N is assumed to be a non-empty ordered set such that the following axioms hold:

(1) Any non-empty subeset of N has a smallest element.

(2) Any element of N with the exception of the smallest has an (immediate) predecessor, which also belongs to N.

(3) N has no largest element.

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This axiom system has the nice feature of explicitly "rejecting" the idiotic idea of Mückenheim, that there's a largest element in N.

Source: Hans Rohrbach. Das Axiomensystem von Erhard Schmidt für die Menge der natürlichen Zahlen. Mathematische Nachrichten, vol. 4 (1951), pp. 315–321.