When evaluating the arguments and claims of others, one should strive to apply a principle of charity, that is, to interpret those arguments and claims as charitably as possible. This would be the opposite of a so-called straw man argument, whereby one assigns untenable views to an opponent and proceeds to argue (successfully, of course) against those untenable views. In the current case, having identified the apparent misconceptions detailed above, one should then try to interpret them in a way that does not make them untenable or incorrect.

In looking at the actual incorrect deductions (using the Peano axioms) that well-ordering implies induction, I think it is hard to be charitable. In this system, using immediate predecessors is simply not warranted. I think, however, that there are two main points at which one can try to afford charity. First, perhaps something else is meant by “equivalence,” for example in [Gun11], where “the equivalence between WO and P5” (p. 62) is discussed. Second, perhaps there is some reasonable interpretation of statements to the effect that some property is “assumed for the natural numbers.”

Equivalence Between Axioms

Admittedly, the sources listed above are not primarily axiomatic treatments of the natural numbers, and so perhaps a more colloquial sense of the word “equivalence” is intended. That is, instead of claiming that the axioms (1)–(5) single out the same model as the axioms (1)–(4) together with (\(5'\)) do, perhaps the intended sense is, for example, that the induction principle and the well-ordering principle are equally useful in proving theorems. Another possible intended meaning might be that for the natural numbers (maybe as given explicitly by some concrete model in set theory), both principles are equally true (that is, they are actually true, plain and simple).

If any of these other possible interpretations are what is actually intended, then I think that it would serve the reader better to state this in less misleading terms. I believe that the intuitive idea most mathematicians have, and that we should want to instill in students, is that the equivalence of two axioms is something relative to another base set of axioms. The two paradigmatic examples of this, which I believe shape mathematicians’ understanding of equivalence of axioms, are the different versions of the parallel axiom in Euclidean geometry (equivalent relative to the rest of the standard axioms), and the axiom of choice versus, for instance, Zorn’s lemma, in Zermelo–Fraenkel set theory. Therefore, regarding the statement that the well-ordering principle and the induction principle are “equivalent,” I would think that the generally perceived meaning among readers is that they are equivalent relative to the other axioms, and I suspect that this is also usually the intended meaning on the part of the authors.

Taking Naive Set Theory as an example, it would seem that what is intended to be claimed is that the well-ordering principle and the induction principle are simultaneously true for the natural numbers, but that there are other structures (the ordinals, for example) in which they are not simultaneously true. Here, it could also be that what is intended is that transfinite induction in the case of the natural numbers coincides with ordinary induction.

Properties “Assumed for the Natural Numbers”

If it is assumed that the natural numbers are somehow at hand, by whatever mechanism, then “assuming” some property about them that is true for the natural numbers is certainly vacuous. Any arguments purporting to uncover some basic characterizing property for the natural numbers proceeding from this assumed property must then also be circular. This echoes the historical attempts to “prove the parallel axiom” in Euclidean geometry, where some “obvious” properties of the geometry were often tacitly assumed. In order to be charitable, one must therefore assume that what is intended by expressions such as “the well-ordering property, which we assume for the positive integers” is something like “assumed for this structure, which we are trying to pin down.” In this case, there must still be some base set of axioms that is already in place.

As indicated above, using axiom (\(3'\)) instead of (3) actually makes the well-ordering principle equivalent to the induction principle, relative to the first four axioms. So perhaps this is what the authors quoted above actually had in mind. However, as remarked in [Gun11], “Peano’s axioms are generally now accepted by the mathematical community as a starting point for arithmetic.” In the sources I have seen that explicitly state the first four axioms, it is invariably axiom (3) that is used, not axiom (\(3'\)). This goes toward the argument that the mere existence of a base set of axioms relative to which (5) and (\(5'\)) are equivalent does not warrant the unspecified claim that they are equivalent.

In any case, formulations such as “assumed for the natural numbers” are questionable even in a slightly less formal treatment of the axiomatic grounds of numbers.

As noted above, I think it would be most interesting to see a more thorough historical investigation into these issues. Additionally, I have searched for, but not found, some source giving a more detailed overview of alternative ways of introducing and characterizing the natural numbers, perhaps also including an analysis of the relative strength of some different selections of axioms. This was investigated in a bachelor’s thesis [SL16] that I supervised on the topic. An end goal would be for these issues to be as widely known as alternative axiomatizations of planar geometry and set theory.