In his book Elements of Set Theory, Herbert Enderton defines (p. 70) a Peano system as a triple $(N, S, e)$ where $N$ is a set, $S$ is an $N-$valued function defined on $N$ and $e$ is a member of $N$ such that

  • $e \notin S(N)$;
  • $S$ is one-to-one;
  • Any subset $A \subset N$ that contains $e$ with $S(x)\in A$ whenever $x \in A$ equals $N.$

Then, on page 76 he has the following cryptic remark:

There is a classic erroneous proof of the recursion theorem that people have sometimes tried (even in print!) to give.

Of course, he gives some explanation of why these "proofs" are wrong (they do not use all the three conditions given above) but anyway: can someone give some concrete examples that Enderton could have in his mind of such an flawed proof?


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