Table Of ContentA Many-Sorted Variant of Japaridze’s
∗
Polymodal Provability Logic
Gerald Berger1, Lev D. Beklemishev2,3,4, and Hans Tompits1
1Institute for Information Systems, Vienna University of Technology
[email protected],[email protected]
6
1 2Steklov Mathematical Institute, Russian Academy of Sciences
0 3Moscow M.V. Lomonosov State University
2
4National Research University Higher School of Economics
n [email protected]
a
J
0
2 Abstract
] We consider a many-sorted variant of Japaridze’s polymodal provability logic GLP. In
O this variant, propositional variables are assigned sorts n<ω, where variables of sort n are
L arithmetically interpreted as Πn+1-sentences of the arithmetical hierarchy. We prove that
themany-sorted variant is arithmetically complete with respect to thisinterpretation.
.
h
t
a Keywords: provability logics, mathematical logic, modal logic, formal arithmetic, arith-
m metical completeness
[
2 1 Introduction
v
7 The polymodal provability logic GLP, due to Japaridze [15], has received considerable interest
5
in the mathematical-logic community. For every n≥0, the language of GLP features modalities
8
2 hni that can be arithmetically interpretedas n-consistency, i.e., a modal formula hniϕ expresses
0 underthisinterpretationthatϕisconsistentwiththesetofalltrueΠn-sentences. Thisparticular
. interpretation steered interest in GLP in mainstream proof theory: Beklemishev [3] showed how
1
0 GLPcanactasaframeworkinordertocanonicallyrecoveranordinalnotationsystemforPeano
6 arithmetic (PA) and its fragments. Moreover, based on these notions, he obtained a rather
1 abstract version of Gentzen’s consistency proof for PA by transfinite induction up to ε and he
0
v: formulated a combinatorial statement independent from PA [5].
i This proof-theoretic analysis is based on the notion of graded provability algebra. Let T be
X
an extension of PA and L be an algebra whose elements are equivalence classes of the relation
T
r
a
ϕ∼ψ ⇐⇒ T ⊢ϕ↔ψ.
df
Furthermore, let {ϕ} denote the equivalence class of ϕ with respect to ∼. Equipping L with
T
the standard Boolean connectives and the relation
{ϕ}≤{ψ} ⇐⇒ T ⊢ϕ→ψ
df
∗This work was supported by the Austrian Science Fund (FWF) under grants Y698 and W1255-N23. The
articleiscurrentlyunderreviewforpublicationintheLogicJournaloftheIGPL.
1
turnsL intoaBooleanalgebra,theLindenbaum algebra ofT. Thus,logicalnotionsarebrought
T
into an algebraic setting. The maximal element ⊤ and the minimal element ⊥ of this algebra
are, respectively, the classes of all provable and all refutable sentences of T. Deductively closed
extensions of T correspond to filters of L (see ref. [4] for details).
T
Let hni be a Π -formula that formalizes the notion of n-consistency in arithmetic. The
T n+1
graded provability algebra M of T is the algebraL extendedby operatorshni definedonthe
T T T
elements of L by
T
hni : {ϕ}7−→{hni ϕ}, for n≥0.
T T
Terms in the languageof M canbe identified with polymodalformulas. Furthermore,foreach
T
sound and axiomatizable extension T of PA, Japaridze’s arithmetical completeness theorem for
GLP states that
GLP⊢ϕ(p~) ⇐⇒ M |=∀p~(ϕ(p~)=⊤),
T
where ~p are all the propositional variables from ϕ(p~). The algebra M carries an additional
T
structure in form of a distinguished family of subsets
P ⊂P ⊂···⊂M ,
0 1 T
where P is defined by the class of Π -sentences of the arithmetical hierarchy. This family of
n n+1
subsets is called stratification of M [3]. Since hni is a Π -formula, the operator hni maps
T T n+1 T
M to P . The presence of a stratification admits to turn M into a many-sorted algebra—
T n T
variables of sort n range over arithmetical Π -sentences. The notion of sort can be naturally
n+1
extended to capture polymodal terms. It is the goal of this paper to investigate a modal-logical
counterpart to this many-sortedalgebra.
Tothisend,wedefineamany-sortedvariantofGLP,denotedbyGLP∗,whichhasvariablesof
sortn,foreveryn≥0. Substitutioninthislogicisrequiredtorespectthesortsofvariables. Our
main result is a Solovay-style arithmetical completeness theorem for GLP∗, i.e., for any sound
and axiomatizable extension T of PA we have
GLP∗ ⊢ϕ(p~) ⇐⇒ M |=∀p~(ϕ(p~)=⊤),
T
where p~ are all propositional variables from ϕ(p~) and a quantifier binding such a variable of
sort n only ranges over elements from P . In particular, we show that the principle of Σ -
n n+1
completeness,
hni p→p
T
(where p is of sort n), in addition to the postulates of GLP, suffices to obtain arithmetical
completeness. AlogicthatcontainsthisprinciplehasbeenstudiedbyVisser[18]whointroduced
a Σ -provability logic of PA, i.e., in this logic, variables are arithmetically interpreted as Σ -
1 1
sentences (see also Boolos [12] and Ardeshir and Mojtahedi [1]).
The remainder of the paper is organized as follows. After this introductory section, we
introducebasicnotionsinSection2. InSection3weprovethearithmeticalcompletenesstheorem
for GLP∗. We continue our exposition with some further results in Section 4 and conclude the
paper in Section 5.
2 Preliminaries
The Logics GLP, GLP∗, and J∗. The polymodal provability logic GLP is formulated in
the language of the propositional calculus enriched by unary connectives h0i,h1i,h2i,..., called
2
modalities. The dual connectives [n], for every n ≥ 0, are abbreviations where [n]ϕ stands for
¬hni¬ϕ. The notion of a formula is defined in the usual way. The logic GLP is axiomatized by
the following axiom schemes and rules:1
(i) all tautologies of classical propositional logic;
(ii) hni(ϕ∨ψ)→hniϕ∨hniψ; [n]⊤;
(iii) hniϕ→hni(ϕ∧hni¬ϕ) (Löb’s axiom);
(iv) hmiϕ→[n]hmiϕ, for m<n;
(v) hniϕ→hmiϕ, for m<n (monotonicity); and
(vi) modus ponens and ϕ→ψ/hniϕ→hniψ.
GLP∗ is formulated over a propositional language that contains variables each being assigned a
unique sort α, where 0 ≤ α ≤ ω. By writing pα, we indicate that the variable pα has sort α.
The notion of sort is defined for all formulas as follows:
(i) ⊤ and ⊥ have sort 0;
(ii) ϕ∧ψ and ϕ∨ψ have sort max{α,β} if ϕ and ψ have the respective sorts α and β;
(iii) ¬ϕ has sort α+1 if ϕ has sort α; and
(iv) hniϕ has sort n, for n<ω.
We denote by |ϕ| the sort of ϕ. The sort ω is explicitly included to provide variables that can
explicitly be assigned an arbitrary arithmetical sentence in an arithmetical interpretation. In
contrast,variablesoffiniten<ω canbeassignedarithmeticalΠ -sentences(seebelow). Note
n+1
that if |¬ϕ|=α, then α is a successor ordinal.
The logic GLP∗ is now axiomatized by the following axiom schemes and rules of inference:
(i) all tautologies of classical propositional logic;
(ii) schemes (i), (ii), (iii), and (v) of GLP;
(iii) hniϕ→ϕ, if |ϕ|≤n (Σ -completeness).
n+1
Note that, for m < n, GLP∗ ⊢ hni¬hmiϕ → ¬hmiϕ, whence GLP∗ ⊢ hmiϕ → [n]hmiϕ follows
by propositional logic. Hence, GLP∗ extends GLP in the sense that, for any formula ϕ in the
languageofGLP,ifGLP⊢ϕ,thenGLP∗ ⊢ϕ′,whereϕ′isobtainedfromϕbyarbitrarilyassigning
sorts to propositional variables.
ThelogicGLPis notcompleteforanyclassofKripkeframes[14]. Therefore,Beklemishev[6]
considers a weaker logic J that is complete with respect to a natural class of Kripke frame and
to which GLP is reducible. We do so as well and define a many-sortedcounterpartJ∗ of J which
arises from GLP∗ by dropping monotonicity and adding the scheme
(vii) hmihniϕ→hmiϕ, for m<n.
It is easy to see that this scheme is provable in GLP∗, i.e., GLP∗ extends J∗.
1Usually, GLP is axiomatized byusing [n]instead of hni. However, itis moreconvenient for our purposes to
use hni, since we focus on Πn+1-axiomatized concepts. Note that GLP isclosed under the necessitation rule: if
GLP⊢ϕthenGLP⊢[n]ϕ.
3
Kripke Models. A Kripke frame is a structure A = hW,{R } i, where W is a non-empty
n n≥0
set of worlds and R , for k ≥ 0, are binary relations on W. A is called finite if W is finite and
k
R =∅ for almost every k ≥0.
k
A valuation J·K on a frame A maps every propositional variable p to a subset JpK ⊆ W.
A Kripke model is a Kripke frame together with a valuation on it. Given any Kripke model
A = hW,{R } ,J·Ki, we extend the valuation J·K recursively to the class of all polymodal
n n≥0
formulas:
(i) J⊤K=W; J⊥K=∅;
(ii) Jψ∧χK=JψK∩JχK;
(iii) J¬ψK=W \JψK, and similarly for the other propositional connectives; and
(iv) JhniψK={x∈W |∃y(xR y &y ∈JψK)}.
n
We often write A,x |= ϕ instead of x ∈ JϕK. Similarly, we also write x (cid:13) ϕ instead of x ∈ JϕK
when A is clear from the context. We say that ϕ is valid in A, denoted by A |= ϕ, if A,x |= ϕ
for every x∈W.
A binary relation R on W is conversely well-founded if there is no infinite chain of elements
of W of the form x Rx Rx .... It is easy to see that, for finite W, this condition is equivalent
0 1 2
to R being irreflexive. A Kripke frame A=hW,{R } i is called a J∗-frame if
n n≥0
(i) R is conversely well-founded and transitive, for all k ≥0;
k
(ii) ∀x,y(xR y ⇒∀z(xR z ⇔yR z)), for m<n; and
n m m
(iii) ∀x,y,z(xR y &yR z ⇒xR z), for m<n.
m n m
A J∗-model is a Kripke model that is based on a J∗-frame. Such a model A=hW,{R } ,J·Ki
n n≥0
is called strongly persistent if it satisfies the following two conditions:
(i) if |p|≤n and y (cid:13)p, then x(cid:13)p whenever xR y; and
n
(ii) if |p|<n and y 1p, then x1p whenever xR y.
n
The following facts are easy to establish.
Lemma 2.1. Let A=hW,{R } ,J·Ki be a J∗-model. Then, A is strongly persistent iff for all
n n≥0
formulas ϕ and all n≥0 we have
(i) if |ϕ|≤n, then xR y and y (cid:13)ϕ imply x(cid:13)ϕ; and
n
(ii) if |ϕ|<n, then xR y and y 1ϕ imply x1ϕ.
n
Lemma 2.2. The axiom scheme hniϕ→ϕ is valid in a J∗-model A for all ϕ such that |ϕ|≤n
iff A is strongly persistent.
Theorem2.3. Foranyformulaϕ,J∗ ⊢ϕiffϕisvalidinallfinite,stronglypersistentJ∗-models.
Proof (Sketch). The soundness direction is a straightforward induction on proof length. For
completeness, the standard methods used to construct canonical Kripke models can be used to
show that J∗ is complete for the class of finite and strongly persistent J∗-models. Roughly, the
argument can be sketched as follows.
4
We define an operator ∼, called modified negation, for all formulas ϕ as follows:
ψ, if ϕ=¬ψ for some ψ,
∼ϕ=
¬ϕ, otherwise.
(cid:26)
Fora set∆offormulas,wedefine ℓ(∆):={n|hniϕ∈∆ for some ϕ}. Following[6],we saythat
a set ∆ of formulas is adequate if ⊤∈∆ and it is closed under subformulas, modified negations,
and the operations
hniϕ,hmiψ ∈∆ ⇒ hmiϕ∈∆,
pm ∈∆,n∈ℓ(∆) ⇒ hnipm ∈∆, for all variables pm and n≥m, and
¬pm ∈∆,n∈ℓ(∆) ⇒ hni¬pm ∈∆, for all variables pm and n>m.
We can easily convince ourselvesthat any finite set Γ canbe extended to a finite set Γ′ which is
adequate and such that ℓ(Γ)=ℓ(Γ′).
Letusnowfixafiniteadequateset∆andassumethatallmodalitiesrangewithinΛ:=ℓ(∆).
Define a Kripke frame F =hW,{R } i, where
∆ n n≥0
W :={x|x is a maximal J∗-consistent subset of ∆},
for n∈Λ and x,y ∈W, and xR y holds if the following conditions are satisfied:
n
(i) For any ϕ∈y, if hniϕ∈∆ then hniϕ∈x.
(ii) For any hniϕ∈∆, we have that hniϕ∈y implies hniϕ∈x.
(iii) For any hmiϕ∈∆ such that m<n, we have hmiϕ∈x ⇐⇒ hmiϕ∈y.
(iv) There exists a formula hniϕ∈∆ such that hniϕ∈x\y.
Foranynaturalnumbern6∈Λ,setR :=∅. Furthermore,defineaKripkemodelA =hF ,J·Ki,
n ∆ ∆
where
A ,x|=p ⇐⇒ p∈x
∆ df
for all variables p∈∆ and x∈W. It can be shown that, for all ϕ∈∆,
A ,x|=ϕ ⇐⇒ ϕ∈x.
∆
Moreover, using the axioms of Σ -completeness, A can be easily shown to be a finite and
n ∆
strongly persistent J∗-model.
Using these facts, completeness follows in the usual way. Suppose J∗ 0 ϕ. Then {∼ϕ} is
J∗-consistent. Let ∆ be the smallest finite adequate set containing ϕ. Then A is a finite and
∆
strongly persistent J∗-model. There is a maximal J∗-consistent set x ⊆ ∆ such that ∼ϕ ∈ x,
whence K ,x6|=ϕ.2 (cid:3)
∆
Formal Arithmetic. We consider first-order theories in the language L =(0,+,·,s) of arith-
0
metic (where s is a unary function symbol for the successor function). Theories and formulas
formulatedin(anextensionof)thislanguagewillbecalledarithmetical. Thetheoriesweconsider
will be extensions of Peano arithmetic (PA). It will be convenient to assume that the language
2AdetailedproofcanbefoundintheworkofBerger[11].
5
of PA contains terms for all primitive recursive functions. It is well-known that PA can be con-
servativelyextended so as to containdefinitions of allthese terms. The class of ∆ -formulas are
0
all formulas where each occurrence of a quantifier is of either forms
∀x≤tϕ:=∀x(x≤t→ϕ),
∃x≤tϕ:=∃x(x≤t∧ϕ),
wheretisatermoverL thathasnooccurrenceofthevariablex. Occurrencesofsuchquantifiers
0
arecalledbounded. TheclassesofΣ -andΠ -formulasaredefinedinductively: ∆ -formulasare
n n 0
Σ - and Π -formulas. If ϕ(~x,y) is a Π -formula, then ∃yϕ(~x,y) is a Σ -formula. Similarly,
0 0 n n+1
if ϕ(~x,y) is a Σ -formula, then ∀yϕ(~x,y) is a Π -formula. When an arithmetical theory T is
n n+1
given, we often identify these classes modulo provable equivalence in T.
We recursively define 0 := 0 and n+1 := s(n). The expression n is called the n-th numeral
and represents the number n in L . We assume a standard global assignment p·q of expressions
0
(terms,formulas,etc.) tonaturalnumbers. Givenanyexpressionτ,wecallpτqthecodeorGödel
number of τ. Note that pτq, being a natural number, “lives” in our informal metatheory and
has a natural representation in L through the term pτq. However, when presenting formulas
0
in the arithmetical language, we usually write pτq instead of pτq. We often consider primitive
recursive families of formulas ϕ that depend on a parameter n ∈ ω. In this context, pϕ q
n x
denotes a primitive recursivedefinable term with free variable x whose value for a givenn is the
Gödelnumberofϕ . Inparticular,theexpressionpϕ(x˙)qdenotesaprimitiverecursivedefinable
n
termwhosevaluegivenanynistheGödelnumberofϕ(n),i.e.,theGödelnumberoftheformula
resulting from ϕ when substituting the term n for x. Following [7], it is convenient to assume
a second sort of first-order variables, denoted α,β,..., that range over codes of arithmetical
formulas. Formulas containing such variables can be naturally translated into the one-sorted
setting by making use of a primitive recursive predicate that defines the notion of “being a
formula.”
A theory T is sound if T ⊢ ϕ implies N |= ϕ, for every arithmetical sentence ϕ. For an
axiomatizable extension T of PA, we denote by ✷ (α) the formula that formalizes the notion of
T
provability in T in the usual sense. We write ✷ ϕ instead of ✷ (pϕq). The formula ✷ defines
T T T
the standard Gödelian provability predicate for T. More generally, given a formula Prv(α) with
onefreevariableα,wesaythatPrvisaprovabilitypredicateoflevelnoverT,ifforallarithmetical
sentences ϕ,ψ:
(i) Prv is a Σ -formula;
n+1
(ii) T ⊢ϕ implies PA⊢Prv(pϕq);
(iii) PA⊢Prv(pϕ→ψq)→(Prv(pϕq)→Prv(pψq)); and
(iv) if ϕ is a Σ -sentence, then PA⊢ϕ→Prv(pϕq).
n+1
It is well-known that ✷ , in its standard formulation, is a provability predicate of level 0. A
T
provabilitypredicatePrvissound ifN|=Prv(pϕq)impliesN|=ϕ,foreveryarithmeticalsentence
ϕ. A sequence π of formulas Prv ,Prv ,... is a strong sequence of provability predicates over T
0 1
if there is a sequence r <r <··· of natural numbers such that, for all n≥0,
0 1
(i) Prv is a provability predicate of level r over T; and
n n
(ii) T ⊢Prv (pϕq)→Prv (pϕq), for any arithmetical sentence ϕ.
n n+1
We denotePrv by[n] andabbreviate[n] (pϕq)by[n] ϕ. Moreover,thedualof[n] isdefined
n π π π π
byhni ϕ:=¬[n] ¬ϕ. Givensuchasequenceπ,wedenoteby|π |thelevelofthen-thprovability
π π n
6
predicate of π.
Since provability predicate [n] from π is a Σ -sentence for some k >0, we can associate (in
π k
analogy to the standard Gödelian provability predicate) a predicate Prf (α,y) which expresses
n
the statement “y codes a proof of α” and
T ⊢Prv (α)↔∃yPrf (α,y).
n n
We stress that Prf is chosen in such a way such that every number y codes a proof of at most
n
one formula and that every provable formula has arbitrarilylong proofs.
WedenotebyTrue (α)thewell-knowntruth-definitionfortheclassofallΠ -sentences,i.e.,
Πn n
True (α) expresses the fact “α is the Gödel number of a true arithmetical Π -sentence.” The
Πn n
truth-definition for Π -sentences serves as a basis for a natural strong sequence of provability
n
predicates. Let [0] :=✷ and
T T
[n+1] (α):=∃β(True (β)∧✷ (β →α)), for n≥0.
T Πn T
The formula [n] is a provability predicate of level n. It formalizes the notion of being provable
T
in the theory T +Th (N), where Th (N) is the set of all true Π -sentences.
Πn Πn n
Another strong sequence of provability predicates is defined by [0]ω :=✷PA and
[n+1] :=∃β(∀x[n] β(x˙)∧[n] (∀xβ(x)→α)), for n≥0.
ω ω ω
Thepredicate[n] isoflevel2nandformalizesthenotionofprovabilitybynapplications oftheω-
ω
rule. Japardize originally showed arithmetical completeness for this arithmetical interpretation,
while completeness with respect to the broader class of arithmetical interpretations, defined by
strong sequences of provability predicates, was later established by Ignatiev [14].3
Arithmetical Interpretation. Let π be a strong sequence of provability predicates over T.
An (arithmetical) realization (over π) is a function f that maps propositional variables to
π
arithmetical sentences. The realization f is typed if f (p) is an arithmetical Π -sentence,
π π |πn|+1
provided n = |p| < ω. Any realization f can be uniquely extended to a map fˆ that captures
π π
all polymodal formulas as follows:
(i) fˆ (⊥)= ⊥; fˆ (⊤)= ⊤, where ⊥ (resp., ⊤) is a convenient contradictory (resp., tauto-
π π
logical) statement in the language of arithmetic;
(ii) fˆ (p)=f (p), for any propositional variable p;
π π
(iii) fˆ (·) commutes with the propositional connectives; and
π
(iv) fˆ (hniϕ)=hni fˆ (ϕ), for all n≥0.
π π π
By some simple closure properties of the class of Π -sentences, its follows that |ϕ| = n implies
n
that fˆ (ϕ) is provably equivalent to a Π -sentence in T. Using this, we readily observe:
π |πn|+1
Lemma 2.4. If GLP∗ ⊢ϕ, then T ⊢fˆ (ϕ) for all typed arithmetical realizations f over π.
π π
Proof. The lemma is shown by induction on the length of a proof of ϕ in GLP∗. Most of the
axioms are clear. In particular, the provability of the instances Löb’s axiom is well-known. The
axiom of Σ -completeness follows from our discussion above. The induction step, i.e., closure
n+1
under the rules of inference, is easy to establish. We leave the details to the reader. (cid:3)
3SeeArtemovandBeklemishev[2]forabriefhistoricalbackground.
7
Hence, GLP∗ is sound for the arithmetical semantics thus defined. Completeness holds under
the additional assumption of soundness of the provability predicates involved. Arithmetical
completeness for GLP∗ has first been established by Japaridze [15] and has been significantly
extended and simplified by Ignatiev [14]. Beklemishev [7] provided yet another simplified proof
forthearithmeticalcompletenesstheoremforGLP∗thatisclosetoSolovay’soriginalconstruction
for the logic GL. The next section will be devoted to the proof of the arithmetical completeness
theorem for GLP∗. To this end, we are going to extend the construction for GLP described in
ref. [7].
3 Arithmetical Completeness
ArithmeticalcompletenessproofsusuallyrelyonreasonableKripkesemantics,sincethoseproofs
usually establish the following fact: if ϕ is a formula that has a Kripke model falsifying ϕ in a
certain world, one can find an arithmetical realization such that the arithmetical theory under
consideration does not prove ϕ under this realization. Since GLP is, however, not complete for
any class of Kripke frames, Beklemishev reduces GLP to J and relies on the Kripke semantics of
J in order to prove arithmetical completeness. We proceed analogously in the following.
To this end, we define formulas M(ϕ) and M+(ϕ) as follows [7]. Consider an enumeration
hm iϕ ,hm iϕ ,...,hm iϕ of all subformulas of ϕ of the form hkiψ and let n := max m .
1 1 2 2 s s i≤s i
Define
M(ϕ):= (hjiϕ →hm iϕ ),
i i i
1≤i≤s
^
mi<j≤n
and, furthermore,
M+(ϕ):=M(ϕ)∧ [i]M(ϕ).
i≤n
^
Bythemonotonicityaxioms,itisclearthatGLP∗ ⊢M+(ϕ). Wearegoingtoprovethefollowing
statement.
Theorem 3.1. Let T be a sound axiomatizable extension of PA and π a strong sequence of
provability predicates over T of which every provability predicate is sound. Then, for all many-
sorted formulas ϕ, the following statements are equivalent:
(i) GLP∗ ⊢ϕ;
(ii) J∗ ⊢M+(ϕ)→ϕ; and
(iii) T ⊢fˆ (ϕ), for all typed realizations f .
π π
Before turning to the proof of this theorem, let us introduce some additional notions. A root of
a J∗-model A = hW,{R } ,J·Ki is a world r ∈ W such that for all x ∈ W there is a k ≥ 0
n n≥0
such that rR x or r = x. A model which has a root is called rooted. The following lemma can
k
be proved similarly as in ref. [7].
Lemma 3.2. For any ϕ ∈ L∗, if J∗ 0 ϕ then there is a finite and strongly persistent J∗-model
A with root r such that A,r 6|=ϕ.
WeproceedwiththeproofofTheorem3.1. Thedirectionfrom(ii)to(i)isimmediatesinceGLP∗
extends J∗ and GLP∗ ⊢ M+(ϕ). Furthermore, the direction from (i) to (iii) is the arithmetical
8
soundness of GLP∗ (Lemma 2.4). We show that (iii) implies (ii) by assuming the contrapositive,
i.e., assume that J∗ 0 M+(ϕ) → ϕ. Then there is a finite and strongly persistent J∗-model
A = hW,{R′} ,J·Ki with root r such that A,r |= M+(ϕ) and A,r 6|= ϕ. Without loss of
n n≥0
generality, assume that W = {1,2,...,N} for some N ≥ 1 and r = 1. We define a new model
A =hW ,{R } ,J·Ki, where
0 0 n n≥0
(i) W ={0}∪W;
0
(ii) R ={(0,x)|x∈W}∪R′;
0 0
(iii) R =R′, for k >0; and
k k
(iv) A ,0|=p ⇐⇒ A,1|=p, for all variables p.
0 df
Notice that A is still a finite and strongly persistent J∗-model such that A ,r 6|= M+(ϕ) → ϕ.
0 0
Letmbe theleastnumbersuchthatR 6=∅andR =∅forallk >m. As inref.[7],wedefine
m k
the following auxiliary notions:
R (x):={y |xR y},
k k
R∗(x):={y |y ∈R (x), for some i≥k}, and
k i
R◦(x):=R∗(x)∪ {R∗(z)|x∈R∗ (z)}.
k k k k+1
[
Furthermore, Solovay functions h : ω →W are defined for all n≤m as follows:
n 0
h (0)=0 and h (0)=ℓ , for n>0;
0 n n−1
y, if h (x)R z and Prf (p¬S q,x),
n n n z
h (x+1)=
n
( hn(x), otherwise.
Here, ℓ = x is a formalization of the statement that the function h (defined by a formula
k k
H (x,y)) has as its limit at x, i.e.,
k
ℓ =x ⇐⇒ ∃N ∀n≥N H (n,x).
k df 0 0 k
The defining formulas H can be constructed via a diagonalizationargument; see [7] for details.
k
For x∈W , S denotes the sentence ℓ =x. The following lemmas are established in ref. [7].
0 x m
Lemma 3.3. For all k ≥0,
(i) T ⊢∀x∃!w ∈W : H (x,w);
0 k
(ii) T ⊢∃!w ∈W : ℓ =w;
0 k
(iii) T ⊢∀i,j∀z ∈W (i<j∧h (i)=z →h (j)∈R (z)∪{z}); and
0 k k k
(iv) T ⊢∀z ∈W (∃xh (x)=z →ℓ ∈R∗(z)∪{z}).
0 k m k
Lemma 3.4. The following conditions hold for the sentences S :
x
(i) T ⊢ S and T ⊢¬(S ∧S ) for all x6=y;
x∈W0 x x y
(ii) T ⊢WSx →hkiπSy, for all y such that xRky;
(iii) T ⊢S →[k] (ℓ ∈R◦(x)), for all x6=0; and
x π m k
9
(iv) N|=S .
0
Lemma 3.5. For all k <m, provably in T,
(i) either ℓ =ℓ or ℓ R ℓ ; and
k k+1 k k+1 k+1
(ii) if k <n≤m, then either ℓ =ℓ or ℓ R ℓ , for some j ∈(k,n].
k n k j n
Proof. Item (i) is clear from Lemma 3.3. Item (ii) is proved by an external induction on n
from (i). (cid:3)
Lemma 3.6. For the arithmetical realization f defined by
π
f : p7−→ S
π x
x(cid:13)p
_
it holds that T 0fˆ (ϕ).
π
Proof (Sketch). Lemma 3.6 can be proved by first establishing Solovay’s “commutation lemma”
which says that, for each world x6=0 and each subformula χ of ϕ, it holds that
(i) if A ,x|=χ, then T ⊢S →fˆ (χ); and
0 x π
(ii) if A ,x6|=χ, then T ⊢S →¬fˆ (χ).
0 x π
Indeed, nowsuppose that T ⊢fˆ (ϕ). ThenA,16|=ϕ implies that T ⊢S →¬fˆ (ϕ). Hence, T ⊢
π 1 π
¬S and so T ⊢¬h0i S , whence T ⊢¬S by item (ii) of Lemma 3.4. This contradicts item (iv)
1 π 1 0
of Lemma 3.4. (cid:3)
Itthereforesufficestoshowthatf isatypedarithmeticalrealizationusingtheassumptionthat
π
A is a strongly persistent J∗-model. To this end, we assume a natural arithmetization of the
0
forcing relation (cid:13) on A by bounded formulas.
0
Lemma 3.7. For any variable p of sort k ≤m, provably in T,
f (p) ⇐⇒ ∀w ∈W \JpK: ∀x¬H (x,w).
π 0 k
Proof. Forthedirectionfromlefttoright,wereasoninT asfollows. Assumef (p)and,towards
π
a contradiction, suppose that ∃xh (x) = w for some w ∈ W such that w 1 p. By item (iv)
k 0
of Lemma 3.3, we know that, provably in T, ∃xh (x)=u implies
k
S ∨ S ,
u z
z∈R_∗k(u)
for any u∈W . In particular, we infer
0
S ∨ S .
w u
u∈R_∗k(w)
SinceA isstronglypersistentandw 1p,weknowthatu1pforallu∈R∗(w). Thiscontradicts
0 k
f (p) by item (i) of Lemma 3.4.
π
For the other direction, we reason in T as follows. Assume the right-hand side of the equiv-
alence. We certainly know that ¬S for all u ∈ W such that u 1 p. Now, if ℓ = ℓ , then,
u 0 k m
by item (i) of Lemma 3.4, S for some x ∈ W such that x (cid:13) p and we are thus finished. So
x 0
10
