Is naive set theory capable of describing and pinning down the intended model of arithmetic?
On the one hand, it doesn't seem very likely. With so many more sets out there to cause trouble, and easy ways to make isomorphisms fail (the totality of the ordinals isn't even isomorphic to itself, for instance), the odds of showing that any two peano relations coincide are low.
On the other hand, taking naive set theory as our base opens some intriguing possibilities. In particular, we can stay in a first order language, while at the same time, noting that the axiomatic equivalence
x is a member of X iff Xx
essentially grants us all the expressive power of second (and higher) order logic. And we know that classical second order theories can establish (classical) categoricity.
And so the question of the month is from Peter Smith's Godel book (p. 186). He's asking rhetorically, but in the paraconsistent context it's an open problem:
"If second order arithmetic does pin down the structure of the natural numbers, then -- given that any arithmetic sentence makes a determinate claim about this structure -- it apparently follows that this theory does enough to settle the truth value of every arithmetic sentence. Which makes it sound as if there can after all be a (consistent) negation-complete axiomatic theory of arithmetic..."
Well, it's an open problem if you leave out that "consistency" part...
