Natural number

Natural numbers can be used for counting (one apple, two apples, three apples, ...)

The most primitive method of representing a natural number is to put down a mark for each object. Later, a set of objects could be tested for equality, excess or shortage—by striking out a mark and removing an object from the set.

The addition (+) and multiplication (×) operations on natural numbers as defined above have several algebraic properties:

Even if one does not accept the axiom of infinity and therefore cannot accept that the set of all natural numbers exists, it is still possible to define any one of these sets.

Each natural number is then equal to the set containing just the natural number preceding it. This is the definition of Zermelo ordinals. Unlike von Neumann's construction, the Zermelo ordinals do not extend to infinite ordinals.