The definition of natural numbers and addition can be constructed using set theory, where '1' is represented as the set containing the empty set, and '2' as the set containing '0' and '1'.

In set theory, numbers can be formally constructed. The empty set, denoted '∅' or '{}', represents '0'. The number '1' is then defined as the set containing the empty set: {∅}. The number '2' is defined as the set containing '0' and '1': {∅, {∅}}. Addition is then a function that combines the cardinalities of sets. This construction, largely attributed to John von Neumann, demonstrates how all natural numbers and their operations can be built from the most basic logical primitive: the concept of a set and its members.