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

It is known that any set-theoretic model satisfying the Axiom of Choice (AC) appears countable from a meta-theoretic perspective. Consequently, the set of reals within such a model is metacountable. Assuming this enumerability, we apply a generalized version of Cantor’ s diagonal argument to demonstrate the existence of “non-selectable reals” lying outside the AC model. Our approach employs finite partial functions as the basic units of diagonalization, together with a gluing procedure, to explicitly and constructively exhibit uncountably many reals unreachable by AC-based choice functions. Although the theorems used are classical, their operational reconstruction reveals the logical gap between the model and its meta-theory. We deliberately omit detailed comparisons with existing forcing techniques, focusing instead on enabling the reader to retrace the constructive procedure itself.