ are persistent under realizability under modest conditions.

• The paper demonstrates that adding the axiom of double complement (DCom) to CZF is equivalent to incorporating the full Powerset axiom.
• It employs Kripke and realizability models to rigorously analyze the consistency strength and independence of DCom, NDCom, and ADCom within intuitionistic frameworks.
• The analysis refines our understanding of non-classical set-theoretic operations by clarifying the precise conditions under which double complements exist.

The Axiom of Double Complement and Its Opposites: A Technical Overview

Introduction

The paper "The Axiom of Double Complement and its opposites" (2606.00861) provides a comprehensive analysis of the Axiom of Double Complement ($\mathsf{DCom}$), a principle rooted in the double-negation interpretation of set theory. It also examines two forms of its negation, labeled $\mathsf{NDCom}$ (Non-Double Complement) and $\mathsf{ADCom}$ (Anti-Double Complement). The authors scrutinize these axioms in the context of constructive and intuitionistic set theories ($\mathsf{CZF}$, $\mathsf{IZF}$), focusing on their model-theoretic status, consistency strength, and implications for major set-theoretic axioms such as Powerset.

Formal Definitions and Motivation

$\mathsf{DCom}$ postulates that every set $x$ admits a double complement $x^ = \{z : \lnot\lnot(z \in x)\}$, asserting a form of closure under double negation. The axiom was introduced by Powell to support a double-negation translation of $\mathsf{ZF}$ into intuitionistic frameworks. However, the precise consistency strength of $\mathsf{DCom}$ and its relation to other set-theoretic axioms has remained unresolved.

$\mathsf{NDCom}$0 and $\mathsf{NDCom}$1 provide, respectively, a direct assertion of the existence of sets not admitting a double complement and a maximal anti-complementation regime, whereby only subsets of the singleton $\mathsf{NDCom}$2 can possess a double complement.

Technical Results on Model Theory and Consistency

$\mathsf{NDCom}$3, Powerset, and Constructive Set Theory

A principal result is that $\mathsf{NDCom}$4 implies the full Powerset axiom over $\mathsf{NDCom}$5. Specifically, leveraging structural analysis of the double complement of the set $\mathsf{NDCom}$6 (the two-element set), the paper shows that $\mathsf{NDCom}$7, i.e., the double complement of $\mathsf{NDCom}$8 is the power set of the singleton, hence realizing $\mathsf{NDCom}$9 at the object level. Through an explicit construction and use of $\mathsf{ADCom}$0-bounded separation, the authors demonstrate that the existence of double complements for certain sets entails the existence of power sets for arbitrary sets, provided exponentiation holds. As a consequence, $\mathsf{ADCom}$1 and $\mathsf{ADCom}$2 are equiconsistent; any model of the former admits the latter, and vice versa.

Independence and Kripke/Realizability Models

By employing Lubarsky-style Kripke models over set frames and examining both linear and non-linear partial orders, the paper demonstrates the following:

• Kripke models with linear frames: If $\mathsf{ADCom}$3 is a model of $\mathsf{ADCom}$4 and $\mathsf{ADCom}$5 is a linear frame, then the Kripke model $\mathsf{ADCom}$6 satisfies $\mathsf{ADCom}$7.
• Non-linear Kripke frames: In particular, over the complete binary tree ($\mathsf{ADCom}$8), it is shown that $\mathsf{ADCom}$9 holds, i.e., there exist names (sets in the Kripke model) that do not admit a double complement.
• Models for $\mathsf{CZF}$0: A variant of Lubarsky's construction over the natural numbers is used to produce a model (hereditarily eventually constant Kripke sets) where only subsets of the singleton $\mathsf{CZF}$1 possess double complements, establishing $\mathsf{CZF}$2 and $\mathsf{CZF}$3 are equiconsistent.

Realizability and Persistence

Persistence under realizability is established for all three axioms ($\mathsf{CZF}$4, $\mathsf{CZF}$5, $\mathsf{CZF}$6) under modest hypotheses. Notably, the paper shows that:

• $\mathsf{CZF}$7 is preserved under realizability models when the base theory supports $\mathsf{CZF}$8 separation or the regular extension axiom (REA). The construction requires the Powerset axiom due to the necessity of bounding the cumulative hierarchy.
• $\mathsf{CZF}$9 is absolute in realizability models when $\mathsf{IZF}$0 holds.
• $\mathsf{IZF}$1 is persistent even without Powerset, as long as $\mathsf{IZF}$2 (the class of weakly decidable subsets of $\mathsf{IZF}$3) fails to be a set in the ground model.

Consistency Strength

The analysis provides precise relative consistency results. For instance:

• $\mathsf{IZF}$4
• $\mathsf{IZF}$5
• $\mathsf{IZF}$6 does not increase the consistency strength of $\mathsf{IZF}$7

Additionally, the authors clarify that $\mathsf{IZF}$8 is independent of $\mathsf{IZF}$9, and that adding $\mathsf{DCom}$0 or $\mathsf{DCom}$1 to $\mathsf{DCom}$2 similarly does not alter consistency strength, largely due to the fact that Subset Collection is essential for translating the existence of double complement operation into the existence of general power sets.

Explicit Contradiction and Decidability Structure

A striking claim is the equivalence (in $\mathsf{DCom}$3) between $\mathsf{DCom}$4 and the non-existence of $\mathsf{DCom}$5, a statement about the structure of weakly decidable subsets. In the constructed models, this yields a refined description of the "size" and structure of classes that can possess double complements.

Opposites and Functional Realizability

Through careful study of functional realizability (as in Swan's model and Hahanyan's earlier work), the paper demonstrates that models satisfying $\mathsf{DCom}$6 can be constructed, further confirming the axiom's compatibility with the basic constructive set-theoretic framework (including separation). Importantly, it is shown that functional realizability does not validate full Powerset, and in fact, in models constructed via this route, $\mathsf{DCom}$7 cannot be a set.

Implications and Prospects

Theoretical Impact

This investigation establishes a deep connection between the double complement operation and the Powerset axiom in constructive set theories. As Powerset is a source of predicativity issues and a dividing line in the relative strength of constructive theories, the fact that $\mathsf{DCom}$8 implies and is essentially equivalent to Powerset over $\mathsf{DCom}$9 gives clear demarcation of its utility and limits. The independence from $x$0 and robust independence analyses via Kripke and realizability models reflect the subtlety of negated and non-classical set existence principles in the absence of excluded middle.

Practical Methodology

The techniques—particularly the use of Kripke semantics with fully inductive relations and slicing functions—stand as robust tools for analyzing persistence and absoluteness of non-classical axioms. The work also clarifies how intricate combinatorics of realizability and functional interpretation can finely calibrate the set-theoretic landscape.

Future Directions

Possible avenues include:

• Examination of the status of the Axiom of Subcountability in connection with $x$1.
• Application of these model-theoretic techniques to further non-classical set-theoretic or type-theoretic principles.
• Analysis of how such results transfer to type-theoretic or higher-order frameworks, especially given connections highlighted to second-order arithmetic via Lubarsky's results.

Conclusion

The paper provides a comprehensive, technical analysis of the logical and model-theoretic status of the Axiom of Double Complement and its opposites within intuitionistic and constructive set theories. By bridging Kripke and realizability models, the authors precisely delineate the power and limitations of each axiom, establishing the precise circumstances under which they are compatible or equiconsistent with key constructive set-theoretic frameworks. The results offer both a clarification of open questions and a toolkit for further developments in constructive foundations.