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Paradox on the Countable Axiom of Choice

Babak Jabbar Nezhad

TL;DR

The paper interrogates the compatibility of constructive mathematics with choice principles by presenting a paradox that arises under $CAC$ and $ADC$ within Bishop's framework. It builds a constructive complex-analytic setup, defines a carefully chosen logarithmic framework on algebraic numbers, and constructs an analytic function whose zeros on a compact set of algebraic points contradict a constructive finiteness principle for zeros. The main contribution is demonstrating that, even with $CAC$ and $ADC$, certain algebraic-analytic assumptions lead to inconsistency, highlighting tensions between constructive foundations and classical-like results. This work underscores the delicate interplay between admissible choice principles and foundational stability in constructive analysis and algebra, with potential implications for how algebraic closure and UFD properties behave under these axioms.

Abstract

Bishop's constructive mathematics school rejects the Law of Excluded Middle, but instead vastly makes use of weaker versions of the Choice. In this paper we pioneer an example, which shows that this road is not consistent, as our example provides a paradox. Therefore, rejecting the Law of Excluded Middle, and as an alternative using the Countable Axiom of Choice and the Axiom of Dependent Choice, still does not create a consistent structure. Actually, constructively; the Countable Axiom of Choice is an implication of the Axiom of Dependent Choice.

Paradox on the Countable Axiom of Choice

TL;DR

The paper interrogates the compatibility of constructive mathematics with choice principles by presenting a paradox that arises under and within Bishop's framework. It builds a constructive complex-analytic setup, defines a carefully chosen logarithmic framework on algebraic numbers, and constructs an analytic function whose zeros on a compact set of algebraic points contradict a constructive finiteness principle for zeros. The main contribution is demonstrating that, even with and , certain algebraic-analytic assumptions lead to inconsistency, highlighting tensions between constructive foundations and classical-like results. This work underscores the delicate interplay between admissible choice principles and foundational stability in constructive analysis and algebra, with potential implications for how algebraic closure and UFD properties behave under these axioms.

Abstract

Bishop's constructive mathematics school rejects the Law of Excluded Middle, but instead vastly makes use of weaker versions of the Choice. In this paper we pioneer an example, which shows that this road is not consistent, as our example provides a paradox. Therefore, rejecting the Law of Excluded Middle, and as an alternative using the Countable Axiom of Choice and the Axiom of Dependent Choice, still does not create a consistent structure. Actually, constructively; the Countable Axiom of Choice is an implication of the Axiom of Dependent Choice.

Paper Structure

This paper contains 4 sections, 6 theorems, 24 equations, 1 figure.

Table of Contents

  1. Introduction
  2. A glimpse of foundation
  3. Some facts in constructive mathematics
  4. The example

Key Result

Lemma 2.3

Let $U\subset\mathbb{C}$ be open, and $f:U\rightarrow\mathbb{C}$ be uniformly continuous on each closed sphere well contained in $U$. Then $f$ is continuous on $U$.

Figures (1)

  • Figure 1: The open simply connected set of $\mathcal{D}$.

Theorems & Definitions (9)

  • Remark 2.1
  • Lemma 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • proof
  • Remark 3.4