[Paper Review] Some Brouwerian Counterexamples Regarding Nominal Sets in Constructive Set Theory

Andrew Swan|arXiv (Cornell University)|Feb 6, 2017

Advanced Algebra and Logic4 references3 citations

TL;DR

This paper presents Brouwerian counterexamples in constructive set theory to demonstrate that the existence of least finite supports—a foundational assumption in nominal set theory—cannot be constructively justified. It shows that assuming least finite support leads to non-constructive principles like WLPO, implying such assumptions should be avoided in constructive reasoning with nominal sets.

ABSTRACT

The existence of least finite support is used throughout the subject of nominal sets. In this paper we give some Brouwerian counterexamples showing that constructively, least finite support does not always exist and in fact can be quite badly behaved. On this basis we reinforce the point that when working constructively with nominal sets the use of least finite support should be avoided. Moreover our examples suggest that this problem can't be fixed by requiring nominal sets to have least finite support by definition or by using the notion of subfinite instead of finite.

Motivation & Objective

• To challenge the assumption that nominal sets always possess least finite supports in constructive set theory.
• To demonstrate that assuming least finite support leads to non-constructive principles such as WLPO.
• To argue that the use of least finite support in nominal set theory should be avoided in constructive mathematics.
• To show that even subfinite supports do not resolve the foundational issues arising from the non-constructive behavior of least supports.
• To reinforce the idea that nominal set constructions must be re-evaluated in constructive frameworks to avoid non-constructive consequences.

Proposed method

• Constructs a nominal set using a binary sequence α and a name a, defining an element αₐ with support properties tied to the truth of ∀n∈ℕ.α(n)=0.
• Uses the set L = {x∈{a} | ¬(∀n∈ℕ)α(n)=0} to represent the intersection of all finite supports of αₐ, showing L equals the intersection of all finite supports.
• Employs double negation reasoning to show that if L were finite and decidable, it would imply WLPO, a non-constructive principle.
• Applies lemma 3.4 to show that if the intersection of all finite supports is a support, then ¬¬ϕ→ϕ holds for all bounded formulas ϕ, leading to restricted excluded middle (REM).
• Analyzes the behavior of subfinite supports and shows that even under subfiniteness, the same non-constructive consequences arise.
• Uses realizability and model-theoretic reasoning to show that WLPO is not provable in CZF, thereby establishing the non-constructive nature of least finite support.

Experimental results

Research questions

• RQ1 Can the existence of least finite support be constructively justified in nominal set theory?
• RQ2 What are the logical consequences of assuming that every nominal set element has a least finite support?
• RQ3 Does requiring subfinite supports instead of finite supports resolve the non-constructive issues arising from least finite support?
• RQ4 To what extent does the assumption of least finite support imply non-constructive principles like WLPO or REM?
• RQ5 Can the intersection of all finite supports of an element be a support without implying stronger non-constructive principles?

Key findings

• The intersection of all finite supports of an element in a nominal set is not necessarily a finite support, and this can fail constructively.
• If the intersection of all finite supports were always a support, then ¬¬ϕ→ϕ would hold for all bounded formulas ϕ, implying restricted excluded middle (REM).
• Assuming that every element has a least finite support implies WLPO, a principle not provable in constructive set theory like CZF.
• Even when restricting to subfinite supports, the same non-constructive consequences arise, showing that subfiniteness does not resolve the issue.
• The set L = {x∈{a} | ¬(∀n∈ℕ)α(n)=0} is equal to the intersection of all finite supports of αₐ, and if L were finite, it would imply WLPO.
• The failure of least finite support to exist constructively undermines foundational assumptions in nominal set theory when working in a constructive framework.