The Peano Axioms are about the Successor Function. As Rips et al make clear in their paper, learning the concept of natural numbers is about learning about the Successor Function. They also indicate that learning the Successor Function is about learning the ordered pairs that are in it and are not.
- Zero exists.
- Each number has a successor.
- Except for zero, each number has a unique predecessor.
- Zero has no predecessor.
- A chain of successor ordered pairs (n,n’) is right spreading if (n,n’) a member of the chain implies that (n’,n”) is also a member.
- A right spreading chain of successor ordered pairs of the form (n,n’) that includes (0,0′) contains all the natural numbers n.
We can emphasize the ordered pairs more. The graph of a function is the set of all ordered pairs that are part of the function.
- 0 exists.
- (n,n’) exists for each n.
- (m,n) is a successor link implies m’ = n.
- No pair (m,0) in successor graph.
- A chain of ordered pairs of the form (n,n’) that includes (0,0′) contains all the natural numbers n and is the same as the graph of the successor function.