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Volume 81, Number 4, December 2016

CLASS FORCING, THE FORCING THEOREM AND BOOLEAN COMPLETIONS

PETER HOLY, REGULA KRAPF, PHILIPP L ¨UCKE, ANA NJEGOMIR, AND PHILIPP SCHLICHT

Abstract. The forcing theorem is the most fundamental result about set forcing, stating that the forcing relation for any set forcing is definable and that the truth lemma holds, that is everything that holds in a generic extension is forced by a condition in the relevant generic filter. We show that both the definability (and, in fact, even the amenability) of the forcing relation and the truth lemma can fail for class forcing.

In addition to these negative results, we show that the forcing theorem is equivalent to the existence of a (certain kind of) Boolean completion, and we introduce a weak combinatorial property (approachability by projections) that implies the forcing theorem to hold. Finally, we show that unlike for set forcing, Boolean completions need not be unique for class forcing.

§1. Introduction. The classical approach to generalize the technique of forcing with set-sized partial orders to forcing with class partial orders is to work with countable transitive modelsMof some theory extendingZFand partial ordersP definable overM (in the sense that both the domain ofPand the relationPare definable over the modelM,∈). ByZFwe mean the usual axioms ofZFwithout the power set axiom, however including Collection instead of Replacement.1In this situation, we say that a filterG onPisP-generic overM ifG meets every dense subset ofPthat is definable overM. We letMPdenote the collection of allP-names contained inM. SinceM |=ZF,MPis definable overM. Given aP-generic filter GoverM, we defineM[G] ={G| ∈MP}to be the corresponding class generic extension. Finally, given a formulaϕ(v0, . . . , vn−1) in the languageLof set theory, a conditionpinPand0, . . . , n−1∈MP, we letpMP ϕ(0, . . . , n−1) denote the statement thatϕ(0G, . . . , n−G 1) holds inM[G] wheneverGis aP-generic filter over Mwithp∈G.

The forcing theorem is the most fundamental result in the theory of forcing with set-sized partial orders. The work presented in this paper is motivated by the question whether fragments of this result also hold for class forcing. Given a countable transitive modelM of some theory extendingZF, a partial orderP definable overMand anL-formulaϕ(v0, . . . , vn−1), we will consider the following fragments of the forcing theorem for notions of class forcing.

Received June 26, 2015.

2010Mathematics Subject Classification.03E40, 03E70, 03E99.

Key words and phrases. class forcing, forcing theorem, Boolean completions.

1Note that in the absence of the power set axiom, Collection does not follow from Replacement and many important set-theoretical results can consistently fail in the weaker theory. For further details, consult [6].

c 2016, Association for Symbolic Logic 0022-4812/16/8104-0015 DOI:10.1017/jsl.2016.4

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(1) We say thatPsatisfies the definability lemma forϕoverMif the set {p, 0, . . . , n−1P×MP× · · · ×MP|pMP ϕ(0, . . . , n−1)}

is definable overM.

(2) We say that P satisfies the truth lemma for ϕ over M if for all 0, . . . , n−1 MP and every P-generic filter G over M with the prop- erty thatϕ(0G, . . . , n−G 1) holds in M[G], there is a conditionp G with pMP ϕ(0, . . . , n−1).

(3) We say thatPsatisfies the forcing theorem forϕoverM ifPsatisfies both the definability and the truth lemma forϕoverM.

Another basic result that is fundamental for the development of forcing with set- sized partial orders is the existence of a Boolean completion for separative partial orders,2 its uniqueness up to isomorphism and the equality of the corresponding forcing extensions. This motivates our interest in the following notions.

(4) LetBbe a Boolean algebra that is definable overM (in the sense that both the domain of Band all Boolean operations are definable over the model M,∈). We say thatBisM-completeif supBAexists inBfor everyA⊆B withA∈M.

(5) We say thatP has a Boolean completion inM if there is an M-complete Boolean algebraBand an injective dense embeddingfromPintoB\ {0B} such that bothBandare definable overM.

(6) We say thatP has a unique Boolean completion in M if P has a Boolean completionB0inM and for every other Boolean completionB1ofPinM, there is an isomorphism inVbetweenB0andB1which fixesP.

In standard accounts on class forcing (see [5]) one studies generic extensions with additional predicates for the generic filter and the ground model and focuses on pretame(resp.tame) notions of forcing, i.e., notions of forcing which preserveZF (resp.ZF) with respect to these predicates. In particular, Sy Friedman shows in [5]

that if the ground model satisfiesZF, then every pretame forcing satisfies the forcing theorem. The converse is false: A simple notion of class forcing which does not preserve Replacement is Col(,Ord), the class of all finite partial functions from to Ord, ordered by reverse inclusion. However, this notion of forcing still satisfies the forcing theorem (see [5, Proposition 2.25] or Section 6 of this paper). In this paper, we will mostly investigate properties of nonpretame notions of class forcing.

In the remainder of this introduction, we present the results of this paper. We will later prove these statements in a more general setting than the one outlined above.

This will allow us to also prove results for models containing more second-order objects, like models ofKelley-Morse class theoryKM(see [1]). We will outline this setting in Section 3.

Throughout this paper, we will work in a modelVofZFCand, given a setMand a recursively enumerable theoryTextendingZF, we say that “Mis a model ofT”

to abbreviate the statement thatM satisfies every axiom ofTinVwith respect to some formalized satisfaction relation (as in [3, Chapter 3.5]). Note that, in general,

2A partial order (or, more generally, a preorder)Pisseparativeif for allp, qP, ifpqthen there existsrpsuch thatrq.

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the assumption that such a modelVcontaining a transitive countable setM ofT exists is stronger than the assumption thatTis consistent. IfTZF, then the results of this paper can also be proven in the setting of [10, Chapter VII,§9, Approach (1b)], where one works with a language that extends the language of set theory by a constant symbol ˙M, and a model of a theory in this language that extendsTby the scheme of axioms stating that every axiom ofTholds relativized to ˙M. The consistency of this theory is equivalent to the consistency ofT. For this paper, we have nonetheless chosen the first approach, because it makes many arguments more intuitive and easier to state.

1.1. Positive results. The results of this paper will show that the forcing theorem (in fact both the definability of the forcing relation and the truth lemma), the amenability of the forcing relation and the existence of a (unique) Boolean completion can fail for class forcing. The following two positive results show that nontrivial implications hold between some of these properties. The first result shows that a failure of the forcing theorem already implies a failure of the definability lemma for atomic formulae. This result is proven by carefully mimicking the induction steps in the proof of the forcing theorem for set forcing.

Theorem1.1. LetMbe a countable transitive model ofZFand letPbe a partial order that is definable overM. IfPsatisfies the definability lemma for either “v0∈v1 or “v0=v1” overM, thenPsatisfies the forcing theorem for allL-formulae overM. In Section 6, we will present a criterion that will allow us to show that many notions of class forcing satisfy the definability lemma for atomic formulae and thus, by the above result, the full forcing theorem (that is, the forcing theorem for all L-formulae).

The next result shows that the existence of a Boolean completion is equivalent to the validity of the forcing theorem for allL-formulae. We will prove this result by showing that the forcing relation for the quantifier-free infinitary languageLOrd,0

of set theory, allowing set-sized conjunctions and disjunctions and also allowing reference to a predicate for the generic filter, is definable under either assumption listed in the theorem.

Theorem 1.2. Let M be a countable transitive model of ZF and let P be a separative partial order that is definable overM. If either the power set axiom holds inM or there is a well-ordering of M that is definable overM, then the following statements are equivalent.

(1) Psatisfies the forcing theorem for allL-formulae overM. (2) Phas a Boolean completion inM.

1.2. Negative results. In the following, we present results showing that each of the properties considered above can fail for class forcing. The first result shows that there is always a notion of class forcing that does not satisfy the definability lemma.

The proof of this result uses a notion of class forcing that was introduced by Sy Friedman and that is mentioned in [12, Remark 1.8]. We will present and study this notion of forcing in detail in Section 2.

Theorem 1.3. Let M be a countable transitive model of ZF. Then there is a partial orderPthat is definable overM and does not satisfy the forcing theorem for atomic formulae overM.

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Our next result shows that even stronger failures of the definability lemma are possible for the above forcing. Its proof relies on so-calledParis models, i.e., - structures M with the property that each ordinal of Mis definable in M by a formula without parameters. Such models have been considered by Ali Enayat in [4]. The stronger concept ofpointwise definable models(that is,∈-structures over which each of their elements is definable by a formula without parameters) was studied in depth in [9]. Note that the existence of a countable transitive model of ZFCyields the existence of a countable transitive Paris model satisfying the axioms ofZFC– this follows from [9, Theorem 11], where it is shown that every countable transitive model ofZFChas a pointwise definable class forcing extension. However, in Section 8, we will, for the benefit of the reader, sketch a simplified argument to verify (the weaker statement) that certain countable transitive models ofZFChave class forcing extensions which are Paris models.

Theorem1.4. LetMbe a countable transitive Paris model withM|=ZF. Then there is a partial orderPsuch thatPis definable overMand theP-forcing relation for

“v0=v1” is notM-amenable, i.e., there is a setx∈Msuch that {p, , ∈P×MP×MP |pMP =} ∩x is not an element ofM.

Next, we consider failures of the truth lemma. The witnessing forcing notion for the next theorem will be the two-step iteration of the above notion of class forcing that has a generic extension which is a Paris model and the aforementioned notion of class forcing of Sy Friedman.

Theorem1.5. Assume thatM is a countable transitive model ofZF such that eitherM is uncountable inL[M]andM L[M]⊆M, orM satisfiesV =Land there is a countable subsetCofP(M)such thatM,Cis a model ofKM. Then there is a partial orderPthat is definable overMand that does not satisfy the truth lemma for “v0=v1” overM.

Finally, we show that the existence and uniqueness of a Boolean completion, in a countable transitiveM |=ZF of which there exists a well-order of order-type OrdM that is definable overM, of a notion of forcingPthat is definable overM, are equivalent toPhaving the Ord-chain condition(or simply Ord-cc) overM, that is the property that every antichain ofPthat is definable overMis already an element ofM. This will easily yield the following result.

Theorem 1.6. Let M be a countable transitive model of ZF and suppose that there exists a global well-order of order typeOrdM that is definable overM. Then there is a notion of class forcing which has two nonisomorphic Boolean completions inM.

§2. Some notions of class forcing. In this section, we introduce several notions of class forcing that will later be used to verify the negative results listed in Section 1.

Notation. Since we will frequently use names for ordered pairs, we introduce the notation

op(, ) ={{,½P},½P,{,½P,,½P},½P}

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for, ∈MP andα Ord. Clearly,op(, )is the canonical name for the ordered pairG, G.

Definition2.1. LetM be a countable transitive model ofZF.

(1) Let Col(,Ord)Mdenote the partial order Col(,OrdM), i.e., Col(,Ord)M is the partial order whose conditions are finite partial functionsp : −−→par OrdM ordered by reverse inclusion.

(2) Define Col(,Ord)Mto be the (dense) suborder of Col(,Ord)Mconsisting of all conditionspwith dom(p).

(3) Let Col(,Ord)M be the notion of forcing whose conditions are finite partial functionsp : −−→par OrdM ∪ {≥α|α∈OrdM}, where≥α is an element ofMwhich is not in OrdMfor everyα∈OrdM, and whose ordering is given byp≤qif and only if dom(p)dom(q) and for everyn∈dom(q), either

p(n) =q(n) or

q(n) is≥αfor someα OrdM and there is ≥αsuch thatp(n) {,≥}.

Note that all of these partial orders are definable over the corresponding modelM. Notation. LetPbe a partial order, letbe aP-name, and letpbe a condition in P. Then we define thep-evaluation ofto be

p = {p| ∃q P[, q ∈ p≤Pq]}.

The next lemma gives some basic properties of the different collapse forcings defined above.

Lemma2.2. LetMbe a countable transitive model ofZF.

(1) IfGis aCol(,Ord)M-generic filter overM, then for every ordinal inMthere is a surjection from a subset ofonto that ordinal inM[G].

(2) IfGis aCol(,Ord)M-generic filter overM, thenM =M[G].

(3) No nontrivial maximal antichain of Col(,Ord)M or Col(,Ord)M is an element ofM.

(4) IfMis a model ofZFC, then no nontrivial complete suborder ofCol(,Ord)M or ofCol(,Ord)M is an element ofM.

(5) Col(,Ord)M is the union ofOrdM-many set-sized complete subforcings.

Proof.(1) Pick OrdM. Givenα∈OrdM, define

Dα = {p∈Col(,Ord)M | ∃n∈dom(p) [p(n) =α]}.

Then eachDαis dense in Col(,Ord)M and definable overM. This implies that if G is Col(,Ord)-generic overM, then for everyα ∈M∩Ord there is ann <

with{n, α} ∈G. This shows that

= {op( ˇn,αˇ),{n, α} |α < , n < } is a name for a surjection from a subset ofonto .

(2) Let be a Col(,Ord)M-name inM. Then ran(p) rank() holds for every conditionpin tc()Col(,Ord)M. If we define

D = {p∈Col(,Ord)M |rank()ran(p)},

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thenD is dense in Col(,Ord)M and definable overM. Moreover, by the above observation, we have G = p M, whenever G is an M-generic filter on Col(,Ord)M andp∈D∩G, because suchpeither extends or is incompatible to any condition in tc().

(3) Let Col denote either Col(,Ord)Mor Col(,Ord)M. Assume thatA∈Mis an antichain of Col which is not equal to{½}. Picka∈A. Now for anyb ∈A\{a}, the domains ofaandbcannot be disjoint by incompatibility. Definec∈Col with dom(c) = dom(a) and for everyn dom(c), letc(n) = sup{b(n)| b A}+ 1.

Hencecis incompatible with every element ofA, showing thatAis not maximal.

(4) This statement follows from the above results because our assumptions imply that set-sized partial orders inMcontain nontrivial maximal antichains.

(5) Let for everyα∈OrdM, Col(, α) denote the subforcing of Col(,OrdM) consisting of finite partial functionsp:−−→par α∪ {≥ | ≤α}with the induced ordering. Clearly,

Col(,Ord)M =

α∈OrdM

Col(, α).

It remains to be checked that for every α OrdM, Col(, α) is a complete subforcing of Col(,Ord)M. LetAbe a maximal antichain of Col(, α) and letp∈Col(,Ord)M. Consider the condition ¯p∈Col(, α) which is obtained frompby replacingp(n) by≥αwheneverp(n)≥αorp(n) is of the formfor some > α. SinceAis a maximal antichain, there isa∈Asuch thata and ¯pare compatible. Let ¯q Col(, α) be a common strengthening of ¯panda. But then the conditionq obtained from ¯q by replacing ¯q(n) byp(n) for everyn dom(p) such that ¯q(n) is of the form≥αwitnesses thatpandaare compatible.

The above computations show that, contrasting the situation with set-sized partial orders, forcing with a dense suborder of a notion of class forcingPcan produce different generic extensions than forcing withPdoes.3In a subsequent paper [7], we will in fact show that for any notion of class forcingP, the property that all forcing notions which containPas a dense subforcing produce the same generic extensions asP, is essentially equivalent to the pretameness ofP.

Corollary2.3. IfMis a countable transitive model ofZF, then there are partial ordersP andQdefinable over M such thatQis a dense suborder of P andM = M[G∩Q]M[G]wheneverGis aP-generic filter overM.

In Section 6, we will show that all of the partial orders that we have mentioned so far satisfy the forcing theorem.

The following notion of class forcing due to Sy Friedman is mentioned in [12, Remark 1.8]. It will be crucial for the proofs of the negative results listed in Section 1.

Definition2.4. LetMbe a countable transitive model ofZF. DefineFMto be the partial order whose conditions are triplesp=dp, ep, fpsatisfying

(1) dpis a finite subset of,

(2) epis a binary acyclic relation ondp,

3Note that it is still true that generic filters forPinduce generic filters for its dense suborders and vice versa.

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(3) fpis an injective function with dom(fp)∈ {∅, dp}and ran(fp)⊆M, (4) if dom(fp) = dp andi, j ∈dp, then we havei ep j if and only iffp(i)

fp(j),

and whose ordering is given by

p≤FM q ⇐⇒ dq ⊆dp ep(dq×dq) =eq fq ⊆fp. Note thatFM is definable overM.

Lemma2.5. The set of all conditionspinFM withdom(fp) =dpis dense.

Proof.Pickp∈FM with dom(fp) =∅ =dp. We inductively define (using that ep is acyclic) a functionfas follows. For everyj∈dplet

f(j) ={f(i)|i ep j} ∪ {{∅, j}}.

Using that ∅ ∈ range(f), it is easy to inductively verify that ¯p = dp, ep, f satisfies conditions (3) and (4) above, and hence is an extension ofpinFM with

dom(fp¯) =dp¯.

Lemma2.6. IfM is a countable transitive model ofZF andG is anFM-generic filter overM, then there is a binary relationEonsuch thatE ∈M[G], and the models, EandM,∈are isomorphic inV.

Proof.Define anFM-name ˙E∈M by setting

E˙ = {op( ˇi,jˇ), pi,j |i, j ∈, i=j},

where pi,j denotes the condition in FM with dpi,j = {i, j}, epi,j = {i, j} and fpi,j =∅. LetG be anFM-generic filter overM and putE = ˙EG ∈M[G]. Note thatE =

{ep | p∈G}. DefineF =

{fp |p∈G}. By Lemma 2.5, the sets Dn={p∈FM |n∈dom(fp)}are dense inFM. Since these sets are definable over M, we can conclude thatF is injective and that dom(F) =. In order to see that F is surjective, we claim that for everyx M, the set{p FM | x ran(fp)} is dense. In order to show this, letp =dp, ep, fp FM such thatx /∈ran(fp).

Using Lemma 2.5, we may assume that dom(fp) = dp. Choosej \dp and definedq = dp ∪ {j}, eq = ep ∪ {i, j | fp(i) x} ∪ {j, i | x fp(i)} and fq =fp∪ {j, x}. Thenq=dq, eq, fqis an extension ofpwithx∈ran(fq).

It remains to be checked thatF is an isomorphism between the models, E andM,∈. Takei, j < such thati E j, i.e.,pi,j ∈G. By the above computations, there is a conditionp∈Gwithi, j dom(fp). We then havei epj and by (4) in Definition 2.4 we haveF(i) =fp(i)∈fp(j) =F(j). For the converse, suppose thatx, y M such thatx ∈y. By the above computations, there is a condition p G and i, j dp withF(i) = fp(i) = x y = fp(j) = F(j). By (4) in Definition 2.4, this impliesi epjand thereforei E jholds.

§3. The general setting. In the following, we outline the general setting of this paper. That is, we will actually make use of an approach that is slightly more general than the one presented in Section 1, namely one that works with models that might contain more second-order objects than just the definable ones, and moreover we will work with preorders instead of partial orders.

Notation. (1) We denote byGB the theory in the two-sorted language with variables for sets and classes, with the set axioms given by ZF with class

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parameters allowed in the schemata of Separation and Collection, and the class axioms of extensionality, foundation and first-order class comprehension (i.e., involving only set quantifiers). Furthermore, we denote the theory GB enhanced with the power set axiom by GB(this is the common collection of axioms of G ¨odel-Bernays set theory).

(2) We letKMdenote the axiom system of Kelley-Morse class theory. That is, in addition to the usual ZFC axioms for sets with class parameters allowed in the schemata of Separation and Collection, one also has the class axioms of Foundation, Extensionality, Replacement, (second order) Comprehension and Global Choice. In particular, class recursion holds in models ofKM. For a detailed axiomatization ofKM, see [1].

(3) By a countable transitive model ofGB(orGB,KM), we mean a modelM= M,CofGB(resp.GB,KM) such thatMis transitive and bothMandCare countable inV.

Example3.1. (1) Let M be a countable transitive model of ZF and let Def(M) be the set of all subsets ofM that are definable overM,∈. Then M,Def (M)is a model ofGB.

(2) LetMbe a countable transitive model ofZFsatisfyingV=L[A] and Replace- ment for formulae mentioning the predicateA, and letCbe Def(M, A). Then M,Cis a model ofGB. This is the approach used in [5].

(3) Every countable transitive model ofKMis a model ofGB.

Fix a countable transitive modelM=M,CofGB. By anotion of class forcing (forM) we mean a preorderP =P,≤Psuch thatP,≤P∈ C. We will frequently identifyPwith its domainP. In the following, we also fix a notion of class forcing P=P,≤PforM.

We callaP-nameif all elements ofare of the form, p, whereis aP-name andp P. DefineMP to be the set of all P-names that are elements ofM and defineCP to be the set of allP-names that are elements ofC. In the following, we will usually call the elements ofMP P-namesand we will call the elements ofCP classP-names. If∈MPis aP-name, we define

rank = sup{rank+ 1| ∃p∈P[, p ∈]} to be itsname rank.

We say that a filterG onP isP-generic over MifG meets every dense subset ofPthat is an element ofC. Given such a filterG and aP-name, we define the G-evaluationofas

G={G | ∃p∈G[, p ∈]},

and similarly we define ΓGfor Γ∈ CP. Moreover, ifGisP-generic overM, then we set M[G] ={G| ∈MP}andC[G] ={ΓG|Γ∈ CP}, and callM[G] =M[G],C[G] aP-generic extensionofM.4

For alln < , we letLndenote the first-order language that extends the language of set theoryLby unary predicate symbols A0, . . . , An−1. Given anLn-formula

4While it does not really play any role for the present paper which second order objects we allow for in our generic extensions, we will argue in a subsequent paper [8] that the above choice (namelyC[G]) is canonical.

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ϕ(v0, . . . , vm−1), a tupleΓ = Γ0, . . . ,Γn−1(CP)n, a conditionp∈Pand names 0, . . . , m−1∈MP, we write

pM,P Γϕ(0, . . . , m−1)

to denote that ϕ(0G, . . . , m−1G ) holds in the structure MΓ[G] = M[G],

∈,ΓG0, . . . ,ΓGn−1wheneverGis aP-generic filter overMwithp∈G. Whenever the context is clear, we will omit the superscripts and subscripts.

Our choice of considering preorders instead of partial orders is due to the reason that in the case of a two-step iterationPQ˙ of notions of class forcing, as defined in [5] (see also Section 8 of the present paper), we will have conditions of the formp,q˙ forp∈Pandpforcing that ˙q Q˙. In general, there will be distinct pairsp,q˙0 andp,q˙1such thatpPq˙0= ˙q1, i.e., one naturally obtains a preorder that is not antisymmetric. However, in some contexts it will become crucial for our orderings to be antisymmetric. In that case we will use the following additional property:

Definition 3.2. We say that a model M,C of GB satisfies representatives choice, if for every equivalence relationE ∈ Cthere isA∈ Cand a surjective map : dom(E)→AinCsuch thatx, y ∈Eif and only if(x) =(y).

Using representatives choice, given a preorderP =P,≤P ∈ C, by considering the equivalence relation

p≈q iff p≤Pq∧q≤Pp,

we obtain a partial orderQ∈C and a surjective map:PQinCsuch that for allp, q∈P,p≈qif and only if(p) =(q).

Clearly, representatives choice follows from the existence of a global well-order.

Furthermore, ifMsatisfies the power set axiom, then we also obtain representatives choice, since we can use Scott’s trick to obtain the sets [p] = {q P | q ≈p∧

∀r[q≈r→rank(q)rank(r)]} ∈M forp∈P.

§4. The forcing theorem. In this section, we fix a countable transitive model M = M,Cof GB and a notion of class forcing P = P,≤P forM. We will show that in order to obtain the forcing theorem for allLn-formulae, it suffices that the forcing relation for either the formula “v0∈v1” or the formula “v0=v1” is definable, thus proving Theorem 1.1.

Definition4.1. Letϕ≡ϕ(v0, . . . , vm−1) be anLn-formula.

(1) We say thatPsatisfies the definability lemma forϕoverMif

{p, 0, . . . , m−1 ∈P×MP× · · · ×MP|pM,P Γϕ(0, . . . , m−1)} ∈ C for allΓ(CP)n.

(2) We say thatPsatisfies the truth lemma forϕoverMif for all0, . . . , m−1 MP,Γ(CP)nand every filterGwhich isP-generic overMwith

MΓ[G]|=ϕ(0G, . . . , m−1G ), there isp∈GwithpM,P Γϕ(0, . . . , m−1).

(3) We say thatPsatisfies the forcing theorem forϕoverMifPsatisfies both the definability lemma and the truth lemma forϕoverM.

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Our goal is to prove a generalization of Theorem 1.1 in our general setting which allows for second-order objects. The first step to achieve this is to show that the definability lemma for some atomic formula already implies the truth lemma to hold for all atomic formulae.

Lemma 4.2. Assume that P satisfies the definability lemma for “v0∈v1” or

“v0=v1” overM. ThenPsatisfies the forcing theorem for all atomic formulae.

Proof.Suppose first that the definability lemma holds for “v0∈v1.” We denote bypM,∗P the statement that for all, r ∈and for allq≤Pp, r, the set

D,={s P|sMP ∈}

is dense belowq inP. Furthermore, letp M,∗P = denote thatp M,∗P and p M,∗P . We show by induction on the lexicographic order on pairs rank() + rank(),rank()that the following hold for eachp∈P:

(1) pMP if and only if the set

E,={q P| ∃, r ∈[q P r∧q M,∗P =]} ∈ C is dense belowpinP.

(2) p MP if and only ifpM,∗P . In particular,p MP =if and only ifpM,∗P =.

(3) There is a dense subset ofPinCthat consists of conditionspinPsuch that eitherpMP orpMP /∈.

(4) There is a dense subset ofPinCthat consists of conditionspinPsuch that eitherpMP orpMP .

To start the induction, note that if rank() + rank() = 0, then (1)–(4) trivally hold. Note that (3) implies that the truth lemma holds for “v0∈v1.” Furthermore, (4) implies the truth lemma for “v0⊆v1” and hence also for equality. Suppose now that (1)–(4) are satisfied for all pairs of names,¯ ¯ inMPfor whichrank( ¯) + rank( ¯),rank( ¯)is lexicographically less thanrank() + rank(),rank(), that is rank( ¯) + rank( ¯) rank() + rank() and in case of equality, we have that rank( ¯)<rank().

In order to prove (1), pick a conditionp Pwith p MP andq P p. LetG beP-generic overMwithq ∈G. ThenG G by assumption and hence there is, r ∈withr∈GandG=G. By our inductive assumption, property (4) yields a condition s G with s MP = which by (2) is equivalent to s M,∗P = . SinceG is a filter, there ist G witht P q, r, s. In particular, t E,. For the other direction, suppose that E, is dense belowp. Let G be P-generic overMwithp∈G. By density ofE,, we can takeq∈Gand, r ∈ such thatq P r andq M,∗P =. Thenr G and soG G. Thus by our inductive assumption, condition (2) implies thatG =G G.

For (2), suppose first thatp MP , let, r ∈ and letq P p, r. Take a P-generic filterGwithq ∈G. ThenG G G. By our inductive assumption, we can finds G so that s MP . Given anyq P q, by strengthening s if necessary, we can find such s P q, as desired. Conversely, assume that pM,∗P and letGbeP-generic overMwithp∈G. Let, r ∈withr∈G. We have to show thatG G. Letq ∈G be a common strengthening ofpandr.

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Then by assumption, the set D, is dense below q. By genericity, we can take s∈D,∩G. Using our inductive assumption, this shows thatG G, as desired.

For (3), consider the set

D={p∈P| ∀, r ∈∀q Pp, r[q MP =]}.

Then our inductive assumptions imply thatD∈ C. Moreover, condition (1) states thatDis nonempty below everyp PwithpMP . Hence it suffices to show thatp MP /∈for everyp∈D, since thenD∪ {p∈P|p MP ∈} ∈ Cis a dense set of conditions deciding . So takep∈Dand suppose thatpMP /∈. Then there is aP-generic filterGcontainingpsuch thatG G. Then there must be, r ∈withr ∈GandG =G. By our inductive assumption, we can find q ∈G withq MP =. By possibly strengtheningq using thatG is a filter, we may assume thatq≤Pp, r. But this contradicts thatp∈D.

In order to verify (4), we define

E={p∈P| ∃, r ∈[p≤Pr∧ ∀q≤Pp(q MP )]}.

As above,Eis inCinductively, and it is nonempty below every condition which does not force . As in the proof of (3) it remains to be checked thatpMP for eachp∈E. Assume, towards a contradiction, that there isp∈EwithpMP . Then there is aP-generic filter withp ∈G andG G. Let, rwitness that p∈E. Thenr∈G and soG G G. Using (3) inductively, we obtainq ∈G withqMP . But then there iss Pp, q, contradicting thatp∈E.

If the definability lemma holds for “v0=v1,” we can define theP-forcing relation for “v0∈v1” by stipulating (as above) thatpMP if and only if the set

{q∈P| ∃, r ∈[q P r∧q MP =]}

is dense belowp.

Theorem 4.3. If P satisfies the definability lemma either for “v0∈v1” or for

“v0=v1” overM, thenPsatisfies the forcing theorem for everyLn-formula overM. Proof.By the previous lemma, we already know thatPsatisfies the forcing the- orem for all atomic formulae. Let us next consider formulas of the form “v0∈V1,” involving a class variableV1.

Let∈MPand Γ∈ CP. We claim thatpM,ΓP Γ if and only if the set D={q∈P| ∃, r ∈Γ [q≤Pr∧qMP =]}

is dense belowp. Note thatD∈ Csince the forcing relation for equality is definable.

First assume thatp M,ΓP Γ and letq P p. LetG be aP-generic filter with q∈G. ThenGΓG, i.e., there is, r ∈Γ such thatr∈GandG =G. By the truth lemma for “v0=v1,” there iss ∈G such thats MP =. But then every t P q, r, s is inD. Conversely, ifDis dense inPandG isP-generic overMwith p∈G, then we findq P pinD∩G. By definition ofDthere is, r ∈ Γ such thatq P randq P =. Thus using thatq, r∈G we getG =G ΓG. The truth lemma for “v0∈V1” follows from the truth lemma for equality.

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For composite Ln-formulae, we can define the forcing relation by the usual recursion:

pM,P Γ(ϕ∧)(0, . . . , m−1) ⇐⇒ pM,P Γϕ(0, . . . , m−1) andpM,P Γ(0, . . . , m−1), pM,P Γ¬ϕ(0, . . . , m−1) ⇐⇒ ∀q≤Pp[qM,P Γϕ(0, . . . , m−1)], pM,P Γ∀xϕ(0, . . . , m−1, x) ⇐⇒ ∀∈MP[pM,P Γϕ(0, . . . , m−1, )], where0, . . . , m−1 MP andΓ (CP)n. The truth lemma can be verified as for

set forcing in each case.

§5. Boolean completions. In set forcing, every partial order has a unique Boolean completion whose elements are the regular open subsets of the partial order. In this section, we will investigate the relationship between the existence of a Boolean completion and the forcing theorem for notions of class forcing. LetM=M,Cbe a fixed countable transitive model ofGBand letP=P,≤Pbe a notion of class forcing.

LetLOrd,0denote the infinitary quantifier-free language that allows for set-sized conjunctions and disjunctions. ByLOrd,0(P, M) we denote the language of infinitary quantifier-free formulae in the forcing language ofPoverM, that allows reference to the generic predicateG. More precisely, its constants are all elements ofMP, and it has an additional predicate ˙G. We defineLOrd,0(P, M) and the class FmlOrd,0(P, M) of G ¨odel codes ofLOrd,0(P, M)-formulae by simultaneous recursion:

(1) AtomicLOrd,0(P, M)-formulae are of the form =, or ˇp G˙ for , ∈MPandp∈P, where ˙G ={p, p |ˇ p∈P} ∈ CPis the canonical class name for the generic filter. G ¨odel codes of atomicLOrd,0(P, M)-formulae are given by

pˇ ∈G˙ =0, p, ==1, , ,

=2, , .

(2) Ifϕis anLOrd,0(P, M)-formula, then so is¬ϕ, and its G ¨odel code is given by

¬ϕ=3,ϕ.

(3) If I M and for every i I, ϕi is an LOrd,0(P, M)-formula such that ϕi|i ∈I ∈M, then so are

i∈Iϕi and

i∈Iϕiand their G ¨odel codes are given by

i∈I

ϕi=4, I,{i,ϕi |i ∈I},

i∈I

ϕi=5, I,{i,ϕi |i ∈I}.

Now define FmlOrd,0(P, M) ∈ C to be the class of all G ¨odel codes of infinitary formulae in the forcing language ofPoverM. IfGis aP-generic filter overMand

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ϕis anLOrd,0(P, M)-formula, then we writeϕGfor the formula obtained fromϕby replacing eachP-nameoccurring inϕby its evaluationG, and by evaluating ˙G asG. Note thatϕGis a formula in the infinitary languageLOrd,0with an additional predicate for the genericG. Given anLOrd,0(P, M)-formulaϕandp∈P, we write pMP ϕto denote thatM[G],∈, G |=ϕG wheneverGis aP-generic filter overM withp∈G.

Definition 5.1. We say that P satisfies the uniform forcing theorem for LOrd,0(P, M)-formulaeif

{p,ϕ ∈FmlOrd,0(P, M)|pMP ϕ} ∈ C

andPsatisfies the truth lemma for everyLOrd,0(P, M)-formulaϕoverM, i.e., for every P-generic filter G over M, if M[G] |= ϕG then there is p G such that pMP ϕ.

The following lemma will allow us to infer that the uniform forcing theorem for infinitary formulae is equivalent to the forcing theorem for equality.

Lemma5.2. There is an assignment

FmlOrd,0(P, M)→MP×MP; ϕϕ, ϕ such that{ϕ, ϕ, ϕ |ϕFmlOrd,0(P, M)} ∈ Cand

½P P (ϕ↔ϕ =ϕ) (1) for everyϕ∈ LOrd,0(P, M).

Proof.We will argue by induction that, given namesandsatisfying (1) for every proper subformulaofϕ, we can, uniformly inϕ, defineϕ andϕ

such that (1) holds.

Observe that since ¬

i∈Iϕi

i∈I¬ϕi and ¬

i∈Iϕi

i∈I¬ϕi, we can assume that all formulae are in negation normal form, i.e., the negation operator is applied to atomic formulae only. Next, due to the equivalences

=≡∨,

,p∈

( /∈∧pˇ∈G˙), /∈≡

,p∈

(=∨p /ˇ∈G˙),

we can further suppose that the only negated formulae are of the form ˇp /∈G˙. For the atomic cases, let

p∈ˇ G˙ ={ˇ0, p}, p∈ˇ G˙ = ˇ1, p /ˇG˙ =∅, p /ˇG˙ ={ˇ0, p},

= =, ==,

=, =∪ {,½P}.

It is easy to check that (1) holds for all atomic formulae.

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If ϕ is a conjunction of the form

i∈Iϕi and ϕi, ϕi have already been defined fori ∈I, let

ϕ={op(ϕi,iˇ),½P |i ∈I}and ϕ={op(ϕi,iˇ),½P |i ∈I}.

IfG isP-generic overM andM[G] |=ϕG, thenM[G]|= ϕGi for all i I. By assumption, this means thatϕGi =Gϕifor everyi ∈I, thus alsoϕG =Gϕ. The converse is similar.

Next suppose thatϕis of the form

i∈Iϕi. Let ¯ϕi =op(ϕi,iˇ) and ¯ϕi= op(ϕi,iˇ) for eachi ∈I. Let

ϕ={op( ¯ϕi,¯ϕi),½P |i ∈I} ∪ {op( ¯ϕi,¯ϕi),½P |i ∈I}, ϕi =ϕ\ {op( ¯ϕi,¯ϕi),½P}.

Now we define

ϕ={ϕi ,½P |i ∈I}, ϕ=ϕ∪ {ϕ,½P}.

IfG isP-generic andM[G] |= ϕG, there is some i I such that M[G] |= ϕiG. By induction, this implies that M[G] |= ϕGi = Gϕi. Thus Gϕ = (ϕi )G andϕG = Gϕ. For the converse, suppose that there is a generic G such that M[G]|=¬ϕG, hence for everyi ∈I,M[G]|=¬ϕiG. But then inM[G], for every i ∈I, we haveϕGi =Gϕi. Therefore, Gϕ is not of the form (ϕi )G for any

i ∈I, which shows thatGϕGϕ\ϕG .

Next, we will use the above lemma to provide a characterization of notions of class forcing which satisfy the forcing theorem, using Boolean completions.

Definition5.3. IfBis a Boolean algebra, then we say thatBisM-completeif the supremum supBAof all elements inAexists inBfor everyA∈MwithA⊆B. Definition 5.4. We say that P has a Boolean completion in M if there is an M-complete Boolean algebraB=B,0B,1B¬,∧,∨such thatB, all Boolean oper- ations ofBand an injective dense embedding from PintoB\ {0B}are elements ofC.

Theorem5.5. Assume thatMsatisfies representatives choice and letP=P,≤P be a separative and antisymmetric notion of class forcing forM. Then the following statements (overM) are equivalent:

(1) P satisfies the definability lemma for one of the L-formulae “v0∈v1” or

“v0=v1.”

(2) Psatisfies the forcing theorem for allL-formulae.

(3) Psatisfies the uniform forcing theorem for allLOrd,0(P, M)-formulae.

(4) Phas a Boolean completion.

Moreover, separativity and antisymmetricity ofPare only necessary for the impli- cation from (3) to (4), in particular the equivalence of (1)–(3) and those being a consequence of (4) holds as well without these assumptions. In fact, without these assumptions, (4) implies thatPis separative and antisymmetric.

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