[FOM] Weak categorical theories of arithmetic
Aatu.Koskensilta at uta.fi
Tue Jun 5 09:21:59 EDT 2012
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
More information about the FOM