Giuseppe Peano’s axioms for the natural numbers are as follows. In this discussion, we will follow tradition and use the notation ′ $ ^{\prime } $ https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9781315146898/9ce47cf6-5c02-4bc8-a80d-2160001f85a9/content/inline-math6_1.tif"/> to denote the “successor” of a natural number. For instance, the successor of 2 is 2 ′ $ 2^{\prime } $ https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9781315146898/9ce47cf6-5c02-4bc8-a80d-2160001f85a9/content/inline-math6_2.tif"/> . Intuitively, the successor of n is the number n + 1 $ n + 1 $ https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9781315146898/9ce47cf6-5c02-4bc8-a80d-2160001f85a9/content/inline-math6_3.tif"/> . However, addition is something that comes later; so we formulate the basic properties of the natural numbers in terms of the successor function.