Proving that the natural numbers don’t contain cycles starting from the Peano Axioms is harder than it sounds. David Groisser’s notes call for students to try to find a simpler way. Rips, Bloomfield and Asmuth say that no cycles in naturals is a key step in understanding math concepts over mechanical learning for children. (See paper a few blog posts back.)
Let a thousand flowers bloom for proving no cycles in natural numbers. This is best done before defining addition or proving the recursion theorem using a hard to understand proof.
Uniqueness is linked to no cycles. Consider following tentative argument, possibly a proof.
Assume Peano Axioms and that each natural number is unique, i.e. different from all others. So from axioms, we have predecessors and successors are unique.
Suppose we have a natural number i, and a chain from i goes to i’ and eventually comes back to i.
If the chain goes to j and j’ is i, then it would appear that i has two predecessors, j, and the ‘i encountered in the chain from 0 to i. This is a contradiction so this is ruled out. Is this a proof for this case?
Now suppose a chain goes from i to j, and then j’ is some h and there is a chain from h to i. Then for h, we are in the same situation as the previous lemma? So this would result in h having two predecessors, so that is ruled out.
Is this a valid proof? Is there a hidden assumption about initial segments from 0 to i?
What about trying to show that uniqueness more directly rules out a cycle? Using induction?
These questions are good for projects even if the results are inconclusive. They help sharpen the understanding of the issues in setting up the naturals. They also emphasize, that everything about the naturals is about order.
Are there other axiom sets that make these proofs easier? Or make a seemingly easy proof valid?