In [35] Niebergall identifies certain features a good notion of theory reduction should have and are arguably accepted by most in the community32. He then presents a formalisation of these features in an unspecified base theory33, that should be as weak as possible. The
following axioms axiomatise the notation SρT (read as “S is reducible to T ”), where S, T, U, E, F range over theories as formalised objects in the unspecified base theory.
1. ∀S, T (S ⊂ T ⇒ SρT )
2. ∀S, T, U (SρT ∧ T ρU ⇒ SρU )
3. ∀S, T (SρT ⇒ (T is consistent ⇒ S is consistent )) 4. ∀S, T (SρT ⇒ ∀ψ ∈ S∃ϕ ∈ T {ψ}`ρ{ϕ}`)
5. ∀E, F finitely axiomatisable
EρF ⇒ IΣ1 ` Con(F ) → Con(E).
The most distinguishing characteristic of the axiomatic approach is, that it does not rest on the idea that a reduction has to be given by a function.34 This accords with some people’s
view that reduction should not be a term-term, predicate-predicate or formula-formula relation, as stated in [35].
We do not wish to argue here for the axioms 1-4, in fact we doubt whether 1-3 would be subject to any objections anyway. Axiom 5 is chosen that way, because provable
32He calls them linguistic intuitions.
33In [35] on page 52 Niebergall explains that, since his reducibility logic (a modal logic like provability
logic) is too weak to distinguish interpretability from proof-theoretical reduction, a “pure” axiomatization does not seem fruitful.
34Niebergall links the lack of a function to the question whether a reduction should be structure preserving
consistency-reduction is a wish of the community and the restriction to finite theories rules out non-standard representations of theories in arithmetic.35 The subtheory relation, interpretability and local interpretability obviously satisfy the axioms above. Moreover a very surprising property of this axiom system is that it even almost defines interpretability.
Theorem 2.3.1 If SρT is governed by axiom 1-5, IΣ1 ⊂ T and T is reflexive36, then
• If SρT , then S locT .
• If Q ⊂ S, then: SρT ⇒ S ⊆Π0 1 T .
• If S is axiomatisable, then: if SρT , then S T .
Proof
See [35, p. 41].2
Since S locT satisfies axiom 1-5 we get the following corollary.
Corollary 2.3.2 If Q ⊂ S, S is axiomatisable, IΣ1 ⊂ T and T is reflexive, then:
• S locT ⇒ S ⊆Π0
1 T and S T .
• If E is finitely axiomatisable, then: SρT and S ` Con(E) implies T ` Con(E).
Moreover the close relationship with interpretability is preserved when axiom 5 is weakened to
6. SρT ∧ S ` Con(E) ⇒ T ` Con(E), where E is a finitely axiomatizable theory.
35For in the case of E = {e
0, ..., ek} one can chose the canonical representation x = pe0q∨...∨x = pekq 36A theory T is called reflexive, if it proves the consistency of all its finite fragments.
Theorem 2.3.3 If SρT is governed by axiom 1-4 and 6, IΣ1 ⊂ T and S is a reflexive
extension ofQ, then
• SρT ⇒ S ⊆Π0 1 T
• If SρT and S is axiomatisable, then S T .
Proof
See [35, p. 42].2
However there are some differences with interpretability. For instance the property of interpretability given by Theorem 2.2.5 does not hold in general for a relation governed by axioms 1-5 as shown in [35, p. 44].
More impotently the above axiom system is not satisfied by proof-theoretical reduction, because proof-theoretical reduction fails to satisfy axiom 4. For instance it is a standard result of proof-theory that ACA0 ≤ P A. But if ρ is governed by axioms 1-5, then
ACA0 is not ρ-reduceable to P A, which is an easy result from the theorem following the
definition of ACA0.
Definition 2.3.4 ACA0is formulated inL2({0, =, S, +, ·}) with all axioms of Q together
with the single second order induction axiom
X(0) ∧ (∀x)[X(x) → X(S(x))] → (∀x)(X(x)), the axiom of extensionality
X = Y ↔ (∀x)(X(x) ↔ Y (x)) and arithmetical comprehension
(∃X)(∀x)(X(x) ↔ ϕ(x)), whereX and second-order quantification do not occur in ϕ.
Theorem 2.3.5 1. ACA0 ≤ P A
2. ACA0 6 P A
Proof
ACA0 ≤ P A can be easily seen by working in IΣ1. If ACA0 ` ϕ for a sentence
ϕ ∈ L1Q, then there is a deduction in a Gentzen system for some sequent Γ ⇒ ϕ, where Γ ⊂ ACA0. As explained in Section 2.1, cut-elimination can be done in IΣ1, therefore
we have a cut-free deduction of Γ ⇒ ϕ. By the subformula property all second order variables occurring in the deduction already occur in Γ ⇒ ϕ. Next we substitute for every second-order variable which occurs in an instance of a comprehension axiom of Γ, its comprehension formula. The comprehension axioms therefore becomes logical valid formulas, which can be cut out. For any remaining X(y) we substitute y = 0. Note that there cannot be any second-order variables left after this procedure. Hence, since P A embraces induction for any formula of L1Q, we get a sequent ∆ ⇒ ϕ with ∆ ⊂ P A, which leads to P A ` ϕ.
It remains to show that ACA0 6 P A. We only defined for first order theories but
it should be clear how one can extend it to second order. However in the second-order case Theorem 2.2.6 proceeds in the same way. Towards a contradiction let’s assume that ACA0 P A. Then by a version of Theorem 2.2.6 for any finite subtheory of
ACA0, let’s call it E, P A ` Con(E). But as we know from [26, p. 154], ACA0 is
finitely axiomatizable. Therefore P A ` Con(ACA0). But since ACA0 ≤ P A, this gives