Infinity is the axiom of paradox. Does the inclusion Infinity complete an otherwise incomplete set of axioms? It solves the halting problem for a finite Turing

Infinity is the axiom of paradox. Does the inclusion Infinity complete an otherwise incomplete set of axioms? It solves the halting problem for a finite Turing Machine. I don't buy the diagonalization proof as anything more than the Pythagoreom Theorom. You have infinite rows, and infinite columns. Infinity is Schrodinger's Cat. Once you check in on the state (nth row by mth column) the only thing you can say about the diagonal number is that is hasn't occurred in the rows up to that point, not beyond, nor in the columns (if n > m). Ergo, Infinity is a paradox, and only mathematical in the absurd.