4 ms·
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
by CapacitorSet 9y ago
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.
- phaedrus 9y agoActually I would think that it divides the domain/codomain of the successor function into two disconnected regions. I just "6 is missing" as meaning succ(5) is undefined and that no x exists such that succ(x) = 7. You could probably still prove a lot of conventional math in such a system, except with extra annotations on equations like "given x != 5" much like how we have to add the annotation "given x != 1" on an equation that uses fraction like "y / (x-1)", to avoid invalid derivations from "y / 0". How do you know we're not already in a universe which is missing a number, which is why we have to add annotations constraining denominators to not be zero?