Erroneous proof of recursion theorem examples

Question

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?

Answer 1

An example, citing Saunders Mac Lane's 'Mathematics Form and Function', is 'Natural Numbers, Integers, and Rational Numbers (Following MacLane)' by Alexander Nita, http://math.colorado.edu/~nita/Numbers.pdf, page 12. I did not check what Mac Lane said, though.

Addendum 1 (2021-03-17): Mac Lane (1986 edition, pages 43 - 46) omits the use of several of the Peano axioms in his proof, hence furnishing an erroneous proof.

Another more subtle example: 'An Innocent Investigation' by D. Joyce, http://homepages.math.uic.edu/~kauffman/EvenOddJoyce.pdf . Here, for the definition of even/odd it is shown that every number besides 1 has a predecessor. It would be necessary, however, to show that it has exactly one predecessor (injectivity of the successor map). In a model with a loop, consisting of 1,2,3,... where the successor of 7 is 3, it is not possible to define the even/odd property recursively for 3 because this number has the predecessors 2 and 7.

Addendum 2 (2021-03-17): An erroneous proof, as of pre-1960, can be found in the brillant article 'On mathematical induction' by Leon Henkin (page 327, in the paragraph starting with 'Clearly (the argument goes), ...'). The (valid) proof methods explained in the article go back to Kalmar (construction by partial functions), Lorenzen (intersections of relations) and Landau (definition of addition and multiplication, with credit to Kalmar).