I think the point of this post is to demonstrate that logical pinpointing is hard. You might think that the first-order Peano arithmetic axioms logically pinpoint the natural numbers, and what this discussion will end up showing is that they just don’t because of general properties of first-order logic (specifically the Löwenheim–Skolem theorem).
If logically pinpointing something as seemingly simple as the natural numbers depends on something as seemingly nontrivial as understanding the distinction between first-order and second-order logic, then (or so I imagine the argument will continue) we shouldn’t expect logically pinpointing something like morality to be any easier. In fact we have every reason to expect it to be substantially harder.
The definition of the order relation is nontrivial. In second-order Peano arithmetic you can define addition from the successor operation by induction, and then you can define a to be less than b if there is a positive integer n such that a + n = b. My understanding is that you cannot define addition this way in first-order Peano arithmetic. Instead it is necessary to explicitly talk about addition in the axioms. From here one could also go on to explicitly talk about the order relation in the axioms.
I think the point of this post is to demonstrate that logical pinpointing is hard. You might think that the first-order Peano arithmetic axioms logically pinpoint the natural numbers, and what this discussion will end up showing is that they just don’t because of general properties of first-order logic (specifically the Löwenheim–Skolem theorem).
If logically pinpointing something as seemingly simple as the natural numbers depends on something as seemingly nontrivial as understanding the distinction between first-order and second-order logic, then (or so I imagine the argument will continue) we shouldn’t expect logically pinpointing something like morality to be any easier. In fact we have every reason to expect it to be substantially harder.
The definition of the order relation is nontrivial. In second-order Peano arithmetic you can define addition from the successor operation by induction, and then you can define a to be less than b if there is a positive integer n such that a + n = b. My understanding is that you cannot define addition this way in first-order Peano arithmetic. Instead it is necessary to explicitly talk about addition in the axioms. From here one could also go on to explicitly talk about the order relation in the axioms.