[FOM] Weak categorical theories of arithmetic

[Aatu Koskensilta]

   Consider the theory T obtained from Robinson arithmetic by going  
second-order and adding as axiom

     If Con'(T) then induction holds.

where Con'(T) is a formula that says that, using some specified rules  
of inference for second-order logic, no contradiction can be derived  
from the axioms of T, where "contradiction", "rule of inference", and  
so on, have their quantifiers restricted to the smallest inductive set  
(a set containing 0 and closed under the successor function). The  
theory T is categorical, on standard semantics for second-order logic,  
but weak in the sense that it fails to prove e.g. basic facts about  
addition, where provability is understood in the sense of the rules  
inference given. T is not proof-theoretically weak, in that we can  
interpret second-order arithmetic in it, simply by restricting all the  
axioms to the smallest inductive set. Is this true of all (finitely?)  
axiomatizable second-order theories of arithmetic that are  
categorical? My hunch is that it is, but here, alas, my poor brains  
short-circuit and I'm unable to come up with a proof...

-- 
Aatu Koskensilta (aatu.koskensilta at uta.fi)

"Wovon man nicht sprechen kann, darüber muss man schweigen"
  - Ludwig Wittgenstein, Tractatus Logico-Philosophicus