AN ABSTRACT BACHMANN-HOWARD PRINCIPLE 77 For Ω < ε(T ) u

α t0 we obtain Ω + s = ωΩ+ ωeγ _< ε(T )u α ω t0 _{+ · · · + ω}tn _{= Ω + t.}

Part (iii) relies on the characterization of ranks in Lemma 2.1.16. In particular, recall |Ω|ε(T )u

α = 0. 

Together with Lemma 2.1.19, it follows that ϑ(Ω + s) is an ε-number for all terms s ∈ ε(T )u_α. The following notation will help to recover the usual Bachmann- Howard construction:

Definition 2.1.22. Given a Bachmann-Howard collapse ϑ : ε(T )uα BH

−−→ α, we define the function

¯ ϑ : ε(T )u_α→ ε(T )u_α∩ Ω, ϑ(s) := i¯ α◦ ϑ(Ω + s) = eϑ(Ω+s). For s ∈ ε(T )u_α, we abbreviate s∗ := iα(|s|ε(T )u α) ∈ ε(T ) u α∩ Ω.

The properties of a Bachmann-Howard collapse can now be stated as follows: Proposition 2.1.23. Assume that Tu and thus ε(T )u is a proto-dilator, and that ϑ : ε(T )u_α −−→ α is a Bachmann-Howard collapse. For s, t ∈ ε(T )BH u

α we have ¯ ϑ(s) <_{ε(T )}u α ¯ ϑ(t) ⇔    either s <ε(T )u α t and s ∗ _< ε(T )u α ¯ ϑ(t), or t <ε(T )u α s and ¯ϑ(s) ≤ε(T )uα t ∗_.

Proof. We begin with “⇐”: First assume s <ε(T )u

α t and s ∗ _< ε(T )u α ¯ ϑ(t). As iα is an embedding, the latter implies |s|ε(T )u

α < ϑ(Ω + t). Using Lemma 2.1.21(iii)

we get |Ω + s|_{ε(T )}u

α < ϑ(Ω + t). Also, Lemma 2.1.21(ii) yields Ω + s <ε(T )uα Ω + t.

Condition (i) of Definition 2.1.3 gives ϑ(Ω+s) < ϑ(Ω+t), and then ¯ϑ(s) <ε(T )u α ϑ(t),¯

as desired. Now assume ¯ϑ(s) ≤ε(T )u α t

∗ _{(the condition t <} ε(T )u

α s is redundant, but

we keep it for symmetry). Using condition (ii) of Definition 2.1.3 we see |t|_{ε(T )}u

α = |Ω + t|ε(T )uα < ϑ(Ω + t).

Applying iα to both sides gives t∗ <ε(T )u α ¯ ϑ(t), and then ¯ϑ(s) <ε(T )u α ¯ ϑ(t). To show “⇒”, we may assume s 6= t. Aiming at the contrapositive, assume that the right side of the equivalence fails. Then we have either t <ε(T )u

α s and t ∗ _< ε(T )u α ¯ ϑ(s); or we have s <ε(T )u α t and ¯ϑ(t) ≤ε(T )uα s

∗_{. In both cases direction “⇐” (with s}

and t interchanged) yields ¯ϑ(t) <_{ε(T )}u α

¯

ϑ(s). The latter implies ¯ϑ(s) 6<_{ε(T )}u α

¯ ϑ(t),

We can now embed the usual notation system for the Bachmann-Howard or- dinal. This will play no official role, but similar constructions will be crucial in Sections 2.3 and 2.4.

Remark 2.1.24. Consider the constant proto-dilator γ 7→ TγX = X for a

well-order X = (X, <X) (cf. Remark 2.1.8). Recall that we have |σ|TX

γ = 0 for

any σ ∈ X = T_γX. Form the associated proto-dilator γ 7→ ε(T )X_γ and assume that ϑ : ε(T )X_α −−→ α is a Bachmann-Howard collapse. I claim that the relativizedBH Bachmann-Howard order ϑX of Rathjen and Valencia Vizca´ıno [71, Definition 2.6]

can be embedded into ε(T )X_α: The term Eσ ∈ ϑX with σ ∈ X = TαX is identified

with Eσ ∈ ε(T )Xα. Note that our rank

Eσ∗= iα(|Eσ|ε(T )X

α) = iα(|σ|TαX) = iα(0) = 0

coincides with Rathjen and Valencia Vizca´ıno’s rank assignment Eσ∗ = 0. Assum-

ing that s ∈ ϑX is identified with s ∈ ε(T )Xα, we can identify ϑs ∈ ϑX with the

element ¯ϑ(s) ∈ ε(T )X_α. Again, our rank assignment ¯

ϑ(s)∗ = eϑ(Ω+s)∗= iα(|eϑ(Ω+s)|ε(T )X

α) = iα(ϑ(Ω + s)) = eϑ(Ω+s) = ¯ϑ(s)

is as required. Also, ¯ϑ(s) = eϑ(Ω+s) ∈ ε(T )Xα behaves like an ε-number below Ω,

just as ϑs ∈ ϑX. Using the previous proposition, it is straightforward to see that

this yields an order embedding of ϑX into ε(T )Xα. Thus the well-foundedness of

ε(T )X_α (see Proposition 2.1.14) implies that ϑX is a well-order. In other words,

the abstract Bachmann-Howard principle implies that X 7→ ϑX is a type-one well-

ordering principle, over primitive recursive set theory. Also, the given embedding restricts to an embedding of ϑX ∩ Ω into ε(T )Xα ∩ Ω ∼= α (the isomorphism comes

from Lemma 2.1.17, as Lemma 2.1.18 gives α = α+). For X = ∅ the structure ϑX∩ Ω is the usual notation system for the Bachmann-Howard ordinal. This shows

that any ordinal α which admits a Bachmann-Howard collapse ϑ : ε(T )X_α −−→ αBH is at least as big as the Bachmann-Howard ordinal. From the assumption that X 7→ ϑX is a type-one well-ordering principle, Rathjen and Valencia Vizca´ıno

deduce that there are ω-models of bar induction. Recall that we want to construct transitive set models of Kripke-Platek set theory. Thus we have reached the correct proof-theoretic strength: The theory of bar induction and Kripke-Platek set theory both correspond to the Bachmann-Howard ordinal. On the other hand, transitive set models are stronger than ω-models. For this reason we have constructed a type-two well-ordering principle: The point is that we do not only get a collapse of

2.1. AN ABSTRACT BACHMANN-HOWARD PRINCIPLE 79

the “constant” order ϑX into some ordinal α ∼= ϑX∩ Ω, but rather a “fixed-point”

α ∼= ε(T )u_α∩ Ω with collapsing structure ϑ : ε(T )X α

BH

−−→ α. The following is also worth pointing out:

Remark 2.1.25. There is considerable freedom in the choice of ε(T )uα. For

example, we could strengthen ε(T )u_α by adding a term εs for each s ∈ ε(T )uα;

the terms εσ for σ ∈ Tαu would then we replaced by terms ϕ2(σ), where ϕ2

refers to the second branch of the Veblen function, which enumerates the fixed- points of α 7→ εα. Note that Proposition 2.1.14 for this stronger system would

still be provable in our base theory PRSω. Conversely, it may be possible to weaken ε(T )u_α: As ∈-induction is automatic in standard models of set theory (see Lemma 1.3.12), we only need to consider Kripke-Platek axioms of bounded com- plexity. Thus we may not need full ordinal exponentiation in our proof-theoretic arguments. Observe that there is a similar degree of freedom in the formulation of Theorem 4.4.6(ii): The statement “any set is contained in an admissible set” is, of course, equivalent to “any set is contained in an admissible set that is itself contained in an admissible set” — even though the corresponding theories KP and KP + “there is an admissible set” have different proof-theoretic strength. So for our purpose, the precise order-type of ε(T )u_α seems less important than the col- lapsing structure. The given definition of ε(T )u_α has the advantage that it relates to the familiar notation systems ϑX (from [71], cf. the previous remark) and εX

(from [52] resp. [4]).

In the rest of this section we reformulate Proposition 2.1.23 in terms of supports, rather than ranks (cf. Definition 2.1.9 and Lemma 2.1.10). This formulation will be more suitable for the functorial approach in the next sections.

Definition 2.1.26. Consider primitive recursive functions (u, α) 7→ Tαu and

(u, α) 7→ suppu_α. For a fixed value of u, assume that Tu is a proto-dilator and that suppu is a support for Tu. Define functions

E_αu: ε(T )u_α→ [α]<ω by the following recursion over terms:

E_αu(0) = ∅, E_αu(Ω) = ∅, E_αu(eγ) = {γ}, E_αu(Eσ) = suppuα(σ), Eαu(ωs0 + · · · + ωsn) = [ i≤n E_αu(si). For s ∈ ε(T )u_α we write ¯E_αu(s) := {iα(γ) | γ ∈ Eαu(s)} ⊆ ε(T )uα∩ Ω.

Note that the function (u, α, s) 7→ E_αu(s) is primitive recursive (just as the func- tion (u, α, s) 7→ Lu

α(s), cf. the discussion after Definition 2.1.11). Proposition 1.2.2

tells us that (u, α) 7→ E_αu is primitive recursive as well (and that E_αu is set-sized).

Lemma 2.1.27. If suppu is a support for Tu then Eu is a support for ε(T )u.

Proof. Condition (i) of Definition 2.1.9 is immediate. To establish condi- tion (ii), consider ordinals α < β. The required implication

E_βu(s) ⊆ α ⇒ s ∈ ε(T )u_α

can be verified by induction on s ∈ ε(T )u_β. The most interesting case is s = Eσ. By

assumption we have suppu_β(σ) = E_βu(Eσ) ⊆ α. As suppu is a support for Tu this

implies σ ∈ T_αu, and then Eσ ∈ ε(T )uα. For condition (iii) we also consider α < β.

We need E_αu(s) = Eu_β(s) for s ∈ ε(T )u_α, which is also shown by induction on s. In the crucial case s = Eσ with σ ∈ Tαu we have

E_αu(Eσ) = suppuα(σ) = suppuβ(σ) = Eβu(s),

as suppu is a support for Tu. 

We can now reformulate Proposition 2.1.23 as promised:

Corollary 2.1.28. Assume that Tu is a proto-dilator with support suppu, so that ε(T )u is a proto-dilator with support Eu. Consider a Bachmann-Howard collapse ϑ : ε(T )u

α BH

−−→ α and the associated function ¯ϑ : ε(T )u

α → ε(T )uα∩ Ω. Then,

for all s, t ∈ ε(T )u_α, we have

¯ ϑ(s) <_{ε(T )}u α ¯ ϑ(t) ⇔    either s <ε(T )u

α t and r <ε(T )uα ϑ(t) for all r ∈ ¯¯ E

u α(s),

or t <ε(T )u

α s and ¯ϑ(s) ≤ε(T )uα r for some r ∈ ¯E

u α(t).

Proof. By the proof of Lemma 2.1.10 we have |s|_{ε(T )}u

α = max(E

u

α(s) ∪ {0}).

Applying the embedding iα to both sides yields

s∗ = max<ε(T )u_α( ¯E

u

α(s) ∪ {0}).

As the term ¯ϑ(t) = eϑ(Ω+t) is different from zero, this implies

s∗ <ε(T )u

α ϑ(t)¯ ⇔ r <ε(T )uα ϑ(t) for all r ∈ ¯¯ E

u α(s).

2.2. A PREDICATIVE BACHMANN-HOWARD PRINCIPLE 81