Abstract

=Mathematical paradoxes associated with infinity and foundations, such as those arising in Galileo’s comparison of squares, Cantor’s theory of transfinite cardinalities, Hilbert’s Hotel, Russell’s paradox, and debates surrounding the Axiom of Choice, are not internal inconsistencies of mathematics but persistent cognitive shocks that accompany its development. Despite the formal rigor of modern axiomatic systems, these paradoxes repeatedly emerge and continue to challenge mathematical intuition. This paper argues that their persistence is not accidental but structural. The paradoxes arise from a systematic conflation between symbolic meaning-formation, rooted in human intuition and conceptual language, and formal ontological structure, governed by axioms, definitions, and proof. Applying the Universal Human Grammar of Inversion (UGI), the paper shows that mathematical paradoxes are necessary consequences of symbolic inversion processes rather than failures of mathematical reasoning. Within this framework, paradoxes are shown to be unavoidable, to recur across historical epochs, to demand resolution through meta-axiomatic reframing rather than technical repair, and to persist even in fully consistent formal systems. Mathematics remains coherent and stable at the formal level, while paradox emerges at the symbolic level where meaning is projected beyond its proper domain.