## No cycles in natural numbers teaching counting

Rips, Bloomfield and Asmuth identified that no cycles in the natural numbers is a critical stage in understanding the concept of numbers in a math sense as opposed to mechanical rote learning to count from 1 to 10.

See the following article on their paper.

http://newmathdoneright.com/2012/08/20/re-rips-bloomfield-and-asmuth-from-numerical-concepts-to-concepts-of-number/

Peano Axiom treatments typically ignore this or slight over it.  Moreover, the approach since the Landau book in 1930 is to prove the recursion theorem, define addition recursively, and then define order using addition.  This approach hides the basic concept of order.

5 is bigger than 3 because we can keep counting by one after 3.  In particular, we have 3, 4 and 5 as a chain of succession from 3 to 5.  This chain of succession has the property that 3 and 5 are in it and except for the end 5, it is closed under succession.   To completely characterize it, we have to add that it is the smallest set with this property in the sense that if we take the intersection of all such sets, it is the intersection. Further work is needed from that definition to characterize its properties.  This intersection is non-empty because the Natural Numbers contain the set.

We could also call it a gapless set that contains the predecessor of every number but the start and the successor of every number but the end, the middle numbers are the predecessor and successor of a number in the set, and the start is the predecessor of a number in the set and the end the successor of a number in the set.  This definition requires a proof that such sets exist and are unique.

David Groisser has studied a definition of such sets.  See the article here for a discussion of his article.