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TorqueDrifter comments on Logical Pinpointing - Less Wrong
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I think that this is the sort of case in which it is useful to do some hand-waving to indicate that you've realized that your reasoning was wrong but that you have additional reasoning to back up your conclusion, as otherwise it can appear that you've realized your reasoning was wrong but want to stick to the conclusion anyway, and therefore need to come up with new reasoning to support it.
Anyway, consider the following: let Pn(x) mean "x is the nth successor of 0" (where the 0th successor of a number is itself). Then, by induction, every number x has the property "there exists some n such that Pn(x)".
I don't think that change has an effect, you're just adding "if two numbers are the same number, they have the same successor", right? Which is already true.
Is zero the zeroth successor of zero, by that property? Is that compatible with zero not being a successor of any number?
As I said, the "zeroth successor" of a number is itself. That is, zero is the result of applying the successor function to itself zero times. You have to apply a function at least once in order to have applied the function (and thus obtained a result of applying the function, e.g., calculated a successor).
If you don't like the term, you can think of it this way:
P0(x): x = 0
P1(x): x = S0
P2(x): x = SS0
and so forth.