^Carothers (2000) says: "ℕ is the set of natural numbers (positive integers)" (p. 3)
^Mac Lane & Birkhoff (1999) include zero in the natural numbers: "Intuitively, the set ℕ = {0, 1, 2, ...}
of all "natural numbers" may be described as follows: ℕ contains an "initial" number 0; ...". They follow that with their version of the Peano Postulates. (p. 15)
لمعانٍ أخرى، طالع 289 (توضيح).