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.

http://newmathdoneright.com/2012/05/09/david-groisser-initial-segments-before-addition-from-peano-axioms/

Dedekind follows a somewhat different route that is longer but eventually gets to the existence of initial segments, i.e. chains of succession starting from 0, and their basic properties. In the Dedekind approach, we could then define chains of succession from i to j as the chain of succession from 0 to j minus the points in the chain from 0 to ‘i. Here i has to be non-zero.

Project: Given the above has been defined and obvious properties proven, prove the following. If we assume all the natural numbers are unique, i.e. different from each other, then there is no non-zero length chain of succession that starts from i and comes back to i. Define the terms needed and think carefully about assumptions about obvious properties you may be making.

Warning it is very easy to fool yourself.

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## About New Math Done Right

Author of Pre-Algebra New Math Done Right Peano Axioms. A below college level self study book on the Peano Axioms and proofs of the associative and commutative laws of addition.
President of Mathematical Finance Company. Provides economic scenario generators to financial institutions.