Revision without ordinals

REVISION WITHOUT ORDINALS

EDOARDO RIVELLO

Abstract. We show that Herzberger’s and Gupta’s revision theories of truth can be

recast in purely inductive terms, without any appeal neither to the transfinite ordinal

numbers nor to the axiom of Choice. The result is presented in an abstract and general

setting, emphasising both its validity for a wide range of revision-theoretic constructions

and its independence from truth-theoretic assumptions.

§1. Introduction. In standard Zermelo-Fraenkel set theory (ZF) the Trans-finite Recursion Theorem grants existence and uniqueness of ordinal-length ite-rations of a given function. More precisely, given a function ρ : P → P from aset P into itself, and an operation Λ taking transfinite sequences of elements of Pas arguments and elements of P as values, we can define a unique ordinal-lengthsequence s = 〈sα | α ∈ On〉 such that

• s0 = Λ(∅),• sα+1 = ρ(sα), and• sλ = Λ(sλ),

where λ is a limit ordinal and sλ denotes the restriction of s up to λ.An important special case, ubiquitous in mathematics, is represented by mono-

tone (or non-monotone) inductive constructions, which correspond to choose afunction ρ whose iteration is progressive (i.e., sα ρ(sα), for every α, with re-spect to some fixed partial order on P ), and to take the least upper bound(lub) of the range at limits (Λ(sλ) = lubsα | α < λ).

Revision constructions can be seen as a generalisation of induction (see, forinstance, [12]) based on the corresponding notion of revision sequence: no as-sumptions are made on the function ρ and Λ is allowed to be any operation whichsatisfies the following coherence condition: liminf(s λ) Λ(s λ) (here, liminfdenotes the inferior limit operation on sequences of elements from a partial order〈P,〉. See below for a definition).

Revision sequences was first introduced, in the context of the Semantic theoriesof Truth, as a mathematical way of formalising Herzberger’s [6] and Gupta’s [4]accounts for self-referential truth. Later, revision-theoretic methods were also

The main result in this paper was announced at the “Numbers and Truth” symposium, held

in Gothenburg, 19-21 October 2012. I wish to thank the organisers for the opportunity of giving

the talk. The article was written whilst I was a postdoctoral research fellow at Scuola NormaleSuperiore in Pisa. I express my gratitude to Gabriele Lolli for his advise and support. The final

version benefits from unvaluable feedbacks by Albert Visser, during a visit at the University of

Utrecht, in May 2013, supported by the European Science Foundation for the activity “NewFrontiers of Infinity: Mathematical, Philosophical and Computational Prospects”.

1

2 EDOARDO RIVELLO

employed in the theory of definitions, contrasting them with induction ([5], [14],among others), and in modelling circular objects ([1]).

It is well-known that most of definitions and proofs given by means of induc-tive transfinite iterations, can also be done by “purely inductive means”, i.e., bytechniques substantially based on Zermelo’s 1908 proof of the Wellordering theo-rem and on Kuratowski’s Fixed point theorem1: see [3] for a detailed account ofthis in the context of Kripke’s theory of truth. Accordingly, on a formal level, theuse of transfinite iteration appears to represent a substantive distinction betweeninduction and revision, being avoidable in the former case.

The present work is devoted to show that transfinite sequences can actuallybe avoided even in a wide range of revision-theoretic constructions, Herzberger’sand Gupta’s original proposals included. Moreover, will be emphasised the factthat the axiom of Choice is not necessary in establishing the basic properties ofrevision, a fact not explicitely noticed in most literature on the topic.

The main result, the fact that the key notions of revision, namely stabilityand recurrence, can be defined in ZF by purely inductive means, will be statedand proved in Section 3. Section 2 will be devoted to an analysis of revision inorder to both provide some lemmata needed in the proof of the main theoremand refine some known results about Herzberger and Gupta sequences.

It is not my intention to claim that the eliminability of the ordinals fromHerzberger and Gupta revision implies that the intuitive idea of a revision processdoes not play any important role in developing these theories. The presentexposition sticks on the formal level of revision, leaving to other works the taskof exploring the philosophical considerations, if any, which could be derived fromthe mathematical facts.

1.1. The (semi)-formal setting. Revision sequences are particular trans-finite sequences of objects of some sort; for instance, in the original approachto truth by Gupta [4], these objects were set of sentences. In the literature onrevision constructions we can find different choices of objects in order to definea general notion of revision sequence: for instance, functions (in [5]) or subsetsof natural numbers (mainly in studies about the complexity of revision, see [9]).

For the present purposes, all these abstract settings are substantially equiva-lent. I choose to follow Visser [12] in presenting revision sequences in an order-theoretic fashion, to emphasise the fact that all results mentioned in this paperdo not depend on the internal structure carried by the objects (as functions orsets) but only on the interplay between two order-theoretic structures: the one tobe assumed on the domain of the objects and the one induced by the sequenceson their own ranges.

Let P = 〈P,〉 be any partially ordered set of objects, called hypotheses. ForX ⊆ P , lub(X) and glb(X) will denote the least upper bound and the greatestlower bound of X in P, respectively, when they exist. A sequence from P willbe any function s = 〈sα | α < lh(s)〉 such that the range of s, ran(s), is includedin P , where lh(s) could denote an ordinal number as well as the class On of allordinals. In the former case we say that s is a set sequence, in the latter we saythat s is an ordinal-length sequence.

1See [7].

REVISION WITHOUT ORDINALS 3

For a sequence s we will use both the notations sα or s(α) with the samesense. For every α < β < lh(s) let s α denote the restriction of s up to α,Sαβ = sξ | α ≤ ξ < β, Sα = sξ | α ≤ ξ < lh(s) and Sα = sξ | ξ < α.

The key notion in doing a revision process is the inferior limit operation (see[8, Chap. 4, Sec. 2]). In the current abstract setting we can define this operationon sequences as follows. Let s be any sequence from P .

Definition 1.1 (Inferior Limit). Assuming that all glb’s and lub’s in the fol-lowing formula exist, define

liminf(s) = lubglb(Sα) | α < lh(s).

In this paper we will always assume that the liminf operation is defined onevery sequence from P . For, it is enough to assume the following two conditionson P:

1. lub(X) exists for every nonempty wellordered chain X ⊆ P , and2. glb(X) exists for every nonempty subset X ⊆ P .

For instance, these conditions are met in every ccpo (coherent complet partialorder, [12, Fact 4, p. 191 ]) and, in particular, in the set [A]B of all partialfunctions from a set A into a set B.

A hypothesis h ∈ P is said to be s-coherent if and only if liminf(s) ≤ h.Following [9], we will say that an operation ∆, assigning to every (set) sequences from P of limit length a hypothesis ∆(s) ∈ P , is a bootstrapping policy if ∆(s)is s-coherent for every s in the domain of ∆. Collections Γ of bootstrappingpolicies will be called limit rules. A one-element limit rule will be a limit ruleof the form Γ = ∆, for some bootstrapping policy ∆. The one-element limitrule Γ0 = liminf will be called the fundamental limit rule.

Fix a function ρ : P → P (called the revision operator) and let Γ be alimit rule. We say that an ordinal-length sequence s from P is a (ρ,Γ)-revisionsequence if

1. for all ordinals α, we have sα+1 = ρ(sα), and2. there is a ∆ ∈ Γ such that for each limit ordinal γ we have sγ = ∆(sγ).

A hypothesis h ∈ P will be said cofinal in s if and only if for every α < lh(s)there exists β < lh(s) such that α ≤ β ∧ sβ = h. The set of all hypothesescofinal in s will be denoted by Cf(s).

A hypothesis h ∈ P will be said (ρ,Γ)-recurring if and only if h occurs cofinallyoften in some (ρ,Γ)-revision sequence. The set of (ρ,Γ)-recurring hypotheses willbe denoted by RecΓ(ρ).

Herzberger and Gupta rules. Herzberger and Gupta sequences are usuallypresented as sequences of subsets of some set, as follows2. Let A be any set andPA = 〈P(A),⊆〉 be its power set ordered by inclusion. Let s be a sequence fromP(A).

An element a ∈ A is said to be positively stable in s if and only if there existsα < lh(s) such that, for all β < lh(s), if α ≤ β then a ∈ sβ . Similarly, a issaid to be negatively stable in s if and only if there exists α < lh(s) such that,for all β < lh(s), if α ≤ β then a /∈ sβ . We denote the set of the positively

2See, for instance, [6] or [9]

4 EDOARDO RIVELLO

stable elements and the set of the negatively stable elements in s by stab+(s)and stab−(s), respectively. The set stab(s) = stab+(s) ∪ stab−(s) will be the setof the elements stable in s.

A constant limit rule ΓX is a one-element limit rule determined by a fixedsubset X ⊆ A as follows:

∆X(s) = stab+(s) ∪ (X − stab−(s)),

and ΓX = ∆X. In particular, we obtain the Herzberger rule ΓH by settingX = ∅. It is straightforward to see that stab+(s) ∪ (∅ − stab−(s)) = stab+(s) =liminfPA(s) hence, in this setting, the Herzberger limit rule ΓH = Γ∅ coincideswith the fundamental limit rule Γ0 = liminfPA.

The Gupta rule ΓG is defined by3:

∆G(s) = stab+(s) ∪ (s0 − stab−(s)),

and ΓG = ∆G. This too is a one-element limit rule, but is not constant, sincethe Gupta policy ∆G = ∆s0 is not of the form ∆X for a fixed X ⊆ A, given that∆G depends on the starting point s0 of s.

We summarise well-known results which hold for both constant and Guptarevision sequences in the following4

Fact 1.2 (Constant and Gupta revision). Let Γ be either a constant limit ruleor the Gupta limit rule and let s be a (ρ,Γ)-revision sequence. Then

Periodicity: s is cyclic, i.e., there exist ordinals γ, δ such that

s(γ + δ · α+ ξ) = s(γ + ξ),

for every α ∈ On and ξ < δ. We call the least such ordinals the initialordinal and the period of s, respectively.

Cofinality: Cf(s) = Sγγ+δ.

Alignment: There exists an ordinal η such that Cf(s) = Cf(sη) and stab(s) =stab(sη). We call the least such ordinal the alignment ordinal of s.

Closure: The initial and the alignment ordinals and the period of s are less thanthe least uncountable cardinal greater than the cardinality of P .

§2. Stability-dependent revision. 5

For the purposes of this section it is more convenient to recast constant andGupta limit rules in a partial functions framework, as follows. Let [A]0, 1 de-note the set of all functions from a subset X ⊆ A into 0, 1 and let P[A] =

〈[A]0, 1,⊆〉. As customary, we can identify every subset X ⊆ A with its cha-racteristic function χ(X) ∈ A0, 1. Hence, the elements of P(A) are in one-one

3See [13, p. 349]4For further details and credits about Fact 1.2 and its proofs, see next section.5Most of the statements proved in this section follow from standard facts established in the

general theory of ordinal arithmetic: for notations, definitions and proofs of such facts, we

refer to any textbook in set theory and, in particular, to [11]. Throughout this paper, the

usual notations for the arithmetical operations of addition, multiplication and exponentiationwill always denote the corresponding operations on ordinals: no risk of confusion with cardinal

arithmetic is possible.


correspondence with the total functions in [A]0, 1, i.e., with the maximal ele-ments of P[A]. For this reason we reserve the name of hypotheses to these latter.

P[A] meets the conditions for the existence of the inferior limit of every sequence

from [A]0, 1, so let denote this operation by liminf [A]. For every sequence s from[A]0, 1, liminf [A](s) itself is a partial function taking arguments in A and valuesin 0, 1, so we feel free to write liminf [A](s)(x) = y for 〈x, y〉 ∈ liminf [A](s).

We now redefine the notions of constant and Gupta limit rules in the newsetting provided by P[A]. A constant limit rule Γh is a one-element limit rule

determined by a fixed hypothesis h ∈ A0, 1 as follows:

∆h(s) = liminf [A](s) ∪ 〈x, y〉 ∈ h | x /∈ dom(liminf [A](s)),and Γh = ∆h. We call Herzberger rule, denoted by ΓH, the constant limit rulegiven by h = χ(∅) = 〈x, 0〉 | x ∈ A.

The Gupta limit rule ΓG is the one-element limit rule defined by:

∆G(s) = liminf [A](s) ∪ 〈x, y〉 ∈ s0 | x /∈ dom(liminf [A](s)),and ΓG = ∆G.

Given a sequence s from P(A) denote by s∗ the corresponding sequence ofhypotheses from [A]0, 1 defined by s∗(α) = χ(s(α)), for every α < lh(s). Then:

1. stab+(s) = x ∈ A | liminf [A](s∗)(x) = 1.

2. stab−(s) = x ∈ A | liminf [A](s∗)(x) = 0.

3. stab(s) = dom(liminf [A](s∗)).

4. χ(∆X(s)) = ∆h(s∗), where h = χ(X) and χ(∆s0(s)) = ∆s∗0(s∗).

Hence, in P[A], the given definitions of constant limit rule Γh and of Gupta ruleΓG faithfully correspond to the notions of constant limit rule ΓX and of Guptarule, respectively, previously defined in PA. However, in the setting P = P[A],the Herzberger rule ΓH no longer coincides with the fundamental rule Γ0 =liminf [A]. On the other hand, every bootstrapping policy of the form ∆h or∆G can be seen as a function of the starting point and of the inferior limitoperation. For, let c be the two-arguments function defined by

c(g, p) = p ∪ 〈x, y〉 ∈ g | x /∈ dom(p),for every g ∈ A0, 1 and p ∈ [A]0, 1 and, for every h ∈ A0, 1, put ch(g, p) =c(h, p). Then

• ∆h(s) = c(h, liminf(s)) = ch(s0, liminf(s)), and• ∆G(s) = c(s0, liminf(s)).

Therefore, it is natural to generalise this situation to an arbitrary settingP = 〈P,〉, as in the following

Definition 2.1 (Stability dependence). An operation ∆, taking as argumentslimit length sequences from P , is said to be stability-dependent if and only if thereexists a function f : P × P → P such that ∆(s) = f(s0, liminf(s)), for every sin the domain of ∆.

We say that Γ is a generalised limit rule if Γ is a collection of operations de-fined on the class of the set sequences from P of limit length. The adjective“generalised” refers to the fact that we no longer restrict ourselves to only con-sider collections Γ of bootstrapping policies, i.e., limit length sequences from P

6 EDOARDO RIVELLO

satisfying the coherence condition, even though this is the intended situationwhen we speak about revision. We say that a generalised limit rule Γ is stability-dependent if all operations in Γ are stability-dependent. Obviously, all constantlimit rules and the Gupta rule are stability-dependent.

In this section we will prove, in the abstract setting P, a version of Fact 1.2for stability-dependent generalised limit rules. Even though proofs of this resultfor Herzberger and Gupta revision are already available in the literature, thereare two reasons for giving here the details: first, some steps of the proof and,in particular, the Unfolding Lemma 2.15, will be needed in proving the maintheorem in the next section; secondly, this version of the theorem and of theproof slightly improves the published results under several aspects 6:

• The statement is more general, since applies to stability-dependent (gen-eralized) limit rules, a concept which stricly includes constant and Guptalimit rules. Moreover, this provide an alternative, uniform way of han-dling both Herzberger and Gupta revision, a way which can be contrastedwith, for instance, Visser’s approach of considering Gupta sequences as aparticular case of Herzberger ones ([12, p. 222]).

• The proof proceeds by identifying some notions about sequences whichallow to separate the facts depending on the order-theoretic structure as-sumed on P from the facts which only depends on the ordering induced bythe sequence on its own range.

• The arguments used in the proof will be entirely formalisable in Zermelo-Fraenkel (ZF) set theory, making no use of the axiom of Choice.

A last remark about the extent of Fact 1.2. In the partial functions frame-work P[A], the hypotheses are usually intended to be total functions. In theabstract setting, this would correspond to be maximal elements in P. Besides,constant and Gupta limit rules are supposed to be (one-element) collections ofboostrapping policies, namely of operations ∆ satisfying the coherence condition:liminf(s) ≤ ∆(s). We stress the fact that our definition of stability-dependentrevision do not assume neither the maximality of the hypotheses nor the cohe-rence of the bootstrapping policies, making explicit the observation that Fact1.2 does not depend on these assumptions.

Formalisation. Given the revision operator ρ and the limit rule Γ, every(ρ,Γ)-sequence is an ordinal-length sequence, hence a proper class. As a conse-quence

recurrence is not prima-facie first-order definable, since the definitionuses a quantification over proper classes ([9, p. 26]).

In the literature, the formalisation of the notion of recurrence for Herzbergerand Gupta revision is achieved either as a by-product of the Alignment andClosure points of Fact 1.2 (by restricting the quantification to set sequences of afixed length determined by the Closure property) or, as in [9, p. 26], by providingan equivalence between recurrence and some form of reflexivity.

6Actually, most of Fact 1.2 was established in [6] without proof. The reader of [6] is referred

to Herzberger’s unpublished typescript Notes on periodicity, May 1980. Unfortunately, I hadno opportunity to read Herzberger’s notes. Hence, my following comments only apply to the

published proofs of Fact 1.2 mentioned in the Historical note paragraph below.


Actually, if we regard Herzberger and Gupta limit rules as (one-element)stability-dependent rules, the formalisation of the notion of (ρ,Γ)-recurrence inZF (without Choice) can also be obtained from the informal definition as follows.The Transfinite Recursion Theorem provides a formula Φ(p, ρ, f, α, y) which uni-formly defines, with the element p ∈ P and the two functions ρ : P → P andf : P × P → P as parameters, all ordinal-length sequences sp,ρ,f such that

• sp,ρ,f (0) = p,• sp,ρ,f (α+ 1) = ρ(sp,ρ,f (α)), and• sp,ρ,f (γ) = f(p, liminf(sp,ρ,f γ)), for γ limit,

by putting sp,ρ,f (α) = y ↔ Φ(p, ρ, f, α, y).Hence, Cf(sp,ρ,f ) will be a single formula depending on the parameters p, ρ

and f :

Cf(sp,ρ,f ) = h ∈ P | ∀α ∃β (α ≤ β ∧ sp,ρ,f (β) = h).

Every stability-dependent limit rule Γ is, by definition, determined by a setFΓ of functions from P × P into P . Hence we can take, as our official definitionof (ρ,Γ)-recurrence, the formula

RecΓ(ρ) = h ∈ P | ∃p ∈ P,∃f ∈ FΓ (h ∈ Cf(sp,ρ,f )).

For instance, if we take Γ = ΓH (the Herzberger limit rule in the partialfunctions setting), then FΓ = c0, where c0 : P × P → P is the function

c0(g, q) = q ∪ 〈x, 0〉 | x ∈ P − dom(q).

Hence h is (ρ,ΓH)-recurring if and only if there exists p ∈ P such that h ∈Cf(sp,ρ,c0).

Cofinally invariant sequences. Let us back to the semi-formal setting. Thekey notions of the revision theory are, for each given sequence s from P , stability(involved in the revision sequence construction) and cofinality (on which is basedrecurrence). While the former depends in some way on assuming an order-theoretic structure on the hypotheses (recall that, in the abstract setting P,stabilities in a sequence s are represented by the inferior limit of s), the latteronly arises from the order induced by the sequence on its own range.

Our present goal is to state and prove a version of Fact 1.2 for arbitrarysequences, only involving the notion of cofinality. To do this, first we need somedefinitions and lemmata.

Eventual invariance. Given a sequence s and ordinals α ≤ β ≤ lh(s) wedenote by sα,β the sequence defined as follows:

• lh(sα,β) = β − α.• sα,β(ξ) = s(α+ ξ), for every ξ < lh(sα,β).

When β = lh(s) we write sα for sα,lh(s).

Definition 2.2. We say that two sequences s, t are eventually equivalent ifand only if there exists α < lh(s) and β < lh(t) such that sα = tβ

Definition 2.3. We say that a sequence s is eventually invariant if and onlyif, for every γ, δ < lh(s), sγ = sδ whenever sγ and sδ are eventually equivalent.

8 EDOARDO RIVELLO

It is straightforward to see that a sequence s is eventually invariant if and onlyif sα = sβ → sα(ξ) = sβ(ξ), for every α ≤ β < lh(s) and ξ < lh(s)− β.

Definition 2.4. We say that a sequence s is eventually cofinal if and only ifthere exists α < lh(s) such that Cf(s) = Sα.

Lemma 2.5. Let s be an eventually invariant sequence. Suppose there existβ < γ < lh(s) such that sβ = sγ and let δ = γ − β. Then

(1) s(β+δ ·k+ξ) = s(β+ξ), for every k < ω and ξ ≤ δ such that (β+δ ·k+ξ) <lh(s).

(2) If (γ + δ ·ω) ≤ lh(s) then s(γ + δ ·ω) = s(β + δ ·ω) is eventually cofinal,witnessed by both γ and β.

Proof. (1) By induction on k. k = 0 is trivial. Let k = n + 1. By theinductive hypothesis, s(β+δ ·n+δ) = s(β+δ) = s(γ) = s(β). Hence, by eventualinvariance, s(β+ δ ·k+ ξ) = s(β+ δ · (n+ 1) + ξ) = s(β+ δ ·n+ δ+ ξ) = s(β+ ξ).a

(2) First, observe that δ+δ ·ω = δ ·(1+ω) = δ ·ω, hence γ+δ ·ω = β+δ+δ ·ω =β + δ · ω. Let x ∈ Sγγ+δ·ω and γ ≤ ζ < (γ + δ · ω). Hence x = s(γ + δ · n + ξ)

and ζ = γ + δ · n′ + ξ′, for some n, n′ < ω and ξ, ξ′ < δ. Let k = n′ + 1. By (1),x = s(β + ξ) = s(γ + δ · k + ξ) and (γ + δ · k + ξ) > ζ. By the arbitrarity of ζ,

x ∈ Cf(s(γ+δ ·ω)). By the arbitrarity of x, Sββ+δ·ω = Sγγ+δ·ω ⊆ Cf(s(γ+δ ·ω)),

so s (γ + δ · ω) = s (β + δ · ω) is eventually cofinal, witnessed by both γ andβ. a

The Unfolding operation. Let Lim∗ denote the class of all zero or limitordinals. For any sequence s, define

µs(γ) = minδ < lh(s) | δ ∈ Lim∗ ∧ sδ = sγ,

for γ ∈ lh(s) ∩ Lim∗ (we will drop the subscript s from µs when the sequence sis clear from the context).

It immediately follows from the definition that sµ(γ) = sγ , µ(γ) ≤ γ andµ(µ(γ)) = µ(γ), for every γ < lh(s).

Define µ(s) = γ ∈ lh(s) ∩ Lim∗ | µs(γ) = γ.

Definition 2.6 (Unfolding operation). Let s be any sequence. Let θ be alimit ordinal less than lh(s). Define 〈θn, ζn, δn | n ∈ ω〉 by ω-recursion as follows:

• θ0 = θ.• ζn = µ(θn) if θn < lh(s). Otherwise ζn = θn.• δn = θn − ζn.• θn+1 = θn + δn · ω.

The sequence 〈θn, ζn, δn | n ∈ ω〉 is well defined since ζn ≤ θn for every n,so θn − ζn makes sense. By definition, θn, δn are limit, θ ≤ θn ≤ θn+1 ands(ζn) = s(θn) for every n ∈ ω.

Let p = p(θ) denote the first natural number such that ζp+1 ≥ ζp (it must existsince, otherwise, 〈ζn | n < ω〉 would be a strictly descending infinite sequence ofordinals).

Define θ0 = ζp = µ(θp) and θ∗ = θp+1.


The definition of the map θ 7→ θ∗ depends on µ = µs, hence depends on s.However we will keep implicit this fact, in order to simplify notations.

Remark 2.7. By ordinal arithmetic, θ∗ ≤ θ · ωp+1.

Lemma 2.8. Let s be an eventually invariant sequence. Let θ < lh(s) be alimit ordinal such that µ(θ) < θ. Then, for every n ≤ p,

(1) ζn < θ < θn < θn+1.(2) s(θn+δn·k+ξ) = s(ζn+ξ), for every k < ω, ξ ≤ δn such that (θn+δn·k+ξ) <

lh(s).

(3) Sζnθn = Sζnθ .

(4) Sβθn+1= Sζnθ , for every ζn ≤ β < θn+1, if θn+1 < lh(s).

In particular, for n = p,

• θ0 < θ < θ∗.• Sβθ∗ = Sθ

0

θ for every θ0 ≤ β < θ∗, if θ∗ < lh(s).

Moreover, if θ∗ < lh(s) then µ(γ) < γ for every θ ≤ γ < θ∗. In particular, eitherµ(θ∗) < θ or µ(θ∗) = θ∗.

Proof. (1). By induction on n. ζ0 < θ by hypothesis. Let ζn+1 ≤ p.By definition of p and by the inductive hypothesis, ζn+1 < ζn < θ. Sinceζn < θ ≤ θn, δn > 0 for every n ≤ p. Hence θn < (θn + δn · ω) = θn+1 for everyn ≤ p. So θ = θ0 < θ1 and, by the inductive hypothesis, θ < θn+1 for everyn ≤ p. a

(2). By Lemma 2.5.(1). a(3). By induction on n. Sζ0θ0 = Sζ0θ by definition. Suppose n + 1 ≤ p. Since

ζn+1 < ζn, Sζn+1

θn+1= S

ζn+1

ζn∪ Sζnθn+1

. By definition, Sζnθn+1= Sζnθn ∪ S

θnθn+1

=

Sζnθn ∪ Sθnθn+δn·ω. By (2) , s(θn + δn · k + ξ) = s(ζn + ξ) for every k ∈ ω and

ξ ≤ δn. Hence Sθnθn+1= Sζnθn . By the inductive hypothesis, Sζnθn = Sζnθ . So,

Sζn+1

θn+1= S

ζn+1

ζn∪ Sζnθ = S

ζn+1

θ . a(4). Claim 1 : If θn+1 < lh(s) then Sθnθn+1

= Sζnθ .

Proof of Claim 1. Let x ∈ Sθnθn+1. Hence x = sα for some θn ≤ α < θn+1.

Since α = θn + δn · k + ξ, with k ∈ ω and ξ < δn, by (2), x = s(ζn + ξ), with

ξ < δn so, by (3), x ∈ Sζnθn = Sζnθ .

Conversely, let x ∈ Sζnθ . Hence x = s(ζn+ξ) for some ξ < δn. Take α = θn+ξ.

Then, by (2) (with k = 0), x = sα ∈ Sθnθn+1, since θn+ξ < θn+δn < θn+δn ·ω =

θn+1. aBy (3) and Claim 1, if θn+1 < lh(s) then Sζnθn+1

= Sζnθn ∪ Sθnθn+1

= Sζnθ ∪ Sζnθ =

Sζnθ .

Claim 2 : If θn+1 < lh(s) then Sβθn+1⊆ Sθnθn+1

, for every θn ≤ β ≤ θn+1.

Proof of Claim 2. Sβθn+1⊆ Sθnθn+1

since θn ≤ β. θn ≤ β < θn+1 implies

β = θn + δn · k + ξ for some k ∈ ω and ξ < δn. Let x ∈ Sθnθn+1. Hence x = sα for

some α = θn+ δn ·m+η, with m ∈ ω and η < δn. Take α′ = θn+ δn · (k+1)+η.

Then, by (2), x = s(α) = s(ζn + η) = s(α′) and β < α′ < θn+1, so x ∈ Sβθn+1. a

10 EDOARDO RIVELLO

Proof of (4). By the claims 1 and 2, Sβθn+1= Sθnθn+1

= Sζnθ . aProof that µ(γ) < γ for every θ ≤ γ < θ∗, assuming that θ∗ < lh(s). Let

θ ≤ γ < θ∗ and suppose, towards a contradiction, µ(γ) = γ. θ ≤ γ < θ∗ impliesθn ≤ γ < θn+1 for some n ≤ p. θn ≤ γ < θn+1 implies γ = θn+δn ·k+ξ for somek < ω and ξ ≤ δn and, by (2), s(θn + δn · k + ξ) = s(ζn + ξ) for every k ∈ ω andξ < δn. Hence there exist n ≤ p and ξ ≤ δn such that s(γ) = s(ζn + ξ). Since γis limit also ξ and ζn + ξ are limit. By (1), ζn < θ ≤ θn, so ζn + ξ < γ = µ(γ),contradicting the minimality of µ(γ). a

Cofinal invariance.

Definition 2.9. We say that two sequences s, t are cofinally equivalent if andonly if they are eventually equivalent or both lh(s) and lh(t) are limit ordinalsand there exist α < lh(s) and β < lh(t) such that:

∀γ < lh(s) [α ≤ γ → ∃δ < lh(t) (β ≤ δ ∧ Sγ = T δ)],

and

∀δ < lh(t) [β ≤ δ → ∃γ < lh(s) (α ≤ γ ∧ Sγ = T δ)].

Lemma 2.10. Let s and t be cofinally equivalent. Then Cf(s) = Cf(t).

Proof. If s and t are eventually equivalent then thesis is trivial. Let lh(s)and lh(t) be limit and let α, β witness that s and t are cofinally equivalent. Letx ∈ Cf(s) and fix θ < lh(t). Let δ = maxβ, θ. By the hypothesis, there existsγ < lh(s) such that α ≤ γ and Sγ = T δ. Since x ∈ Cf(s) there exists γ′ < lh(s)

such that γ ≤ γ′ ∧ x ∈ Sγ′ ⊆ Sγ = T δ. Hence x ∈ T δ, i.e., x = tδ′ for someδ′ ≥ δ ≥ θ. By the arbitrarity of θ, x ∈ Cf(t), so Cf(s) ⊆ Cf(t). By a symmetricargument, also Cf(t) ⊆ Cf(s) holds, thus Cf(s) = Cf(t). a

Lemma 2.11. Let s and t be two eventually cofinal sequences. Then s and tare cofinally equivalent if and only if (a) lh(s) and lh(s) are both successor orboth limit and (b) Cf(s) = Cf(t).

Proof. The left-to-right direction is Lemma 2.10.On the other direction, if lh(s), lh(s) are both successor then Cf(s) = Cf(t)

implies that s and t are eventually (thus also cofinally) equivalent. Let lh(s) andlh(t) be both limit, let α < lh(s) and β < lh(t) witness that s and t are eventuallycofinal, respectively, and assume Cf(s) = Cf(t). Hence Sα = Cf(s) = Cf(t) = T β .If α ≤ γ < lh(s), then Sγ = Sα = T β . Conversely, if β ≤ δ < lh(t), thenT δ = T β = Sα. Hence α and β witness that s and t are cofinally equivalent. a

Definition 2.12. We say that a sequence s is cofinally invariant if and onlyif, for every γ, δ < lh(s), sγ = sδ whenever sγ and sδ are cofinally equivalent.

Lemma 2.13. Let s be a cofinally invariant sequence. Suppose there exist β <γ < lh(s), γ limit, such that γ + δ · ω < lh(s) and s(β) = s(γ) = s(γ + δ · ω),where δ = γ − β. Then

s(β + δ · α+ ξ) = s(β + ξ),

for every α and ξ < δ such that β + δ ·α+ ξ < lh(s). In particular, ran(s) = Sγ .


Proof. Since s is cofinally invariant, s is also eventually invariant, so forα < ω the statement holds by Lemma 2.5.(1). We will prove the statement forα ≥ ω by induction on α and on ξ.

Let ξ = 0. Since δ is limit so is δ · α.If α = η+ 1 then, by the inductive hypothesis on α, ∀ζ < δ (s(β + δ · η+ ζ) =

s(β + ζ)). Hence β + δ · η and β witness that s (β + δ · α) = s (β + δ · η + δ)and s(β + δ) are eventually equivalent, so s(β + δ · α) = s(β + δ) = s(β).

If α is limit then, by the inductive hypothesis on α, s(β+ δ · η+ ζ) = s(β+ ζ),for every η < α and ζ < δ.

Claim: s(β + δ · α) is eventually cofinal, witnessed by β.

Proof of the claim. Let x ∈ Sββ+δ·α and β ≤ θ < β + δ · α. Hence x =

s(β+ δ · η+ ζ) and θ = β+ δ · η′+ ζ ′, for some η, η′ < α and ζ, ζ ′ < δ. Since α islimit, let η′′ = η′+1 < α. By the inductive hypothesis on α, x = s(β+δ ·η+ζ) =s(β + ζ) = s(β + δ · η′′ + ζ) and β + δ · η′′ + ζ > θ. By the arbitrarity of θ,

x ∈ Cf(s (β + δ · α)). By the arbitrarity of x, Sββ+δ·α ⊆ Cf(s (β + δ · α)), so

s(β + δ · α) is eventually cofinal, witnessed by β. aBy Lemma 2.5.(2), s(β+ δ ·ω) is also eventually cofinal, witnessed by β. By

the inductive hypothesis on α, Sββ+δ·α = Sββ+δ = Sββ+δ·ω, so Cf(s (β + δ · α)) =

Cf(s (β + δ · ω)). Hence, by Lemma 2.11, s (β + δ · ω) and s (β + δ · α) arecofinally equivalent. Thus, by the hypothesis, s(β + δ · α) = s(β + δ · ω) = s(β),since β + δ · ω = γ + δ · ω. This proves the statement for ξ = 0.

Let ξ > 0. By the inductive hypothesis on ξ, β + δ · α and β, respectively,witness that s (β + δ · α + ξ) and s (β + ξ) are eventually equivalent, sos(β + δ · α+ ξ) = s(β + ξ). a

Let x ∈ ran(s), so x = s(θ) for some θ < lh(s). Since δ > 0, if θ ≥ γthen, by the ordinal division property, ∃α, ξ < δ (θ = γ + δ · α + ξ). Thereforex = s(θ) = s(β + ξ) ∈ Sγ . a

Lemma 2.14. Let s be a cofinally invariant sequence. Let θ < lh(s) be a limitordinal such that θ∗ < lh(s), µ(θ) < θ and µ(θ∗) < θ∗. Then

s(µ(θ∗) + δ · α+ ξ) = s(µ(θ∗) + ξ),

where δ = θ∗−µ(θ∗), for every α and ξ < δ such that (µ(θ∗) + δ ·α+ ξ) < lh(s).Moreover, ran(s) = Sθ.

Proof. By definition and by Lemma 2.8, s(θp) = s(θ0), δp = θp − θ0, θ∗ =θp+1 = θp+δp·ω, θ0 ≤ µ(θ∗) < θ.

Let γ = θ∗, β = µ(θ∗), δ = γ− β. By Lemma 2.5.(2), sγ+ δ ·ω is eventuallycofinal witnessed by γ and sγ = sβ + δ is eventually cofinal witnessed by θp.

By Lemma 2.5.(1), Sγγ+δ·ω = Sγγ+δ = Sβγ . By Lemma 2.8, Sβγ = Sθ0

θ = Sθpγ .

Since s(γ+ δ ·ω) and sγ both are eventually cofinal witnessed, respectively,

by γ and θp, and Sγγ+δ·ω = Sθpγ , by Lemma 2.11 s(γ+δ ·ω) and sγ are cofinally

equivalent, hence s(γ + δ · ω) = s(γ). Thus, by Lemma 2.13, s(β + δ · α + ξ) =s(β+ξ), for every α and ξ < δ such that (β+δ ·α+ξ) < lh(s), and, in particular,ran(s) = Sγ .

Since, by Lemma 2.8, Sθγ = Sθ0

θ ⊆ Sθ, ran(s) = Sγ = Sθ ∪ Sθγ = Sθ. a

12 EDOARDO RIVELLO

The Unfolding Lemma. We recall some basic facts and notations aboutwellordered sets. For A = 〈A,<〉 a wellordered set and a ∈ A:

• a<A = b ∈ A | b < a.• a+A denotes the immediate successor of a in A, if it exists, i.e., if a is not

the maximum of A.• a−A denotes the immediate predecessor of a in A, if it exists, i.e., if a<A has

the maximum.• ot(A) denotes the order type of A.

We will drop the subscript A when it is clear from the context. a< will alsodenote the wellordered set 〈a<, <〉. We say that a is a successor point of Aif a< has the maximum a− and that a is a limit point of A if a is neitherthe minimum of A nor a successor point, i.e., if a< is nonempty and has nomaximum. (a−)+ = a for every successor a and (a+)− = a for every a which isnot the maximum of A.

We write B v A when A end extends B, i.e., B is an initial segment of A(b ∈ B ∧ a < b → a ∈ B). We write B @ A when A properly end extends B,i.e., B v A ∧ B 6= A. For B @ A, B> denotes the first a ∈ A such that b < afor every b ∈ B (i.e., the first a ∈ A − B). ∅> = min(A). For every a ∈ A,a< is a proper initial segment of A and (a<)> = a. Conversely, B @ A impliesB = (B>)<.

For any set of ordinals X let sup∗(X) denote the least γ ∈ Lim∗ greater thanall ordinals in X. It follows from the definition that if X = ∅ then sup∗(X) = 0;if X has the maximum ν then sup∗(X) = ν + ω, i.e., the least limit ordinalgreater than ν; otherwise, sup∗(X) = sup(X).

We denote by / the natural wellordering on µ(s), i.e., the natural order onthe ordinals restricted to µ(s). For X,Y initial segments of µ(s), X @ Y →sup∗(X) < sup∗(Y ). For X @ µ(s), sup∗(X) ≤ X., hence µ(sup∗(X)) is defined.If µ(sup∗(X)) = sup∗(X) then sup∗(X) = X..

Let θ 7→ θ∗ be the unfolding operation on lh(s) ∩ Lim defined in Def. 2.6.

Lemma 2.15 (The Unfolding Lemma). Let s be a cofinally invariant sequence.Then

X. =

sup∗(X) if µ(sup∗(X)) = sup∗(X),sup∗(X)∗ if µ(sup∗(X)) < sup∗(X).

for every proper initial segment X of µ(s).

Proof. If µ(sup∗(X)) = sup∗(X) thenX. = sup∗(X). Suppose µ(sup∗(X)) <sup∗(X). If sup∗(X) < X. < sup∗(X)∗ then, by Lemma 2.8, µ(X.) < X., con-tradicting X. ∈ µ(s). Hence sup∗(X)∗ ≤ X. < lh(s), so µ(sup∗(X)∗) is defined.Let γ = µ(sup∗(X)∗). Suppose, towards a contradiction, γ < sup∗(X)∗. ByLemma 2.14, s(γ+ δ ·α+ ξ) = s(γ+ ξ), where δ = sup∗(X)∗−γ, for every α andξ < δ such that (γ+δ ·α+ξ) < lh(s). By Lemma 2.8, γ < sup∗(X). Since X. andδ are limit, X. = γ + δ · α for some α ∈ On. Hence s(X.) = s(γ) = s(γ + δ) =s(sup∗(X)∗). Since γ < sup∗(X)∗, sup∗(X)∗ 6= X., hence sup∗(X)∗ < X..Since sup∗(X)∗ is limit and s(sup∗(X)∗) = s(X.), µ(X.) ≤ sup∗(X)∗ < X.:contradiction.


Hence, γ = sup∗(X)∗, so sup∗(X) ≤ sup∗(X)∗ ∈ µ(s). Since sup∗(X)∗ ≤ X.

and, by definition, X. ≤ sup∗(X)∗, it follows X. = sup∗(X)∗. a

The Cofinality Lemma. For any sequence s, define

Cl(s) = sup∗(µ(s)).

The following is the promised version of Fact 1.2 only involving cofinalityarguments.

Lemma 2.16 (Cofinality Lemma). Let s be a cofinally invariant ordinal-lengthsequence. Let θ = Cl(s), γ = µ(θ∗), δ = θ∗ − γ and η = γ + δ · ω. Then

Periodicity: s is cyclic, i.e., s(γ + δ · α+ ξ) = s(γ + ξ), for every α ∈ On andξ < δ. Moreover, ran(s) = Sθ.


Alignment: Cf(s) = Cf(sη).Closure: Let ℵ(P ) denote the least ordinal non-injectable in P (the Hartogs

ordinal for P ). If ℵ(P ) ≤ ω then the ordinal η (hence, a fortiori, alsoγ, θ, δ and θ∗) is less than ωω; otherwise η < ℵ(P ).

Proof. (Periodicity) µ(θ) and µ(θ∗) are less than θ = Cl(s) hence, by Lemma2.8, µ(θ∗) < θ < θ∗. Then apply Lemma 2.14. a

(Cofinality) Let x ∈ Cf(s). By definition, there exists β ≥ γ such that x =s(β). Let α and ξ < δ be unique such that β = γ + δ · α + ξ. By Periodicity,s(γ + ξ) = s(γ + δ · α+ ξ) = s(β) = x, so x ∈ Sγγ+δ.

Conversely, suppose x = s(β) for some β = γ + ξ, ξ < δ and fix λ ∈ On. Toavoid trivialities, let λ ≥ β ≥ γ. Hence λ = γ + δ · α+ ζ, for some α and ζ < δ.Let β′ = γ + δ · (α+ 1) + ξ. Hence λ < β′ and, by Periodicity, s(β′) = s(β) = x.So, x ∈ Cf(s). a

(Alignment) Since s is eventually invariant and s(γ+δ) = s(γ), by Lemma 2.5s(γ + δ ·ω) is eventually cofinal witnessed by γ (since γ + δ ·ω = γ + δ+ δ ·ω),hence Cf(sη) = Sγη .

By Cofinality and Periodicity, Cf(s) = Sγγ+δ = Sγη = Cf(sη). a(Closure) First we will show that if ℵ(P ) ≤ ω then Cl(s) < ωω and, otherwise,

Cl(s) < ℵ(P ). Let 〈δξ | ξ < λ〉 be the unique isomorphism between λ = ot(µ(s))and µ(s).

Claim: For every ξ < λ, there exists k < ω such that δξ < ωξ+k.Proof of the claim. By induction on ξ. Since ν ≤ ν∗ for every ν limit, by the

Unfolding Lemma 2.15 and Remark 2.7, δξ = (δ/ξ ). ≤ (sup∗(δ/ξ ))∗ ≤ sup∗(δ/ξ )·ωp,for some p < ω. By the inductive hypothesis, for every α < ξ there exists n < ωsuch that δα < ωα+n.

If ξ = 0 then δ0 = 0 < 1 = ω0.If ξ = α+ 1 then δβ ≤ δα < ωα+n < ωξ+n for every β < ξ. Hence sup∗(δ/ξ ) ≤

ωξ+n. Thus, δξ ≤ sup∗(δ/ξ ) · ωp ≤ ωξ+n · ωp < ωξ+k, where k = n+ p+ 1.

If ξ is limit then α < ξ → α + n < ξ, for every n < ω, so δα < ωξ for everyα < ξ. Hence sup∗(δ/ξ ) ≤ ωξ. Thus, δξ ≤ sup∗(δ/ξ ) · ωp ≤ ωξ · ωp < ωξ+k, wherek = p+ 1. a

14 EDOARDO RIVELLO

By the claim, if ℵ(P ) ≤ ω then λ < ω hence, for every ξ < λ, there existsn < ω such that δξ < ωn. Let nξ be the least one. Since λ is finite, δξ < ωn, forevery ξ < λ, where n = maxnξ | ξ < λ. Since ωn is limit, Cl(s) ≤ ωn < ωω.

If ℵ(P ) > ω then ℵ(P ) is an infinite initial ordinal. For every ξ < λ, δξ < ωλ+ω.Since ωλ+ω is limit, Cl(s) ≤ ωλ+ω. Since ℵ(P ) is an infinite initial ordinal, λ, ω <ℵ(P ) implies λ+ω < ℵ(P ) and ωλ+ω < ℵ(P ). Hence Cl(s) = θ ≤ ωλ+ω < ℵ(P ).

By Remark 2.7, θ∗ ≤ θ ·ωp for some p < ω. By definition, η = γ+ δ ·ω, whereγ, δ ≤ θ∗, hence η ≤ θ∗ + θ∗ · ω = θ∗ · ω ≤ θ · ωp · ω = θ · ωp+1. If θ < ωω thenθ ≤ ωn for some n < ω, thus η ≤ ωn · ωp+1 < ωω. Otherwise, η < ℵ(P ) since θand ωp+1 < ℵ(P ) and ℵ(P ) is an infinite initial ordinal. a

Cofinality and stability. From the Cofinality Lemma and some propertiesof the inferior limit operation easily follows that Fact 1.2 holds for stability-dependent revision:

Theorem 2.17 (Stability-dependent revision). Let Γ be a stability-dependentgeneralised limit rule and let s be a (ρ,Γ)-sequence of ordinal length. Then

Periodicity: The sequence s is cyclic, i.e., there exist ordinals γ, δ such thats(γ + δ · α+ ξ) = s(γ + ξ), for every α ∈ On and ξ < δ.


Alignment: There exists an ordinal η such that Cf(s) = Cf(sη) and liminf(s) =liminf(sη).

Closure: The initial and the alignment ordinals and the period of s are less thanthe Hartogs ordinal ℵ(P ), if ℵ(P ) > ω. Otherwise are less than ωω.

The hypotheses of Theorem 2.17 do not require s to be a revision sequence,in the sense of satisfying the coherence condition.

The important point about the Closure thesis is that it provides a bound forthe “closure” ordinals of s (the least ordinal α such that ran(s) ⊆ Sα and theinitial, alignment and period ordinals) which only depends on the set P . TheClosure thesis of Theorem 2.17 is stated in terms of the Hartogs ordinal for Pin order to make the theorem and its proof Choice-free. Assuming the axiom ofChoice, this formulation of the Closure thesis is equivalent to that given in Fact1.2.

To prove Theorem 2.17 we first need to establish some facts about the inferiorlimit operation and stability-dependent sequences.

Lemma 2.18. Let s be any sequence from P . Let X be a class of ordinalscofinal in lh(s). Then

liminf(s) = lubglb(Sγ) | γ ∈ X.

Proof. For every α < lh(s) let γα be the first ordinal in X greater than orequal to α. α ≤ γα → Sγα ⊆ Sα → glb(Sα) ≤ glb(Sγα), so lubglb(Sγ) | γ ∈ Xis un upper bound of glb(Sα) | α < lh(s). Hence, liminf(s) ≤ lubglb(Sγ) | γ ∈X. Since liminf(s) is an upper bound of glb(Sγ) | γ ∈ X, the converse alsoholds. aIn particular, for every α < lh(s), liminf(s) = lubglb(Sβ) | α ≤ β < lh(s) =liminf(sα) and liminf(s) = sβ if lh(s) = β + 1.


Lemma 2.19. liminf is cofinally invariant, i.e., if two sequences s and t arecofinally equivalent then liminf(s) = liminf(t).

Proof. If s and t are eventually equivalent, witnessed by α < lh(s) andβ < lh(t), respectively, then liminf(s) = liminf(sα) = liminf(sβ) = liminf(t). Letlh(s), lh(t) be limit and let s, t be cofinally equivalent witnessed by α < lh(s) andβ < lh(t), respectively. Let α ≤ γ < lh(s). By cofinal equivalence, there existsδ < lh(t) such that β ≤ δ and Sγ = T δ. Hence glb(Sγ) = glb(T δ) ≤ liminf(t).Therefore, glb(Sγ) ≤ liminf(t) for every α ≤ γ < lh(s) so, by Lemma 2.18 andby definition of lub, liminf(s) = lubglb(Sγ) | α ≤ γ < lh(s) ≤ liminf(t). By asymmetric argument, also liminf(t) ≤ liminf(s) holds, so liminf(s) = liminf(t). a

Lemma 2.20. For every sequence s from P , liminf(s) ≤ glb(Cf(s)). If s iseventually cofinal, then liminf(s) = glb(Cf(s)).

Proof. By definition Cf(s) ⊆ Sα for every α < lh(s). Hence, glb(Sα) ≤glb(Cf(s)), for every α < lh(s), so liminf(s) = supglb(Sα) | α < lh(s) ≤glb(Cf(s)). If s is eventually cofinal then, by definition, there exists α < lh(s)such that Cf(s) = Sα. Moreover, Sβ ⊆ Sα = Cf(s) ⊆ Sβ for every β < lh(s)such that α ≤ β. Hence, by Lemma 2.18, liminf(s) = lubglb(Sβ) | α ≤ β <lh(s) = glb(Cf(s)). a

Proof of Theorem 2.17. Let Γ be a stability-dependent generalised limitrule and let s be a (ρ,Γ)-sequence of ordinal length. Hence sα+1 = ρ(sα) forevery α and there exists f : P × P → P such that sν = f(s0, liminf(s ν)) forevery ν limit. Therefore, by Lemma 2.19, s is cofinally invariant.

By applying the Cofinality Lemma 2.16 we have ordinals θ, γ, δ and η = γ+δ ·ωsuch that:

1. s(γ + δ · α+ ξ) = s(γ + ξ), for every α ∈ On and ξ < δ, and ran(s) = Sθ.2. Cf(s) = Sγγ+δ = Cf(sη).

3. If ℵ(P ) > ω then η (and, a fortiori, also γ, θ and δ) is less than ℵ(P ).Otherwise, η < ωω.

Hence Periodicity, Cofinality and the first part of the Alignment thesis hold.Since s is an ordinal-length sequence, s is eventually cofinal (see [5, p. 170]).

By Lemma 2.5.(2) also S η is eventually cofinal (by applying the lemma with γin the role of β and θ∗ in the role of γ). Thus, by Lemma 2.20,

liminf(s) = glb(Cf(s)) = glb(Cf(sη)) = liminf(sη).

Hence, η is the alignment ordinal for s. Therefore, Closure and the second partof the Alignment thesis also hold. a

Historical note. Fact 1.2 was first established in [6] for Herzberger sequences,without proof. Proofs of Fact 1.2, for both Herzberger and Gupta sequences,appeared in [2], [12] (first edition, 1989), [10] and [15]. Visser [12] establishesall results in an order-theoretic setting quite similar to that of Theorem 2.17.In [5], the Cofinality statement is proved for arbitrary ordinal-length sequences.Lemma 2.13 is proved in [15] for constant limit rules.

The upper bound given by the Closure statement is established, in the above-mentioned works, by different proofs, all involving cardinality arguments whichimply a substantial use of the axiom of Choice.

16 EDOARDO RIVELLO

§3. Order-theoretic revision. Every stability-dependent sequence s is de-termined, via the inferior limit operation and the Transfinite Recursion Theorem,by three parameters: the starting hypothesis h, the revision operator ρ : P → Pand the limit operation f : P × P → P . We write s = Σ(h, ρ, f) to denote theunique ordinal-length stability-dependent sequence s such that

• s0 = h,• sα+1 = ρ(sα), and• sλ = f(s0, liminf(sλ)), for λ limit.

Accordingly, also the cofinality Cf(s) of s only depends on the same three pa-rameters and, as a consequence, also the inferior limit of the full sequence since,by Lemma 2.20, liminf(s) = glb(Cf(s)). We write Cf(h, ρ, f) = Cf(Σ(h, ρ, f)) =Cf(s) and liminf(h, ρ, f) = liminf(Σ(h, ρ, f)) = liminf(s), to make this point ex-plicit.

In this section we will show that it is possible to give an alternative definitionof Cf(h, ρ, f) and liminf(h, ρ, f), making no use of the intermediate ordinal-lengthsequence s = Σ(h, ρ, f) and, more in general, of the machinery of the transfiniteordinal numbers.

Generalised orbits. Given a function ρ : P → P and an element a ∈ P wedenote by ay the orbit of ρ starting from a, i.e., the smallest (under inclusion)subset of P closed under ρ and containing a as an element.

A wellordering from P is a wellordered set A = 〈A,<〉 such that A ⊆ P(note that the wellorder < does not need to be related in any way to the partialordering ≺ of P).

An orbit ay (in P ) is determined by a pair (ρ, a), where ρ : P → P anda ∈ P . This observation motivates the following generalisation:

Definition 3.1. A generalised orbit (in P ) is a pair (ρ,A), where ρ : P → Pand A = 〈A,<〉 is a wellordering from P .

Every orbit (ρ, a) is a generalised orbit, by identifying a with the uniquewellordering of the singleton A = a. When A has more than one element,we say that (ρ,A) is a transfinite orbit.

We fix ρ so that, in the following, a generalised orbit will be simply identifiedwith a wellordering from P . Given any set P , W(P ) will denote the set of allwellorderings from P .

Theorem 3.2 (Zermelo). 7 Let P be any set. Let Θ : P(P )→ P . Then thereis a unique W = 〈W,<〉 ∈ W(P ) such that:

(a) For every x ∈W , Θ(x<W) = x, and(b) Θ(W ) ∈W .

Corollary 3.3. Let G : W(P )→ P . Then there is a unique W = 〈W,<〉 ∈W(P ) such that:

(a) For every x ∈W , G(x<W) = x, and(b) G(W) ∈W .

7See [7, Theorem 2.1, p. 292]


Proof. Say that a subset X ⊆ P is a G-set if and only if there exists awell-ordering R of X such that G〈y ∈ X | y Rx, R〉 = x, for every x ∈ X.By comparability of wellorderings, we can show that any G-set X has a uniquewittnessing wellorder which we denote by RX (the argument runs exactly as inthe proof of Zermelo’s Theorem).

Define Θ : P(P )→ P as follows:

Θ(X) =

G〈X,RX〉 if X is a G-set,G〈∅, ∅〉 otherwise.

Let W = 〈W,<〉 be the wellordering definable from Θ according to Zermelo’sTheorem. We will show that W satisfies (a) and (b) of the Corollary.

(a) Suppose, towards a contradiction, that there exists x ∈ W such thatG(x<W) 6= x and let x be the first one. Hence G(y<W) = y for every y < x, so x<Wis a G-set witnessed by <. Therefore,

x = Θ(x<W) = G(x<W , <),

contradicting the assumption on x.(b) By (a), W is a G-set, so G(W) = Θ(W ) ∈W .Finally, let 〈W,<〉 and 〈W ′, <′〉 be two wellorderings which satisfy (a) and

(b). By (a) both are G-sets, so we can assume, without loss of generality, thatW is an initial segment of 〈W ′, <′〉. Suppose, towards a contradiction, that Wis a proper initial segment. Since 〈W ′, <′〉 is a wellordering, W is of the formy ∈W ′ | y <′ x for a unique x ∈W ′. Hence, by (a), G〈W,<′〉 = x /∈W . But,since 〈W,<′〉 = 〈W,<〉, by (b) G〈W,<′〉 ∈W : contradiction. aFix the function f : P × P → P . We will define an operation on generalisedorbits (depending on the parameters h, ρ and f) which, in the next section,we will show to correspond, in a strict sense, to the inferior limit operationon sequences. In view of this, for A ∈ W(P ), let us denote the result of thisoperation by liminf(A), again.

First, we define liminf(A) when A = ay is an orbit. Let Q be the transitiveclosure of ρ, i.e., the smallest transitive relation on P extending ρ.

Definition 3.4. Let a ∈ P . For every x ∈ ay let xQ = y ∈ ay | y =x ∨ xQy. We define:

liminf(ay) = lubglb(xQ) | x ∈ ay.

Now, we extend the definition of liminf to all generalised orbits.

Definition 3.5. For every generalised orbit (ρ,A) define, by recursion on thelength of A:

(1) For every a ∈ A,

a→A =

⋃XB | B @ A ∧ a ∈ B ∪ by if b = max(A).⋃XB | B @ A ∧ a ∈ B if A has no maximum.

(2)

A0 =

f(h, liminf(by)) if b = max(A).f(h, lubglb(a→A ) | a ∈ A) if A has no maximum.

18 EDOARDO RIVELLO

(3) tA : ω → P , by recursion on ω.

tA(0) = A0.

tA(n+ 1) =

f(h, glb(tA(n)→A )) if tA(n) ∈ A.tA(n) otherwise.

(4)

qA =

0 if A0 /∈ A.minn | tA(n+ 1) /∈ A ∨ tA(n) ≤ tA(n+ 1) if A0 ∈ A.

A∗ = tA(qA).

XA =

by if A0 /∈ A ∧ b = max(A).∅ if A0 /∈ A ∧ A has no max.(A∗)→A otherwise.

liminf(A) =

liminf(by) if A0 /∈ A ∧ b = max(A).lubglb(a→A ) | a ∈ A if A0 /∈ A ∧ A has no max.glb((A∗)→A ) otherwise.

By applying Corollary 3.3 to the function G : W(P )→ P defined by:

• G(∅, ∅) = h, and• G(A) = f(min(A), liminf(A)), when A 6= 〈∅, ∅〉,

for every h, ρ, f there exists a unique generalised orbit W = 〈W,<〉, denoted byΩ(h, ρ, f), such that

• min(W) = h,• a = f(h, liminf(a<)), for every a ∈W , a 6= h, and• f(h, liminf(W)) ∈W .

The definition of the wellorder W = Ω(h, ρ, f) only relies on the recursivedefinition of the inferior limit operation for generalised orbit and on Zermelo’sTheorem 3.2. Therefore, Ω(h, ρ, f) is defined by purely inductive means, makingno use neither of the transfinite ordinal numbers nor of the axiom of Choice:actually the definition can be done in Zermelo’s set theory, not assuming neitherthe Foundation nor the Replacement axioms. In the next section we will showthat, for the sake of the revision theory, the definition of the wellorder W =Ω(h, ρ, f) is equivalent to the definition of the ordinal-length revision sequences = Σ(h, ρ, f).

Generalised orbits and sequences. Given any sequence s, define

• W (s) = sγ | γ ∈ lh(s) ∩ Lim∗.• µ(x) = minγ < lh(s) | γ ∈ Lim∗ ∧ sγ = x, for x ∈W (s).

For x ∈ W (s) and γ ∈ lh(s) ∩ Lim∗, sµ(x) = x and µ(sγ) = µ(γ), hence µ :W (s)→ µ(s) is a bijection whose inverse is sµ(s).

We say that W (s) is wellordered by first limit occurrence (denoted by /, again)when it is endowed by the wellordering induced on W (s) by the bijection s µ(s) together with the wellorder 〈µ(s), /〉. It follows that s µ(s) is the uniqueisomorphism between 〈µ(s), /〉 and W(s) = 〈W (s), /〉.


Theorem 3.6 (Main Theorem). Let s = Σ(h, ρ, f) andW = Ω(h, ρ, f). Then

(1) W(s) =W.(2) Cf(s) = (W ∗)→.(3) liminf(s) = liminf(W).

To prove Theorem 3.6 we need to establish some facts about sequences andgeneralised orbits.

Proposition 3.7. Let s be the ω-sequence generated by ρ and the startingpoint a, i.e., s is the unique sequence of length ω such that s0 = a and sn+1 =ρ(sn) for every n < ω. Then

liminf(s) = liminf(ay).

Sketched proof. By the definitions, ran(s) = ay and bQ c ↔ ∃m,n ∈ω (b = sm ∧ c = sn ∧ m < n), for every b, c ∈ ran(s). Moreover, it is straight-forward to see that Sn = sQn , for every n ∈ ω. Hence, by Lemma 2.18,

liminf(s) = lubglb(Sn) | n ∈ ω = lubglb(xQ) | x ∈ ay = liminf(ay).

a

Lemma 3.8. a ∈ a→A and a ≤ c→ c→A ⊆ a→A , for every a, c ∈ A.

Proof. For the first claim, if a = maxA then a ∈ ay ⊆ a→A . Otherwise,B = a≤ is a proper initial segment of A, a ∈ B and a ∈ ay ⊆ XB. For thesecond claim, let x ∈ c→A . If b = maxA and x ∈ by then x ∈ c→A by definition.Otherwise, x ∈ XB for some B @ A such that c ∈ B. Since B is an initialsegment of A, a < c implies a ∈ B, hence x ∈ a→A . Therefore c→A ⊆ a→A . a

Remark 3.9. If A = 〈∅, ∅〉 then A0 = f(h,min(P )) /∈ A, qA = 0, A∗ = A0,XA = ∅ and liminf(A) = min(P ). If b = maxA, then by = b→A ⊆ a→A for everya ∈ A. In particular, by ⊆ XA.

Lemma 3.10. Let B v A and a ∈ B. Then a→B ⊆ a→A .

Proof. Suppose, without loss of generality, B @ A. Let x ∈ a→B . If thereexists C such that C @ B, a ∈ C and x ∈ XC , then x ∈ a→A , since C @ A. Ifb = maxB then, by Remark 3.9, x ∈ by ⊆ XB. Thus x ∈ a→A , since B @ A. a

For A ⊆ W (s), a ∈ W (s), set ν(A) = sup∗(µ ′′A) and ν(a) = ν(a/). Byisomorphism we have the following facts:

1. A v W(s)→ µ ′′A v µ(s) and X v µ(s)→ s ′′X v W(s).2. B @ A v W(s)→ µ ′′B @ µ ′′A→ ν(B) < ν(A).3. µ(A.) = (µ ′′A). and s(X.) = (s ′′X)..4. µ ′′a/ = µ(a)/ and s(γ)/ = s ′′γ..5. ν(A) = minγ ∈ Lim∗ | ∀x ∈ A (µ(x) < γ).6. a = min(W(s))↔ µ(a) = 0↔ a = s0.7. (Unfolding Lemma) For every a ∈ W (s), µ(a) = ν(a) if µ(ν(a)) = ν(a);

otherwise µ(a) = ν(a)∗.

Lemma 3.11. Suppose A v W(s) has no maximum and let θ = ν(A). Thenthe set µ(a) | a ∈ A is cofinal in θ.

20 EDOARDO RIVELLO

Proof. Suppose, towards a contradiction, that there exists α < θ such thatµ(a) ≤ α for every a ∈ A. Let α = δ + n for a unique limit δ ≤ α. Since µ(a)is limit, µ(a) ≤ δ for every a ∈ A. So we can assume that δ is the first limitsuch that µ(a) ≤ δ for every a ∈ A. Since δ 6= θ, there exists a ∈ A such thatµ(a) = δ. So a = maxA: contradiction. a

Lemma 3.12. Let A v W(s) and θ = ν(A). Then

(1) Sµ(a)θ = a→A , for every a ∈ A.

(2) sθ = A0.

Moreover, if µ(θ) < θ then

(3) sθ0 = A∗ ∈ A.

(4) Sθ0

θ = XA.

Proof. By induction on the length of A.Proof of (1). If A = a then µ(a) = 0, θ = ω and S0

ω = sy0 = ay = a→A .Suppose b = maxA. Let γ = µ(b). Hence θ = γ +ω. Let a ∈ A and ξ = µ(a).

Let x ∈ Sξθ , i.e., x = sα for some ξ ≤ α < θ = γ + ω.Case I. γ ≤ α < γ + ω. Then x ∈ by ⊆ a→A , by Remark 3.9.Case II. ξ ≤ α < γ. Then α < µ(b), with b ∈ A. Let c be the first in A such

that α < µ(c). Let B = c<. Since µ(a) ≤ α < µ(c), a ∈ B. Since c ≤ b, B @ A.Hence XB ⊆ a→A . Let η = ν(B).

Case IIa. If c is successor in A, then a ≤ c− < c, hence µ(c−) ≤ α < µ(c).Let d = c− and δ = µ(d).

Case IIa1. If µ(c) = δ + ω then x ∈ dy and d = max(B). Since a ∈ B,x ∈ dy ⊆ a→B ⊆ a→A by Remark 3.9 and Lemma 3.10.

Case IIa2. If µ(c) > δ + ω then µ(η) < η = δ + ω. Since B @ A, by the

inductive hypothesis, Sη0

η = XB. η0 < η ≤ δ + ω implies η0 ≤ δ ≤ α < η, hence

x = sα ∈ Sη0

η = XB ⊆ a→A .Case IIb. Let c be limit in A. Since c< has no maximum, by Lemma 3.11

η ≤ α < µ(c). Hence, by the Unfolding Lemma, µ(η) < η. Since B @ A, by the

inductive hypothesis, Sη0

η = XB. By the Unfolding Lemma , µ(c) = η∗, hence,

by Lemma 2.8, α ∈ Sηη∗ = Sη0

η . Therefore x ∈ XB ⊆ a→A . aConversely, let x ∈ a→A . If x ∈ by then x ∈ Sγγ+ω = Sγθ ⊆ S

ξθ , since ξ = µ(a) <

θ = γ + ω implies ξ ≤ γ. Otherwise, there exists C @ A such that a ∈ C andx ∈ XC . Let η = ν(C). C @ A implies η < θ. By the inductive hypothesis,C0 = sη.

Suppose µ(η) = η. Hence C0 /∈ C. Since XC 6= ∅, it follows from the definitionof XC that C has the maximum, say c, and that XC = cy. Hence x = sα for

some µ(c) ≤ α < µ(c) +ω = η. Since µ(c) < η, ξ ≤ µ(c) ≤ α < η < θ, so x ∈ Sξθ .

If µ(η) < η then, by the inductive hypothesis, XC = Sη0

η . If ξ ≤ η0 then

Sη0

η ⊆ Sξη ⊆ Sξθ . Otherwise, by Lemma 2.8, Sη0

η = Sξη∗ . Suppose, towards acontradiction, θ < η∗. By Lemma 2.8, µ(δ) < δ for every η ≤ δ < η∗, henceµ(a) < η for every a ∈ A, contradicting the minimality of θ. Thus, η∗ ≤ θ.

Therefore, x ∈ Sη0η = Sξη∗ ⊆ Sξθ . This ends the proof of (1) when b = max(A).


If A has no maximum the proof is the same as in the Case II in one directionand in the case where there exists C @ A such that a ∈ C and x ∈ XC in theother direction. a

Proof of (2). If A = a then A0 = f(h, liminf(ay)) = f(h, liminf(sω)) = sω.If b = max(A) then A0 = f(h, liminf(by)) = f(h, liminf(sγ+ω)) = sθ, where

γ = µ(b) and θ = γ + ω.If A has no maximum, then, by (1) and by the lemmata 3.11 and 2.18, A0 =

f(h, lubglb(a→A ) | a ∈ A) = f(h, lubglb(Sµ(a)θ ) | a ∈ A) = f(h, liminf(sθ)) =

sθ. aProof of (3) and (4).Claim I : If µ(θ) < θ then, for every n ≤ p,

(a) sζn = tA(n) ∈ A.

(b) Sζnθ = tA(n)→.

Proof of Claim I. Assuming (a), (b) immediately follows from (1). We willprove (a) by induction on n ≤ p, assuming µ(θ) < θ.

(n = 0) By (2), s(ζ0) = s(θ) = A0 = tA(0). Since µ(θ) < θ, s(µ(θ)) = s(ζ0) =tA(0) ∈ A.

(n = m+ 1 ≤ p). By the inductive hypothesis on m:

s(ζn) = s(θn) = f(h, liminf(sθm+1)) =

f(h, glb(Sζmθ )) = f(h, glb(tA(m)→A )) = tA(n).

If b = maxA and γ = µ(b), then θ = γ + ω. Since µ(θn) < θ = γ + ω,µ(tA(n)) ≤ γ = µ(b). Hence tA(n) ≤ b. If A has no maximum then, by Lemma3.11, µ(θn) < θ implies µ(tA(n)) < µ(a) for some a ∈ A. In both cases, tA(n) ≤ afor some a ∈ A. Since A is an initial segment of W(s), tA(n) ∈ A. a

Claim II : p = qA.Proof of Claim II. µ(θ) < θ implies that sθ = A0 ∈ A. Hence

tA(p+ 1) = f(h, glb(tA(p)→)) = f(h, glb(Sζpθ )) =

f(h, liminf(sθp+1)) = sθp+1.

By Lemma 2.8, µ(θp) ≤ µ(θp+1) < θ or θ < µ(θp+1) = θp+1.In the first case, tA(p) ≤ tA(p + 1). In the second case, since µ(tA(p + 1)) =

µ(θp+1), it follows tA(p + 1) /∈ A. If n < p then tA(n) ∈ A and µ(tA(n + 1)) =ζn+1 < ζn = tA(n). Hence p = minn | tA(n+1) /∈ A ∨ tA(n) ≤ tA(n+1) = qA.a

By the two claims, putting n = p = q it follows

(3) sθ0 = A∗ ∈ A.

(4) Sθ0

θ = XA.

aProof of the main theorem. (1) (W(s) = W). For A v W(S), let θ =

ν(A). Define γ(A) = θ, if µ(θ) = θ, and γ(A) = θ∗ otherwise.Claim: f(h, liminf(A)) = s(γ(A)).Proof of the claim. By Lemma 3.12, s(θ) = A0. If µ(θ) = θ then A0 /∈ A,

hence f(h, liminf(A)) = A0 = s(θ) = s(γ(A)).

22 EDOARDO RIVELLO

If µ(θ) < θ then A0 ∈ A, since A is an initial segment of W(s). There-

fore, by Lemma 3.12, liminf(A) = glb(XA) = glb(sθ0

θ ). Hence, by Lemma 2.8,

f(h, liminf(A)) = f(h, glb(sθ0

θ )) = f(h, liminf(sθ∗)) = s(θ∗) = s(γ(A)). aIn particular, when A = a/, for a ∈ W (s), a 6= h, then, by the Unfolding

Lemma, γ(a/) = µ(a), therefore f(h, liminf(a/)) = s(γ(a/)) = s(µ(a)) = a.Moreover, f(h, liminf(W)) = s(γ(W)) ∈W , since γ(W) is limit.

Since min(W (s)) = h, a = f(h, liminf(a/)), for every a ∈ W , a 6= h, andf(h, liminf(W(s))) ∈ W (s), it follows, by uniqueness, that W(s) = Ω(h, ρ, f) =W. a

(2) (Cf(s) = (W ∗)→). Let θ = ν(W (s)) = sup∗(µ(s)) = Cl(s), γ = µ(θ∗) andδ = θ∗ − γ. By Theorem 2.17, Lemma 2.8 and Lemma 3.12, Cf(s) = Sγγ+δ =

Sθ0

θ = XW = (W∗)→, since W0 = sθ ∈W . a(3) (liminf(s) = liminf(W)). By (2) and Lemma 2.20, liminf(s) = glb(Cf(s)) =

glb((W∗)→) = liminf(W), since W0 = sθ ∈W . a

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[2] John P. Burgess, The Truth is never simple, The Journal of Symbolic Logic, vol. 51(1986), no. 3, pp. 663–681.

[3] Melvin Fitting, Notes on the mathematical aspects of Kripke’s theory of truth, NotreDame Journal of Formal Logic, vol. 27 (1986), no. 1, pp. 75–88.

[4] Anil Gupta, Truth and paradox, Journal of Philosophical Logic, vol. 11 (1982), no. 1,

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[6] Hans G. Herzberger, Notes on naive semantics, Journal of Philosophical Logic,vol. 11 (1982), no. 1, pp. 61–102.

[7] Akihiro Kanamori, The mathematical import of Zermelo’s well-ordering theorem, TheBulletin of Symbolic Logic, vol. 3 (1997), no. 3, pp. 281–311.

[8] Kazimierz Kuratowski and Andrzej Mostowski, Set theory. With an introductionto descriptive set theory, North-Holland, 1968, second, 1976 completely revised edition.

[9] Benedikt Lowe and Philip D. Welch, Set-theoretic absoluteness and the Revisiontheory of truth, Studia Logica, vol. 68 (2001), no. 1, pp. 21–41.

[10] Vann McGee, Truth, vagueness, and paradox. An essay on the logic of truth,

Hackett Publishing Company, Indianapolis, Cambridge, 1991.[11] Wac law Sierpinski, Cardinal and ordinal numbers, PWN-Polish Scientific Publish-

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[13] Philip D. Welch, On Gupta-Belnap Revision theories of truth, Kripkean fixed points,and the next stable set, The Bulletin of Symbolic Logic, vol. 7 (2001), no. 3, pp. 345–360.

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