Natural Numbers No Cycles more

This is a continuation from the prior post.  Suppose we argue against cycles in the natural numbers as follows.

For the following lemma, we assume as part of our axioms that each successor of a natural number is different from all other natural numbers, i.e. is unique.

Lemma: Any chain of succession from i to j results in a j different from i.

Proof?   This is because we assume each successor is unique and thus different from all other naturals.  Since j is a success of ‘j, then j is different from i, because we assume any successor differs from all other natural numbers.

This seems convincing as a proof given this assumption.

Can we drop the assumption that a successor is unique, i.e. different from all other naturals, and prove that from the Peano Axioms without having to prove the recursion theorem or addition?  And possibly before using the David Groisser approach to initial segments?  Or just after that?

 

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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.
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