The Universal Property Perspective: Initial Successor Structures

We develop the universal-property perspective on the natural numbers, presenting the NNO (N, 0, succ) as the initial pointed set with endomorphism. From this universal property we derive primitive recursion, dependent induction, Lambek's fixed-point theorem, and the rigidity of N up to unique isomorphism. We then extend the analysis to other constants (π, e, R, Q), dualizing through coalgebraic stream types and final coalgebras.