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Abstract
<jats:p>This article examines mathematical paradoxes not as errors or curiosities, but as "protectors," the limits of applicability of mathematical theories. Drawing on J. Wigner's concept of the incomprehensible effectiveness of mathematics, the author demonstrates that it is the presence of paradoxes (from Russell's paradox to Gödel's incompleteness theorems) that guarantees the consistency and reliability of mathematical knowledge within the framework of accepted axioms. Paradoxes perform diagnostic, restrictive, and constructive functions, preventing the creation of inconsistent theories and pointing to directions for clarifying the foundations of mathematics. The question "What if there were no paradoxes?" receives an answer: mathematics would then lose the ability to distinguish truth from falsehood, and its "effectiveness" would become illusory.</jats:p>