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arXiv · 2607.17306

Paradoxes Are Not Contradictions: Re-examining the Third Mathematical Crisis

H. Y. Yuan
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Abstract

Russell's paradox was proposed in the early 20th century to address loopholes in set theory, which directly triggered the third mathematical crisis. This paper proposes and elaborates a perspective distinct from previous studies: paradoxes do not give rise to contradictions; instead, they constitute a M\"obius strip-style self-consistent logical structure via self-reference and negation under the logical rules within a system. G\"odel's incompleteness theorems indicate that such structures universally exist in formal logical systems. Turing proved the undecidability of the halting problem by first assuming the existence of a halting program and subsequently refuting this assumption through paradox construction, and this paper demonstrates flaws inherent to such proof strategy. Cases of paradoxes within three-valued logical systems are further discussed in this work, where the undecidability of paradoxes is rigorously proven. Finally, inspirations drawn from paradoxes for the real world are explored: two opposing factors can be integrated through the joint mechanism of self-reference and negation. A representative example is the wave-particle duality of light, whose essence may be interpreted as a paradox of waves and particles.

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H. Y. Yuan. 2026-07-19. Paradoxes Are Not Contradictions: Re-examining the Third Mathematical Crisis. https://arxiv.org/abs/2607.17306

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