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

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 Möbius strip-style self-consistent logical structure via self-reference and negation under the logical rules within a system.

Gödel'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.