Construction of Non-Selectable Reals Outside AC Models via a Generalization of Cantor' s Diagonal Argument

A set-theoretic model satisfying the Axiom of Choice (AC) is, by Skolem’s theorem, countable when viewed from a metatheoretical perspective.From this, we assume that the reals within the model can be enumerated metatheoretically, and by applying Cantor’s diagonal argument, we are led to the existence of “non-selectable reals” outside the model.Building upon this observation, the present paper constructively and explicitly demonstrates, through a generalization of Cantor’s diagonal argument based on finite partial functions and their amalgamation procedure, that uncountably many reals inaccessible by AC-based choice functions exist outside the AC model.Thus, we provide a method for explicitly constructing objects that were previously known to exist only “non-constructively.”Although the theorems employed are classical, their operational reconstruction renders visible the logical gap between the inside and outside of the model. This paper deliberately refrains from detailed comparison with existing forcing methods, focusing instead on enabling the reader to directly retrace the procedure of the construction.