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Reflections on Russell's antinomy

Paola Cattabriga

Abstract

We present Russell's antinomy using three distinct deductive systems, which are then compared to deepen the logical deductions that lead to the contradiction. Some inferential paths are then presented, alternative to the commonly accepted one, that allow for the formal assertion of the antinomy without deriving the contradiction, thus preserving the coherence of the system. In light of this, the purpose of this article is to propose a review of the consequences of asserting Russell's antinomy and, by extension, the widespread belief that any attempt to resolve a paradox is doomed to failure.

Reflections on Russell's antinomy

Abstract

We present Russell's antinomy using three distinct deductive systems, which are then compared to deepen the logical deductions that lead to the contradiction. Some inferential paths are then presented, alternative to the commonly accepted one, that allow for the formal assertion of the antinomy without deriving the contradiction, thus preserving the coherence of the system. In light of this, the purpose of this article is to propose a review of the consequences of asserting Russell's antinomy and, by extension, the widespread belief that any attempt to resolve a paradox is doomed to failure.
Paper Structure (5 sections, 7 theorems, 36 equations, 7 figures)

This paper contains 5 sections, 7 theorems, 36 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Another inferential path
  3. (In)-Discernibility
  4. Uniqueness
  5. Ending Note

Key Result

Theorem 1

There exists no set which contains exactly those elements which do not contain themselves, in symbols $\neg\exists y\forall x(x\in y \iff x\notin x)$.

Figures (7)

  • Figure 1: Semantic tree proving (\ref{['antirussell']})
  • Figure 2: A semantic tree for (\ref{['russell']})
  • Figure 3: Semantic tree proving (\ref{['nisba']})
  • Figure 4: Semantic tree proving (\ref{['ext']})
  • Figure 5: Semantic tree for (\ref{['paola']})
  • ...and 2 more figures

Theorems & Definitions (16)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Lemma 2.1
  • proof
  • Theorem 3
  • proof
  • Remark 1
  • Lemma 3.1
  • ...and 6 more