(参考) https://en.wikipedia.org/wiki/Natural_number Natural number Contents 2.7 Infinity 3 Generalizations 4 Formal definitions 4.1 Peano axioms 4.2 Constructions based on set theory 4.2.1 Von Neumann ordinals 4.2.2 Zermelo ordinals
Generalizations Two important generalizations of natural numbers arise from the two uses of counting and ordering: cardinal numbers and ordinal numbers.
For finite well-ordered sets, there is a one-to-one correspondence between ordinal and cardinal numbers; therefore they can both be expressed by the same natural number, the number of elements of the set. This number can also be used to describe the position of an element in a larger finite, or an infinite, sequence.
A countable non-standard model of arithmetic satisfying the Peano Arithmetic (that is, the first-order Peano axioms) was developed by Skolem in 1933. The hypernatural numbers are an uncountable model that can be constructed from the ordinary natural numbers via the ultrapower construction.
Constructions based on set theory Von Neumann ordinals The standard definition, now called definition of von Neumann ordinals, is: "each ordinal is the well-ordered set of all smaller ordinals."
