> Perhaps you can tell me what use "3 ∩ 4 = 3" has.
As I said:
> a handy set theoretic implementation of the min() function.
i.e. if you wanted (for whatever reason) to define min(a, b) directly and briefly in your set theoretic reconstruction of the natural numbers, you can just use intersect operator and define it as "a ∩ b".
Perhaps because in terms of the interesting distinction you introduce:
> you have to distinguish things that are true of natural numbers as an algebraic structure, from things that just happen to be the case because you picked some specific representation to use for natural numbers
this particular operation seems to be part of the former rather than of the latter.
As I said:
> a handy set theoretic implementation of the min() function.
i.e. if you wanted (for whatever reason) to define min(a, b) directly and briefly in your set theoretic reconstruction of the natural numbers, you can just use intersect operator and define it as "a ∩ b".
Perhaps because in terms of the interesting distinction you introduce:
> you have to distinguish things that are true of natural numbers as an algebraic structure, from things that just happen to be the case because you picked some specific representation to use for natural numbers
this particular operation seems to be part of the former rather than of the latter.