It seems to me that Turnage-Butterbaugh's argument implicitly referenced Peano's axioms: if there's no number 6, then its successor doesn't exist, nor does its successor's successor [an existence which would be otherwise guaranteed by, ironically, Axiom 6]. Then it claims that "all the other integers are out", which may be a reference to the reverse: if 6 is the successor of 5, and 6 doesn't exist, then neither does 5, and therefore neither do 4 or 3.

I think the professors were basing themselves on Peano, but it was omitted by either the editor or the professors themselves for the sake of clarity.