Forcing as a computational process
aa r X i v : . [ m a t h . L O ] S e p FORCING AS A COMPUTATIONAL PROCESS
JOEL DAVID HAMKINS, RUSSELL MILLER, AND KAMERYN J. WILLIAMS
Abstract.
We investigate how set-theoretic forcing can be seen as a compu-tational process on the models of set theory. Given an oracle for informationabout a model of set theory h M, ∈ M i , we explain senses in which one maycompute M -generic filters G ⊆ P ∈ M and the corresponding forcing exten-sions M [ G ]. Specifically, from the atomic diagram one may compute G , fromthe ∆ -diagram one may compute M [ G ] and its ∆ -diagram, and from the ele-mentary diagram one may compute the elementary diagram of M [ G ]. We alsoexamine the information necessary to make the process functorial, and con-clude that in the general case, no such computational process will be functorial.For any such process, it will always be possible to have different isomorphicpresentations of a model of set theory M that lead to different non-isomorphicforcing extensions M [ G ]. Indeed, there is no Borel function providing genericfilters that is functorial in this sense. Introduction
The method of forcing, introduced by Paul Cohen to show the consistency of thefailure of the continuum hypothesis, has become ubiquitous within set theory. Inthis paper we analyze this method from the perspective of computable structuretheory. To what extent is forcing an effective process?
Main Question.
Given an oracle for a countable model of set theory M , to whatextent can we compute its various forcing extensions M [ G ] ? We answer this, considering multiple levels of information which might be givenby the oracle.
Main Theorem 1 (Forcing as a computational process) . (1) For each countable model M of set theory and for any forcing notion P ∈ M there is an M -generic filter G ⊆ P computable from the atomic diagram of M .(2) Given the ∆ -diagram of M and a forcing notion P ∈ M , we can uniformlycompute such a G , the atomic diagram of M [ G ] , and moreover the ∆ -diagram of M [ G ] . Mathematics Subject Classification.
Key words and phrases.
Forcing, computable structure theory.We thank the anonymous referee for their helpful comments.The second author was supported by NSF grant We will give an overview of the construction during Section 4. For full details we refer thereader to [Sho71] or [Kun80]. See also [Cho09] for a conceptual overview. ∆ in the sense of the L´evy hierarchy, as described below. (3) From P and the elementary diagram of M , we can uniformly computethe elementary diagram of M [ G ] , and this holds level-by-level for the Σ n -diagrams. These statements are proved in Section 4 as Theorems 8, 10, and 11 below. InSection 5 we will extend these results to look at the generic multiverse of a countablemodel of set theory, i.e. those models obtained from the original model by takingboth forcing extensions and grounds, where taking grounds is the process inverse tobuilding forcing extensions. In Section 6 we also consider versions of these resultsfor class forcing instead of set forcing.We remark here that in this context the ∆ -diagram of M refers to the setof formulae true in M that are ∆ in the L´evy hierarchy. In this hierarchy, thestandard one used within set theory, ∆ -formulae are those whose only quantifiersare bounded by sets, that is, of the form ∃ x ∈ y or ∀ x ∈ y . This differs from ∆ in the arithmetical hierarchy, whose formulae’s quantifiers are only those boundedby the order relation on ω , that is of the form ∃ x < y or ∀ x < y . In Section 2 wewill show that very little can be computed from the atomic diagram for a model ofset theory. From the diagram we are not even able to compute relations as simpleas x ⊆ y . We view this as evidence that for the computable structure theory ofset theory the L´evy ∆ -diagram is an appropriate choice of the basic informationto be used. The usual signature of set theory—just equality and the membershiprelation—is too spartan to say anything of use. In Section 3 we introduce anexpansion of the signature for set theory which captures the strength of the L´evy∆ -diagram. We show that the Σ n -formulae in the L´evy hierarchy are preciselythose that are Σ n in the arithmetical hierarchy with the expanded signature.We end the paper, in Sections 7 and 8, by investigating how much informationis required to make the process of computing M [ G ] from M functorial. Withoutsignificant detail about the dense subsets of P , it will not be so. Recall that a pre-sentation of a countable structure M is simply a structure isomorphic to M whosedomain is ω . The following theorem emphasizes the importance of not conflatingthe isomorphism type of M with a specific presentation of M . Main Theorem 2 (Nonfunctoriality of forcing) . There is no computable procedureand indeed no Borel procedure which performs the tasks of Main Theorem 1 in auniform way so that distinct presentations of the model M will result in isomorphicpresentations of the extension M [ G ] . We also show that forcing can be made a functorial process. One way to dothis is to add extra information to the signature of the model. Another way is torestrict to a special class of models, namely the pointwise definable models.2.
The atomic diagram of a model of set theory knows very little
In this section, we show that very little about a model of set theory can becomputed from its atomic diagram. In particular, many basic set-theoretic relationsare not decidable from the atomic diagram. As a warmup, let us first see that theatomic diagram does not suffice to identify even a single fixed element.
Proposition 3.
For any countable model of set theory h M, ∈ M i and any element b ∈ M , no algorithm will pick out the number representing b uniformly given anoracle for the atomic diagram of a copy of M . ORCING AS A COMPUTATIONAL PROCESS 3
For example, given the atomic diagram of a copy of M , one cannot reliably findthe empty set, nor the ordinal ω , nor the set R of reals. Proof.
Fix b ∈ M and fix an oracle for the atomic diagram of a copy of M . Supposewe are faced with an algorithm purported to identify b . Run the algorithm until ithas produced a number that it claims is representing b . This algorithm inspectedonly finitely much of the atomic diagram of M . The number b it produced hasat most finitely many elements in that portion of the atomic diagram. But M has many sets that extend that pattern of membership, and so we may find analternative copy M ′ of M whose atomic diagram agrees with the original one onthe part that was used by the computation, but disagrees afterwards in such a waythat the number for b now represents a different set. So the algorithm will get thewrong answer on M ′ . (cid:3) This idea can be extended to characterize which relations on M are computablyfrom the atomic diagram. In particular, any such relation must contain both finiteand infinite sets. Theorem 4.
Let h M, ∈ M i | = ZF be a countable model of set theory and let X ⊆ M n for some ≤ n < ω . The following are equivalent. (In the following, Σ in all casesrefers to the arithmetical hierarchy, not the L´evy hierarchy. )(1) X is uniformly relatively intrinsically computably enumerable in the atomicdiagram of M . That is, there is a single computably enumerable operatorwhich given the atomic diagram of a presentation of M will output the copyof X for that presentation.(2) Membership of each single ~a in X is witnessed by a finite pattern of ∈ inthe transitive closures of { a } , . . . , { a n − } , and the list of finite patternswitnessing membership is enumeration-reducible to the Σ -diagram of M .(3) There is a L ω ,ω -formula ϕ ( x ) so that ϕ ( x ) is Σ and defines X , and theset of (finite) disjuncts in this formula is enumeration-reducible to the Σ -diagram of M .Proof. (1 ⇔
3) is a standard fact in computable structure theory, first establishedin [AKMS89], relativized here since M does not have a computable presentation.(2 ⇒
1) For notational simplicity we will present the argument for the n = 1case. Suppose membership of a in X ⊆ M is witnessed by a finite pattern of ∈ inthe transitive closure of { a } . That is, a is in X if and only if one of a certain list offinite graphs can be found in the pointed graph (TC( { a } ) , a, ∈ M ). By hypothesisthis list is e -reducible to the Σ -diagram of M , which we can enumerate, since wehave the atomic diagram of M as an oracle. Therefore, we can list out these finitegraphs, one by one. Meanwhile, we enumerate ∈ M and M , and continually checkwhether the most recent pair in ∈ M has completed a copy of a graph from this list.If such happens, then we search through the a ∈ M we have already enumeratedand check whether a has one of these graphs appearing below its transitive closure.We output all the a for which this happens. We also check whether the most recent a ∈ M has a copy of one of the graphs in the portion of its transitive closureso far enumerated. If that happens, we output a . This process will find every a ∈ M which has a copy of this graph below its transitive closure, so it will exactlyenumerate X . See Section 3 for further discussion of the distinction.
JOEL DAVID HAMKINS, RUSSELL MILLER, AND KAMERYN J. WILLIAMS (1 ⇒
2) Again, we will we present the proof for the n = 1 case for notationalsimplicity. Assume that X is uniformly relatively intrinsically computable enumer-able from the atomic diagram of M . That is, the decision of whether a ∈ X mustbe made by a finite portion of the graph ∈ M where we are only allowed to know theindex of a itself. We can separate this information into two pieces, that inside thetransitive closure of { a } and that outside. We claim that only the first informationcan be relevant, which then implies (2). To see this, observe that every possiblefinite pattern of finite graphs (with a as a constant) occurs outside of the transitiveclosure of { a } in every model of set theory. So the information outside is true forevery a ∈ M and thus cannot have any effect on the decision of whether to output a in the enumeration. (cid:3) As a consequence of this theorem, the atomic diagram of a model of set theorydoes not know any of the predicates and relations named in Lemma 7 below, whichare each easily checked to fail item (2) of Theorem 4. In short, the atomic diagramof a model of set theory knows little about the model. Before addressing forcing,therefore, we now present a natural expansion of the signature, under which theatomic diagram will present the information that set theorists normally consider tobe basic. 3.
The L´evy diagram
The signature ordinarily used in set theory is simple: it consists of the binaryrelation ∈ along with equality. This signature suffices all by itself to express all ofmathematics (possibly using assumptions beyond ZFC). But from the perspectiveof computable structure theory we cannot say much using just the atomic diagramin this signature. As we see it, the lesson here is that this atomic diagram is tooweak to take as the basic information for the computable structure theory of modelsof set theory. Instead, we will work with the ∆ -diagram, in the sense of the L´evyhierarchy , in which set-bounded quantification is not counted when determiningquantifier complexity of a formula.This conflicts with the convention in computability theory, where “boundedquantification” refers to bounded in the sense of arithmetic—i.e. of the form ∃ x < y or ∀ x < y —rather than bounded in the sense of a relation of the structure. In short,∆ in the L´evy hierarchy is not the same as ∆ in the arithmetical hierarchy, as ∀ x ∈ a ϕ ( x ) may express infinitely many independent atomic statements.In this section we describe how we can make the L´evy hierarchy line up with thearithmetical hierarchy, by expanding the signature to capture the content of the(L´evy) ∆ -diagram. Let us call this expanded signature the L´evy signature . Thisprovides an alternate way to think of the ∆ -diagram. One may instead work withthe atomic diagram in this expanded signature, with the correspondence betweenthe two being effective. And this correspondence continues upward, with Σ n inthe L´evy hierarchy corresponding to Σ n in the arithmetical hierarchy with thisexpanded signature. In the sequel, we will speak of the ∆ -diagram or Σ n -diagram,referring to the L´evy hierarchy, but the reader who prefers the other approach mayfreely translate over the statements. As further evidence for the naturalness of this expansion of the language, we note that it hasarisen in other contexts. Venturi and Viale [VV19] studied model companions for set theory. Fortheir work they also found it helpful to expand the signature from the spartan {∈} to includesymbols for each ∆ relation. ORCING AS A COMPUTATIONAL PROCESS 5
Definition 5.
The
L´evy signature contains the binary relation symbol ∈ and also,for each ∆ -formula ϕ ( x , . . . , x n ) , an n -ary predicate R ϕ . Structures in the L´evysignature are assumed to satisfy the axiom schema (over all ∆ -formulae): ∀ ~x [ R ϕ ( ~x ) ⇐⇒ ϕ ( ~x )] . For example, since inclusion is defined by a ∆ -formula using only ∈ , we see that M | = b ⊆ c ⇐⇒ M | = ∀ y ∈ b y ∈ c ⇐⇒ M | = R ( ∀ y ∈ x ) y ∈ x ( b, c ) . Since we can enumerate the ∆ -formulae effectively, the language of the L´evy signa-ture is a computable language, and it includes equality (defined by x ⊆ y ∧ y ⊆ x ).Our choice of signature allows us to imitate the usual set-theoretic conventionunder which ∆ is the lowest possible complexity of a formula. Additionally, setsdefined by Σ n -formulae in the L´evy hierarchy will all be arithmetically Σ n in modelsof ZF in the L´evy signature, by the following standard lemma. Lemma 6.
For every formula ϕ ( t , . . . , t n , x, y, z ) in the L´evy hierarchy, ZF ⊢ ∀ ~t [ ∀ x ∈ z ∃ y ϕ ( ~t, x, y, z ) ⇐⇒ ∃ Y ∀ x ∈ z ∃ y ∈ Y ϕ ( ~t, x, y, z )] . Proof. If M | = ∀ x ∈ z ∃ y ϕ ( ~a, x, y, z ) for a model h M, ∈ M i of ZF with parameters ~a , then by the Replacement axiom schema, there is some set Y in M such that M | = ∀ x ∈ z ∃ y ∈ Y ϕ ( ~a, x, y, z ). The reverse direction is trivial. (cid:3) The lemma, applied repeatedly, allows us to turn every formula from the sig-nature with just membership and equality into an equivalent formula which is ina prenex form, with all unbounded quantifiers preceding a ∆ matrix, and whichhas the same complexity (in the L´evy hierarchy) as the original formula. There-fore, every formula of L´evy complexity Σ n may be expressed in the L´evy signatureby a formula of arithmetic complexity Σ n , and there is an effective procedure forproducing the latter from the former.To conclude this section, let us remark that many basic predicates of set theoryare ∆ , and that the basic predicates for forcing are ∆ and hence computable fromthe ∆ -diagram. Lemma 7.
For models M of ZFC , the following predicates are all ∆ . x = ∅ . x ⊆ y . x = { y, z } . x = S y . x = { z ∈ y : ϕ ( z ) } , for ∆ -formulae ϕ ( z ) . x is a (Kuratowski) ordered pair. x is a set of ordered pairs. x is a function. x is transitive. x is an ordinal. x is inductive. x = ω . We omit any proof of these standard facts. More pertinently for this article, letus briefly remark that the relations “ x is a P -name”, p (cid:13) σ ∈ τ , and p (cid:13) σ = τ are all ∆ and hence computable from the ∆ -diagram. This will be important inSection 4 to see that the ∆ -diagram suffices to compute a forcing extension. JOEL DAVID HAMKINS, RUSSELL MILLER, AND KAMERYN J. WILLIAMS Computing forcing extensions
For the sake of the reader who may not be an expert in set theory, we interleaveour proof of the first main theorem with an exposition of forcing. This expositionwill follow the three parts of the main theorem. First we discuss generic filters andwe show that a generic filter may be effectively built from just the atomic diagram.We next discuss how to build the forcing extension M [ G ] from the ground model M and the generic G . Then we show that there is an effective procedure to construct(the atomic diagram of) the forcing extension from the ∆ -diagram of the groundmodel. Finally, we discuss how truth in the forcing extension is determined by theforcing relations in the ground model. Then we can show that elementary diagramof the forcing extension is effectively computable from the elementary diagram ofthe ground model.In the ground universe M , a partially ordered set P gives partial informationabout an object from outside the universe. A condition p is stronger than anothercondition q , written p ≤ q , if it gives more information about this outside object. Itis convenient to assume that P has a maximum element 1 which gives no informa-tion. Two conditions are compatible, written p k q , if there is a condition strongerthan both of them. Otherwise, they are incompatible, written p ⊥ q . Formally, theoutside object is an M -generic filter G ⊆ P . That is, it is downward directed andupward closed, and it meets every dense D ⊆ P in M . Here, D ⊆ P is dense ifevery condition in P can be strengthened to a condition in D . It is a straightfor-ward exercise that if P is splitting—any condition p extends to two incompatibleconditions—then M itself contains no M -generic filter.However, if M is countable, we can always externally construct an M -genericfilter G ⊆ P , and the construction works uniformly for all P (and all ∆ ( M )). Wesimply fix the first-encountered p ∈ P as our starting point (using the externalorder < on the domain ω of M to find it) and then ask of each element s ∈ ω inturn whether the current p s can be extended to an element of P ∩ s . This is a ∆ question, and we either find the least p s +1 ≤ p s lying in s if the ∆ ( M ) indicatesthat one exists, or else set p s +1 = p s . Finally, set G = { q ∈ P : q k p n for some n } .Of course, if s was a dense subset of P , then we ensured that G does meet s , so this G is M -generic.In fact, there exists a generic filter that is computable merely from the atomicdiagram, as we now show. Theorem 8.
Given an oracle for the atomic diagram of a model of set theory h M, ∈ M i | = ZF and a notion of forcing P ∈ M , there is a computable procedure tocompute an M -generic filter G ⊆ P .Proof. Fix an oracle for h M, ∈ M i . That is, we are presented on the oracle tape withan isomorphic copy of M as a binary relation on the natural numbers. Indeed, wemight as well assume that M = N and ∈ M is that relation on N . Let P ∈ M beany notion of forcing in M . More specifically, we have in M an element P ∈ M forthe underlying set of the partial order, and we also have a set ≤ P in M for what M thinks is the set of ordered pairs for that relation. We assume that the Kuratowskipairing function ( p, q ) = { { p } , { p, q } } is used when coding ordered pairs.Notice that from the atomic diagram we can decide whether a given number p represents an element of P or not, since we need only ask the oracle whether p ∈ M P . See Footnote 1 for citations to references with more detail on the forcing construction.
ORCING AS A COMPUTATIONAL PROCESS 7
Further, we can computably enumerate the pairs ( p, q ) of forcing conditions with p ≤ P q . For this we must unwrap the Kuratowski pairing function, but this ispossible as follows. We search for conditions p, q in P and for elements x ∈ ≤ P thatrepresent the pair ( p, q ). This will happen when x = { y, z } , where y = { p } and z = { p, q } . So we can search for the elements y and z which have y ∈ M x and z ∈ M x and p ∈ M y and p, q ∈ M z . When this happens, we can be confident that p ≤ P q . (This kind of computational inspection of the Kuratowski ordered pair alsoarose in [GH17].)But actually, we can fully decide the relation ≤ P , not just enumerate it. Thereason is that M , being a model of set theory, has what it thinks is the set ofpairs ( p, q ) ∈ P with p P q . And so for any pair p, q , we can search for it tobe enumerated by ≤ P or by the analogous procedure applied with P , and therebydecide whether p ≤ P q or not.Next, let D ∈ M be the set that M thinks is the set of all dense subsets of P .From this data, we can enumerate the elements D , D , D , . . . of D , listing all theelements of M that M thinks are dense subsets of P . We simply run through allthe natural numbers d , and ask the oracle whether d ∈ M D , and if so, we put it onthe list.Let us use this enumeration to compute a descending sequence of forcing condi-tions p ≥ P p ≥ P p ≥ P · · · with p n ∈ D n . To begin, we let p be the first-encountered element of D , whichexists because D is dense. Next, given p n , we search through the natural numbersfor the first-encountered condition p n +1 ≤ P p n such that p n +1 ∈ D n +1 . There isalways such a condition because D n +1 is dense.Finally, having constructed the descending sequence, we define G as the set ofconditions q for which p n ≤ q for some n . This is an M -generic filter, because itis a filter and it meets every dense subset of P in M . The elements of G can beenumerated by the processes above, because whenever we find a condition q forwhich p n ≤ P q for some n , then we can enumerate q into the filter.But actually, we claim that G is fully decidable from the oracle, not just enu-merable. To see this, let ⊥ P be the set in M that M thinks is the set of pairs ( p, q )of incompatible conditions p ⊥ q . By genericity, it follows that if q / ∈ G , it must bethat q ⊥ P p n for some n , since it is dense to either get below q or become incompat-ible with it. The algorithms above allow us to enumerate the pairs of incompatibleconditions, and so for any condition q , we search for a p n for which either p n ≤ P q or p n ⊥ P q , and in this way we can tell whether q ∈ G or q / ∈ G , as desired. (cid:3) The algorithm above is non-uniform in several senses. First, the algorithm makesuse not only of the indices of the forcing notion (namely, the set P , the set ≤ P andthe complement of ≤ P in P × P ), but also the index of the set of dense subsetsof P , and the index ⊥ P for the set of incomparable elements. The methods ofTheorem 4 show that it is not possible in general to compute these just from theatomic diagram of h M, ∈ M i and P . Second, a more serious kind of non-uniformityarises from the fact that, even if we are given this additional data and even if weare given the full elementary diagram of the model M , the particular generic filterthat we end up with will depend on the order in which M is represented in thispresentation. The filter G is determined in part by the order in which the densesets D n appear in the presentation of M . If one rearranges the dense sets, then JOEL DAVID HAMKINS, RUSSELL MILLER, AND KAMERYN J. WILLIAMS at a certain stage, one might be led to place a different, incompatible condition onthe sequence, and this will give rise to a different filter. In Sections 7 and 8, weshall examine uniformity in more detail and prove there is no uniform computableprocedure nor even Borel function that always produces the same generic filter G from all isomorphic presentations of the same model M .We also want to remark that it mattered which pairing function we used. Givensets x, y, p where we know p is a Kuratowski ordered pair it requires looking at onlyfinitely many objects to check whether p = ( x, y ). On the other hand, there are∆ definitions for ordered pairs which we cannot computably unravel from just theatomic diagram. Consider the Morse pairing function( x, y ) ⋆ = ( { } × x ) ∪ ( { } × y ) , where the product here is defined via the Kuratowski ordered pair. This definitionis ∆ in the L´evy hierarchy. But to recognize whether p = ( x, y ) ⋆ requires lookingat infinitely many elements of the model when at least one of x and y is infinite,making it not effective (from just the atomic diagram). It follows from Theorem 4that the relation p = ( x, y ) ⋆ is not decidable from the atomic diagram, even if weknow p is a Morse pair.We next turn to the construction of the full forcing extension. Given a genericfilter G , which we have already seen how to construct, we need to determine therest of the sets that will be the forcing extension of M . New sets are given by namesin M , which are then interpreted by the generic. These P -names are recursivelydefined as sets whose elements are pairs ( σ, p ) with σ a P -name and p ∈ P . Thisamounts to a recursive definition on rank, with each P -name having an ordinalrank. Lemma 9.
The property of being a P -name is ∆ , hence decidable uniformly inthe ∆ -diagram of M . Naively, one might attempt to prove that the class of P -names is Σ by, given anelement x , querying the ∆ -diagram to check that all elements of x are pairs whosefirst coordinate is a P -name, which is checked by querying the ∆ -diagram, andrecursively continue this process until it halts. (And similarly to prove the class of P -names is Π .) The trouble with this approach is that this recursive procedure istransfinite; even in the case it is internally finite, M might be ω -nonstandard andthus have internally finite sets which are externally seen to be infinite. Nevertheless,the key of this idea is correct, and it can be made into a proper proof. The pointis that instead of externally carrying out the recursive procedure, we instead lookinside M for a certificate that the recursive procedure was carried out. Proof of Lemma 9.
To see that being a P -name is Σ , given a set x we check whether x is a P -name by searching for a tree witnessing that x satisfies the recursive def-inition of a P -name. Such a tree has x as its root, and the immediate children ofthe root are the first coordinates of elements of x . These nodes then have theirown children by the same process, and this continues downward through the wholetree. This tree is necessarily well-founded (in the sense of M ), because M thinksits membership relation is well-founded. And this tree witnesses that x is a P -name This pairing function has the nice property that if x, y ⊆ V α where α is a limit ordinal, then( x, y ) ⋆ ⊆ V α . ORCING AS A COMPUTATIONAL PROCESS 9 if the second coordinate of each node is in P and no node has an element not repre-sented among its children. It is clear that given a tree T it is ∆ to check whetherit satisfies this property of witnessing that x is a P -name, since this requires onlyquantifying over nodes in the tree and over P . And thus the class of P -names is Σ .To see that it is also Π we carry out a similar procedure, except now we look fora tree witnessing that the recursive definition of being a P -name fails for x . Beingsuch a tree is again ∆ , because failure is witnessed by looking at and below somenode. This then gives that being a P -name is a ∆ property. (cid:3) We would like to remark that this argument generalizes; any property definedby a set-theoretic recursion of ∆ properties will be ∆ .A standard approach to forcing is to define the interpretation of a name σ by G , denoted σ G , to be the set of τ G for ( τ, p ) ∈ σ for some p ∈ G . But thisrecursive interpretation can only be carried out if M is well-founded, limiting whichmodels it can be applied to. Fortunately, there is an alternative construction wecan use, which applies to all models and which matches the recursive interpretationconstruction in case M is well-founded.Every set theorist knows that there are a pair of definable relations, p (cid:13) P σ = τp (cid:13) P σ ∈ τ, which determine membership and equality in the forcing extension. Namely, σ G ∈ τ G if and only if there is p ∈ G so that p (cid:13) P σ ∈ τ , while σ G = τ G if and only ifthere is p ∈ G so that p (cid:13) P σ = τ . We can use this property as the definition of theforcing extension. That is, define the following relations on the class of P -names: σ = G τ ⇐⇒ ∃ p ∈ G p (cid:13) P σ = τσ ∈ G τ ⇐⇒ ∃ p ∈ G p (cid:13) P σ ∈ τ. It is readily checked that the relation = G is an equivalence relation and indeed acongruence relation with respect to the relation ∈ G . The forcing extension M [ G ]can then be presented as the equivalence classes of names [ σ ] G by the relation= G with the membership relation induced by ∈ G . This is essentially the Booleanultrapower manner of constructing the forcing extension, rather than the valuerecursion method (see [HS06] for an account of how these constructions can differ).Let us see that the above described process is indeed effective in the ∆ -diagram. Theorem 10.
There is a uniform computable procedure that, given an oracle forthe ∆ -elementary diagram of a model of set theory h M, ∈ M i | = ZF and a notion offorcing P ∈ M , computes an M -generic filter G ⊆ P and then computes the atomicdiagram of a presentation of the extension M [ G ] . Indeed, with this oracle it candecide the ∆ -elementary diagram of this presentation.Proof. The diagram of the forcing extension M [ G ] will be in the full forcing lan-guage h M [ G ] , ∈ M [ G ] , ˇ M , σ i σ ∈ M P , with a predicate ˇ M for the ground model and constants for all the P -names σ ; wedenote the class of P -names in M by M P . The class M P of P -names was seen in Lemma 9 to be computable from ∆ ( M ). The atomic forcing relations p (cid:13) P σ = τp (cid:13) P σ ∈ τ, have complexity ∆ in the L´evy hierarchy, with P as a parameter, because againthese relations are the result of the solution of a recursion of a ∆ property (see[GHH + ] for explicit discussion of the forcing-relation-as-solution-to-a-recursion per-spective). Similarly the negated atomic forcing relations are also ∆ with P as aparameter. So given the ∆ -diagram and the generic G we can compute = G , andthereby pick out representatives from the equivalence classes and compute a bijec-tion onto ω = dom M [ G ] from the collection of these classes: 0 will denote the classof the < -least P -name, 1 the class of the least P -name not in the class of 0, and soon. Similarly, we can also compute ∈ G , giving us the atomic diagram of M [ G ] (inthe signature with just ∈ ).We delay the argument that we can decide the ∆ -diagram of M [ G ] until aftera discussion of truth in the forcing extension; see the proof of Theorem 11. (cid:3) At last we come to truth in the forcing extension. As a first-order structure,truth in M [ G ] is given by the usual Tarskian recursive construction. A key fact inforcing, however, is that truth in the forcing extension is closely tied to the groundmodel. We have already seen the start of this with the atomic forcing relations p (cid:13) σ ∈ τ and p (cid:13) σ = τ . In general, for any formula ϕ ( x , . . . , x n ) there is adefinable relation p (cid:13) ϕ ( σ , . . . , σ n ) so that M [ G ] | = ϕ (( σ ) G , . . . , ( σ n ) G ) if andonly if there is p ∈ G so that p (cid:13) ϕ ( σ , . . . , σ n ). The map which sends ϕ to theformula p (cid:13) ϕ is effective. And as we explain in the proof, this map does notincrease complexity; if ϕ is ∆ then p (cid:13) ϕ is ∆ and if ϕ is Σ n (respectively Π n )for n ≥ p (cid:13) ϕ is Σ n (respectively Π n ).Using these forcing relations, we can compute the elementary diagram of theextension if we are given the elementary diagram of the ground model. Indeed, thisgoes level by level. Theorem 11.
There is a uniform computable procedure that, given an oracle forthe full elementary diagram of a model of set theory h M, ∈ M i | = ZF and a notionof forcing P ∈ M , decides the full elementary diagram of the presentation of M [ G ] built in Theorem 10. Moreover, this goes level-by-level: given P and an oraclefor the Σ n -elementary diagram of M , for n ≥ , it can decide the Σ n -elementarydiagram of this presentation.Proof. By the process of Theorem 10 we can compute a generic G and the atomicdiagram of M [ G ]. Consider next the forcing relations p (cid:13) P ϕ ( σ , . . . , σ n ), where p ∈ P and ϕ is an assertion in the language of set theory with P -name parameters σ i ∈ M P . For each formula ϕ , the corresponding forcing relation (as a relation on p and the names σ i ) is definable in the ground model M . Furthermore, the proof thatthe forcing relations are definable is uniform, in the sense that from any formula ϕ , we can write down the formula defining the corresponding forcing relation. Soit suffices to prove now that from the Σ n -diagram of the ground model we can We are not claiming that the forcing relations are uniformly definable in M , since indeed as ϕ increases in complexity, the complexity of the definition of p (cid:13) ϕ ( σ ) similarly rises. Rather, weonly claim here that there is a computational procedure that maps any formula ϕ to the formulaforce ϕ ( p, σ ) defining the forcing relation “ p (cid:13) ϕ ( σ )” in M . ORCING AS A COMPUTATIONAL PROCESS 11 compute the Σ n -diagram of the extension. Given the full diagram of the groundmodel, the same process will compute the full diagram of the extension.First, we claim that the forcing relation p (cid:13) ϕ ( σ , . . . , σ n ) for any ∆ -formula ϕ iscomplexity ∆ in the L´evy hierarchy. This can be proved by induction on formulae.The only nontrivial case is the bounded-quantifier case p (cid:13) ∃ x ∈ τ ϕ ( x, σ ), whichby the forcing recursion is equivalent to saying that there is a dense collection ofconditions q ≤ p with some h ρ, r i ∈ τ such that q ≤ r and q (cid:13) ϕ ( ρ, σ ). Thepoint is that all these quantifiers remain bounded, and ∆ is closed under boundedquantification in set theory. This establishes that given the ∆ -diagram of M wecan compute the ∆ -diagram of M [ G ], completing the proof of Theorem 10.We can now prove inductively that the forcing relation p (cid:13) ϕ ( σ ) for Σ n -formulae ϕ has complexity Σ n , for n ≥
1, and similarly the forcing relation on Π n -formulaeis Π n . We can tell if M [ G ] | = ϕ ( σ , . . . , σ n ), for ϕ a Σ n -formula, by looking for acondition p ∈ G such that M satisfies the assertion p (cid:13) P ϕ ( σ , . . . , σ n ). Thus, theΣ n -diagram of the forcing extension M [ G ] can be computed from the Σ n -diagramof M , for n ≥
1, as desired. (cid:3)
We can also prove a version of this theorem for computable infinitary formulae.By definition the Σ c -formulae are the Boolean combinations of atomic formulae—which, since we work in the L´evy signature, means precisely the ∆ -formulae—andthese are also the Π c -formulae. For computable ordinals α , the Σ cα +1 -formulae arethose Σ α +1 -formulae in L ω ,ω with finitely many free variables ~x that are com-putable (countable) disjunctions of formulae ∃ ~y m γ m ( ~x, ~y m ), with each γ m in Π cα .(The length k m of the tuple ~y m = ( y m, , . . . , y m,k m ) of variables may vary over m , but must be computable from m .) The Π cα +1 -formulae are their negations. Forcomputable limit ordinals α , the Σ cα -formulae are computable disjunctions in whicheach individual disjunct is Π cβ for some β < α ; any computable presentation of α may be used in this definition to give the β ’s. Theorem 12.
Fix a computable ordinal α . Then there is a single computablefunction f : ω → ω such that, for every model of set theory h M, ∈ M i | = ZF andevery notion of forcing p = P ∈ M , the function f ( p, · ) : ω → ω is an m -reductionfrom the Σ cα -diagram of the structure M [ G ] produced by the procedure in Theorem10 to the Σ cα -diagram of M itself. (Here p ∈ ω = dom M is the domain element P .)Proof. Knowing that the characteristic function of ∆ ( M [ G ]) is given as Ψ ∆ ( M ) for some Turing functional Ψ, we explain the m -reduction between the Σ c -diagramsclaimed in the theorem. A Σ c -formula _ m ∈ ω ∃ ~y m γ m ([ σ ] G , . . . , [ σ k ] G , ~y m ) , using a computable sequence h γ m i m ∈ ω of ∆ -formulae about a finite tuple from M [ G ], holds in M [ G ] just if there exist m, s ∈ ω , elements [ τ ] G , . . . , [ τ k m ] G in Note that this argument uses the Replacement axiom schema. This is as p (cid:13) ∃ xϕ ( x ) if andonly if there are densely many q ≤ p so that q (cid:13) ϕ ( τ ) for some name τ . It is the Replacementschema that allows us to pull the bounded quantifier over q inside the unbounded quantifiers inthe definition for q (cid:13) ϕ ( τ ) to obtain a Σ n -formula. Recall that in this context we are working in the full forcing language with constants forevery P -name, so it is sensible to ask whether M [ G ] satisfies ϕ ( σ , . . . , σ n ) where the σ i here areconstant symbols referring to ( σ i ) G . M [ G ], and a finite initial segment ρ ⊆ ∆ ( M ) such that Ψ ρ converges within s steps on the G¨odel number of the formula γ m ([ σ ] G , . . . , [ σ k ] G , [ τ ] G , . . . , [ τ k m ] G )and outputs 1, meaning that this ∆ -formula holds in M [ G ]. This constitutes a Σ c statement about M itself, quantifying over the ρ ⊆ ∆ ( M ) which cause the programΨ to halt with value 1 (as well as over m , s , and the P -names). Since we can computean index for this Σ c statement about M from the original formula about M [ G ], wehave an m -reduction from Σ c ( M [ G ]) to Σ c ( M ), as claimed. This same function isan m -reduction from Π c ( M [ G ]) to Π c ( M ), and analogous arguments hold with anylarger computable ordinal α + 1 in place of 1, and also for limit ordinals. (cid:3) The generic multiverse
In the previous section we investigated the computable structure theory of how amodel M of set theory relates to a single forcing extension M [ G ]. We turn now to abroader perspective. Given a model h M, ∈ M i of set theory, the generic multiverse of M is the smallest collection of models of set theory which is closed under extensionby forcing and by grounds, where W is a ground of M if M is a forcing extension of W via a partial order in W . In this section we would like to investigate the extentto which the generic multiverse can be computed from a countable M , extendingthe analysis in Section 4. Let us begin by looking at grounds. Lemma 13.
There is a uniform computable procedure which given an oracle forthe Π -elementary diagram of a model of set theory h M, ∈ M i | = ZFC will computea list of the ∆ -diagrams of the grounds of M .Proof. Because the membership relation of a ground of M is the restriction of themembership relation of M and thus M and its grounds agree on ∆ truth, all weneed to compute is the domains of the grounds. The key fact is the ground modelenumeration theorem [FHR15, Theorem 12], which asserts that the grounds of amodel of ZFC are uniformly definable by a Π -formula. (See also [BHTU16, Sec-tion 2].) That is, there is a Π -formula ϕ ( x, r ) so that for each r either { x : ϕ ( x, r ) } is empty or else it is a ground [FHR15]. So given the Π -elementary diagram of M we can compute whether { x : M | = ϕ ( x, r ) } is nonempty, say by checking whether ϕ ( ∅ , r ) holds. We can thus compute the set { ( n, x ) : M | = ϕ ( x, r n ) } where r n is the n th element r of M , according to the order on ω , so that { x : ϕ ( x, r ) } is nonempty.From this we can get a list of the ∆ -diagrams of the grounds of M . (cid:3) Let us highlight the assumption in the statement of this theorem that M satisfiesthe axiom of choice, an assumption that was missing in the results in Section 4.In [GJ14], Gitman and Johnstone showed for an ordinal δ that DC δ , a version ofthe principle of dependent choice which is weaker than the full axiom of choice,suffices to establish that ground models are definable for a certain class of forcings,namely those with a gap at δ . (See their paper for definitions and details.) Theyconjectured that the ground model definability theorem fails for ZF. This remainsan open problem, but Usuba has recently achieved some partial results [Usu19].We assumed M satisfies the axiom of choice because in this case we do know thatthe grounds are uniformly definable. If Gitman and Johnstone’s conjecture were tobe refuted, then we could improve this theorem to assume the model satisfies onlyZF instead of ZFC. ORCING AS A COMPUTATIONAL PROCESS 13
Corollary 14.
Given an oracle for the full elementary diagram of a model of settheory h M, ∈ M i | = ZFC , there is a computable procedure to compute a list of thefull elementary diagrams of the grounds of M .Proof. This follows using the fact that the translation map on formulae ϕ ϕ W is computable, if W is a definable class, since the translation is merely replacingunbounded quantifiers with quantifiers bounded by W . So from the full elementarydiagram of M we can compute a listing of the elementary diagrams of the grounds W of M . (cid:3) We turn now from the grounds to the full generic multiverse. It is not possible tocompute a listing of all the models in the generic multiverse for the simple reasonthat the generic multiverse is uncountable. Even if we restrict to extensions fromjust a single simple forcing, for instance the forcing to add a Cohen real, therewill still be uncountably many extensions. Nevertheless, the computable genericmultiverse of M , that portion of the multiverse computable from the diagram of M , is close to the full multiverse in a sense we now describe.Usuba’s result that the grounds are strongly downward directed [Usu17] impliesthat every model in the generic multiverse of M is at most two steps away from M , namely it is a forcing extension of a ground of M . While we cannot in generalhope that every model is computable from the elementary diagram of M , we canalways compute a model which is a forcing extension of the same ground by thesame poset. Corollary 15.
Let h ¯ N , ∈ ¯ N i be a model in the generic multiverse of M , where ¯ N = W [ ¯ G ] for a distinguished ground W of M where ¯ G is generic over W for adistinguished poset P . Given an oracle for the full elementary diagram of M thereis a computable procedure to compute a W -generic filter G ⊆ P and decide the fullelementary diagram of N = W [ G ] .Proof. This follows immediately from Corollary 14 and Theorem 11. (cid:3)
On the other hand, there is a different sense in which the computable genericmultiverse of M is far from the full generic multiverse of M . Namely, the com-putable generic multiverse is not dense in the generic multiverse. There are modelsin the generic multiverse so that no extension of them can be computed from thefull diagram for M . Theorem 16.
Let h M, ∈ M i be a countable model of ZF . Then there is M [ G ] aforcing extension of M by the forcing to add a Cohen-generic G ⊆ ω M so thatno outer model of of M [ G ] has a ∆ -diagram computable from the full elementarydiagram of M .Proof. Let us first describe the generic G . Fix any real z , thought of as an ω -lengthbinary sequence, which is not computable from the full elementary diagram of M .From the diagram of M we can compute a list of the dense subsets of Add( ω, M .We use this list to build a generic, in the following manner. Start with p = h z (0) i .Having built p n , first extend to meet the n th dense set, minimizing the length ofthe extension (if there is more than one minimal length extension to the n th denseset then pick arbitrarily). Then put z ( n + 1) on the end to get p n +1 . Because wemet every dense set, G = S n p n is generic. Note that this process works even if M is ω -nonstandard. In this case, we still have a list,whose ordertype is the real ω , of the dense subsets of Add( ω, M . And since for each a ∈ ω M it Assume now that h N, ∈ N i is an outer model of M [ G ], where we think of theuniverse of N as being N with ∈ N being some binary relation on N . Let us see howto compute z from the ∆ -diagram of N and the full diagram for M . Without lossof generality we may assume that the ordinals ≤ ω in M and N are representedby the same natural numbers, as we may compute an isomorphic copy of N withthis property from what we are given. Fix the index of G in N . From the ∆ -diagram of N we can compute z (0), simply by asking what the first bit of G is.From the full diagram of M we know the shortest distance we have to extend past p = h z (0) i to meet the 0th dense set. So we can compute the next coding pointand thereby recover z (1) and p . Continuing this process upward, we can compute z ( n ) for each n . Therefore, if we could compute the ∆ -diagram of N from the fulldiagram of M then we could compute z from the full diagram of M , which wouldbe a contradiction. (cid:3) Next we wish to discuss the extent to which the computable generic multiversehas the same structural properties as the full generic multiverse. Let us start withthe following property, due essentially to Mostowski [Mos76] and recently extendedin [HHK + E of models in the generic multiverse, say that E is amalgamable when there is a model in the generic multiverse which containsevery model in E . Note that if each model in E is a forcing extension of M it isequivalent to ask whether there is a forcing extension of M which contains everymodel in E . Theorem 17 (Mostowski) . Let I be a finite set and let A be a family of subsets of I which contains all singletons and is closed under subsets. Let h M, ∈ M i | = ZF be acountable model of set theory. Then there are reals c i ⊆ ω M for i ∈ I so that each c i is Cohen-generic over M and for A ⊆ I the family { M [ c i ] : i ∈ A } is amalgamableif and only if A ∈ A . The same phenomenon happens within the computable generic multiverse.
Theorem 18.
Let I be a finite set and let A be a family of subsets of I which con-tains all singletons and is closed under subsets. Let h M, ∈ M i | = ZF be a countablemodel of set theory. Then from an oracle for the elementary diagram of M thereis a procedure to compute Cohen reals c i ⊆ ω M generic over M and the elemen-tary diagrams of M [ c i ] for i ∈ I so that for A ⊆ I the family { M [ c i ] : i ∈ A } isamalgamable in the generic multiverse if and only if A ∈ A .Proof. In the language of [HHK + z be a catastrophic real for M ; that is, z is a real so that no outer model of M can contain z . We claim that there is such z computable from the elementary diagram of M . Namely, we can take z to be anisomorphic copy of ∈ M on ω , along with with an isomorphism onto M . Then, bythe Mostowski collapse lemma, any model of ZF which contains z would have tocontain ∈ M itself as a set, which is impossible for an outer model of M .Without loss of generality we may assume that I, A ∈ M . For each A ⊆ I let P A be the forcing Q i ∈ A Add( ω, ∈ M . From the elementary diagram of M wecan compute a listing, in ordertype ω , of all pairs h A, D i with A ∈ A and D ∈ M adense subset of P A . We build the Cohen reals c i by means of a descending sequenceof conditions, which we think of as filling in an ω M × I matrix with 0s and 1s, with is dense in Add( ω, M to have a condition with length ≥ a , the G we produce is unbounded in ω M . ORCING AS A COMPUTATIONAL PROCESS 15 the i th column growing into c i . We will ensure that at each step of the constructionwe have built all columns up to the same height.We start with a completely empty matrix, i.e. with c i = ∅ for all i ∈ I . Nowsuppose we have built up c ni . We are presented with A n ∈ A and D n ⊆ P A dense.Extend the columns with index in A n to collectively meet D n , then pad with 1s toensure the columns all have the same height. Next, pad the remaining columns with0s to build them up to the same height, then extend each column by appendinga row of 1s followed by a row of z ( n )’s. These rows of 1s are the coding pointswhich will be used to recover z ( n ) if we are dealing with A
6∈ A . Note that thisprocess is computable given the ∆ -diagram of M , since from that we can computethe minimal length we need to extend to meet D n and then pick one of the finitelymany extensions of that length. So if we set c i to be the generic determined by h c ni : n ∈ ω i then c i is computable from the diagram of M . And so, once we knowthe c i ’s are generic we know that we can, as before, compute the full diagrams ofthe M [ c i ]’s.It remains only to see that the M [ c i ]’s have the desired amalgamability property.First, suppose that A ∈ A . Then we built up { c i : i ∈ A } so that they met everydense subset of P A in M . So the c i for i ∈ A are mutually generic and so thefamily { M [ c i ] : i ∈ A } is amalgamable, witnessed by M [ c i : i ∈ A ]. In particular,this shows that each c i is generic over M . Now suppose that A ∈ P ( I ) \ A . Then,by the construction, the only rows in which each c i for i ∈ A has value 1 are thecoding points identifying where the bits of z are coded. So no outer model of M which satisfies ZFC can contain each c i for i ∈ A . (cid:3) In [HHK + M there are computable procedures to compute the desiredgenerics for the results from that article. So the properties of the generic multiverseexplored in that article are also enjoyed by the computable generic multiverse.To close out this section, we remark that the existence of many (non-isomorphic)grounds for the same countable model of ZFC implies that in general it is impossibleto recover M effectively—or even non-effectively—from an arbitrary copy of M [ G ]. M has a canonical embedding into M [ G ] by the map sending each x ∈ M to theclass [ˇ x ] G , where the P -name ˇ x is defined by recursion as { h ˇ y, i : y ∈ x } . With a∆ ( M )-oracle, one can compute ˇ x from x , by the methods seen earlier for recursivedefinitions, and thereby compute the canonical embedding of M into M [ G ]. Itsimage will thus be ∆ ( M )-computably enumerable in M [ G ], but it is not defineduniformly across copies of M [ G ].The question of how to recover a copy of M from a copy of M [ G ] is closely tied tothe question of whether ∆ ( M [ G ]) can compute ∆ ( M ), and if so, whether thereis a uniform procedure for doing so from all copies of M [ G ]. The answer is notobvious. Indeed, one can imagine the possibility that the isomorphism type of thestructure M [ G ] may be simpler in some sense than that of M , and that thereforethere may exist a copy of M [ G ] that cannot compute any copy of M . For example,perhaps M [ G ] satisfies GCH, whereas the map κ κ on cardinals in M may havebeen far more chaotic and may have encoded some information not intrinsicallyrecoverable from M [ G ]. There is an analogy here to computable fields. Rabin’s Theorem states that forevery computable field F , the algebraic closure F also has a computable presen-tation, and that both a copy of F and an embedding of F into that copy may becomputed uniformly from the atomic diagram of F , in the signature with + and · .This much is analogous to our results above for a given ( M, P ). However, the uni-formity carries over to countable fields F that are not computably presentable, andin this case the algebraic closure may be far simpler than any presentation of F , asevery countable algebraically closed field has a computable presentation. Takingthe algebraic closure smoothes out a field and eliminates complexity, and we askwhether the same might happen with a forcing extension of a model of set theory. Question 19.
Let M be a countable model of ZFC , and P ∈ M a forcing notion,for which the procedure in Theorem 10 computes a filter G and the atomic diagramof a presentation of M [ G ] . Can there exist a presentation A ∼ = M [ G ] such thatfor every presentation M ∗ ∼ = M , we have ∆ ( M ∗ ) T ∆ ( A ) ? And if so, can thispresentation A be the one computed by our procedure? Class forcing
Elsewhere, we have restricted our attention to forcing notions which are set-sized.We detour in this section to consider proper class-sized forcing notions. There aretwo major approaches to formulate class forcing, and we consider both of them. Thefirst approach, let us call it the first-order approach, is to work over ZF (possiblyassuming more) and deal with a definable class. A generic then has to meet everydefinable dense subclass of the forcing notion.
Theorem 20.
Given an oracle for the full elementary diagram of a countablemodel h M, ∈ M i | = ZF and given a definable pretame class forcing P ⊆ M , there isa computable procedure to compute an M -generic filter G ⊆ P and decide the fullelementary diagram of the forcing extension M [ G ] . See [Fri00, Section 2.2] for a definition of pretameness, which is equivalent tothe preservation of ZF − (in the language with a predicate for the generic filter).The reason to ask P to be pretame is that the pretame forcings are precisely thosewhich have a definable atomic forcing relation [HKS18]. Proof.
From the full diagram of M we can compute a list of the definable densesubclasses of P . So we can compute G as in Theorem 8. Now given the atomicforcing relation for P there is a computable procedure to associate a formula ϕ withthe formula defining the corresponding forcing relation p (cid:13) ϕ . So we can compute M [ G ] and its full elementary diagram as in Theorem 11. (cid:3) This answers the question for the first-order approach. We turn now to theother approach, call it the second-order approach. For this approach classes areactual objects in our models, where we work (in first-order logic) with two-sortedstructures. We work over a second-order set theory, such as G¨odel–Bernays settheory GB or Kelley–Morse set theory KM. We will use italic letters such M to GB is the weaker of the two, stating the existence of classes defined predicatively—quantification is allowed only over sets. On the other hand, KM allows for impredicative com-prehension, defining classes by quantifying over the classes. See [Wil19, Section 2] for preciseaxiomatizations of these two theories, as well as a discussion for their place in the hierarchies ofsecond-order set theories.
ORCING AS A COMPUTATIONAL PROCESS 17 refer to the sets and calligraphic letters such as M to refer to the classes of a modelof second-order set theory. Abusing notation slightly, we will also use M to refer tothe whole model; this is unambiguous, as the sets are definable from the classes. Aclass forcing notion P is then a class in the model and a generic meets every densesubclass of P in the model. We will write M [ G ] for the extension by a generic G ⊆ P . Let ∆ refer to the class of formulae with only set quantifiers. Up to equivalence,this is the same as the class of formulae where all quantifiers are bounded, possiblyby classes, because ∃ x ϕ ( x ) is equivalent to ∃ x ∈ V ϕ ( x ). So this is the second-orderanalogue of ∆ in the first-order L´evy hierarchy. Theorem 21.
Let hM , ∈ M i be a countable model of GB and suppose P ∈ M isa class forcing notion with its atomic forcing relation a class in M . Then, froman oracle for the ∆ -elementary diagram of M there is a computable procedure tocompute an M -generic filter G ⊆ P and the ∆ -diagram of M [ G ] .Proof. Begin by observing that it is ∆ to say that a class is a dense subclass of P .So from the ∆ -diagram for M we can compute a list of all the dense subclasses of P . We can then compute an M -generic filter G ⊆ P as in Theorem 8. Next, notethat being a class P -name is a ∆ property, because a class P -name is a class whoseelements are all pairs of elements of P and a set P -name, and being a set P -nameis a ∆ property. So we can decide which elements of M are P -names. Furtherobserve that the relations p (cid:13) σ ∈ τ, p (cid:13) σ = τ, p (cid:13) σ ∈ T, p (cid:13)
Σ = T are all ∆ . As in the proof of Theorem 11 given the generic G we can define relations= G and ∈ G on the P -names, which are decidable from the ∆ -diagram. And sowe can pick out representatives of the = G -equivalence classes, thereby computingthe atomic diagram of M [ G ]. Finally, observe that for a ∆ -formula ϕ that p (cid:13) ϕ (Σ , . . . , Σ n ) is ∆ , as it is defined from the atomic forcing relation by quantifyingover sets. So we can decide the ∆ -diagram of M [ G ]. (cid:3) As in the set forcing case, from the full elementary diagram of M we can computethe full elementary diagram of the class forcing extension. Theorem 22.
Let hM , ∈ M i be a countable model of GB and suppose P ∈ M is aclass forcing notion with its atomic forcing relation in M . Then, from an oraclefor the second-order elementary diagram of M there is a computable procedure tocompute an M -generic filter G ⊆ P and the second-order elementary diagram of M [ G ] . To clarify, by the second-order elementary diagram of a model of second-order settheory we mean second-order in the sense of allowing quantifying over the classesof the model—that is, in the Henkin semantics—not second-order in some externalsense. Our approach is in first-order logic with two-sorted structures, but as is well-known one canequivalently work in second-order logic with Henkin semantics. In this semantics, one explicitlylists out the classes considered for the second-order part of semantics. Note, however, that to makethis approach amenable to the context of computable structure theory we would have to attachto each model a list of the classes to be used for its second-order semantics. That is, we wouldneed natural numbers to identify each class and include a relation for the set-class membershiprelation. This amounts to the same as the approach in first-order logic.
Proof.
By Theorem 21 we can compute G and the ∆ -diagram of M [ G ]. The resultfor the full diagram then follows from the fact that p (cid:13) ϕ ( T , . . . , T n ) is second-orderdefinable when ϕ is second-order. (cid:3) It is natural to ask whether this goes level-by-level. The Σ n and Π n -formulae areinductively defined from the ∆ -formulae similar to how in first-order set theorythe Σ n and Π n -formulae are defined from the ∆ -formulae. For instance, a formulais Σ if it is of the form ∃ X ∀ Y ϕ ( X, Y ), where both quantifiers are over the classesand ϕ is ∆ . To argue this goes level-by-level as in the set forcing case we wouldneed that if ϕ is Σ n then p (cid:13) ϕ is Σ n . To prove the analogous fact for the setforcing case we used the Replacement schema. The same argument works in theclass forcing case if our model satisfies the Class Collection schema, a second-orderversion of the Replacement schema.
Definition 23.
The
Class Collection axiom schema asserts that if for every setthere is a class satisfying some property, then there is a single class coding the“meta-class” consisting of a witnessing class for every set. Formally, instances ofthis schema take the form ∀ ¯ P (cid:2) ( ∀ x ∃ Y ϕ ( x, Y, ¯ P )) ⇒ ( ∃ C ∀ x ∃ i ϕ ( x, ( C ) i , ¯ P )) (cid:3) , where ( C ) i = { y : ( i, y ) ∈ C } . For n ∈ ω , the Σ n -Class Collection schema is therestriction of Class Collection to Σ n -formulae. It is simple to check that Σ n -Class Collection implies Σ n -Comprehension. Theconverse does not hold. Even KM with the full impredicative Comprehensionschema cannot prove Σ -Class Collection [GHK]. However, adding Class Collectiondoes not increase consistency strength—see [MM75, Theorem 2.5] for the KM/fullClass Collection case and [Rat79] for the level-by-level case. Corollary 24.
Let hM , ∈ M i be a countable model of GB + Σ n -Class Collectionand suppose P ∈ M is a class forcing notion with its atomic forcing relation in M .Then, from an oracle for the Σ n -elementary diagram of M there is a computableprocedure to produce an M -generic filter G ⊆ P and the Σ n -elementary diagram of M [ G ] . (cid:3) Functoriality and Interpretability
In this article we are considering an effective procedure mapping one class ofmodels on the domain ω to another such class (in fact, to the same class). In thissection we recall some known theorems about this scenario and what it says abouteffective interpretability, relating these general facts to our specific case of a model M of set theory inside a forcing extension M [ G ].Theorems from [HTMMM17] and [HTMM18] relate such procedures to the in-terpretability of each output model (in the second class) in the corresponding inputmodel (in the first class). Here we repeat the simplest versions of those theorems.The gist is that interpretations of one structure in another by L ω ,ω -formulae (andwith no fixed arity on the domain of the interpretation) correspond bijectivelyto functors from the category of isomorphic copies of the second structure (with Cf. Footnote 8. Observe that in the presence of Global Choice—the assertion that every class can be well-ordered—we may equivalently ask that this index i for x be x itself. ORCING AS A COMPUTATIONAL PROCESS 19 isomorphisms as the morphisms in the category) into the category of isomorphiccopies of the first. Rather than attempt to define all the terms here, we refer thereader to [HTMMM17, Definition 1.2], [HTMM18, Definition 2.1], and [Mon14,Definition 5.1] for the notions of interpretability, and to [HTMMM17, Definition1.2], [HTMM18, Definition 2.7], and [MPSS18, Definition 3.1] for the notions aboutfunctors.
Theorem 25 (Theorem 1.5 of [HTMMM17]) . Let A and B be countable structures.Then A is effectively interpretable in B if and only if there exists a computablefunctor from the category Iso( B ) of isomorphic copies of B (under isomorphism) tothe corresponding category Iso( A ) . Theorem 26 (Theorem 2.9 of [HTMM18]) . Let B and A be countable structures,possibly in different countable languages. For each Baire-measurable functor F :Iso( B ) → Iso( A ) there is an infinitary interpretation I of A within B , such that F is naturally isomorphic to the functor F I associated to I . Furthermore, if F is ∆ α in the lightface Borel hierarchy, then the interpretation can be taken to usecomputable ∆ α L ω ,ω -formulae and the isomorphism between F and F I can be takento be ∆ α . Those sources also extend these theorems to situations of bi-interpretability be-tween countable structures, and (of greater relevance here) to interpretations thathold uniformly from one class of structures to another, rather than interpretationsmerely of A in B . For us, the point is simply that results about functors and theircomplexity, such as the simple ones we derive in this section, correspond to resultsabout interpretability. For example, a computable functor mapping copies of M tocopies of M [ G ] would correspond to an effective interpretation of M [ G ] in M .A computable functor is a particularly strong kind of operator on countablestructures. Characteristically, attempting to create such an operator requires oneto determine exactly which aspects of the structure are relevant to the operation.The following analysis is not difficult, but it serves as a good example of thisprinciple.We here extend the L´evy signature defined above to a larger signature. Alongwith the symbols from the former, this new signature has countably many constants p , c , and d , d , . . . . The intention (which can be expressed as an L ω ,ω -formula,but not by any finitary axiom) is that p denotes the partial order P used for forcingin this structure, that the constants d j name precisely the dense subsets of P , andthat c names a choice function on the nonempty subsets of P in M . We considermodels of ZFC here, not just ZF, to ensure that M will contain such a choicefunction, although in many cases P will have a readily definable choice function. It isimportant that c be internal to M , and so formally we treat c as a constant symbol,rather than a function symbol. The domain of our functor is the category of modelsof ZFC on the domain ω , in our larger signature, satisfying these conditions. Themorphisms between two such structures are exactly the isomorphisms of structuresin this signature. The range of the functor is the category of models of ZFC onthe domain ω in the L´evy signature, since in the forcing extension, the forcingnotion is no longer relevant. (If desired, one can use the larger signature, with thecheck-name of p M as p M [ G ] and similarly for c and the d j .)The construction of the generic G is now done effectively by a prescribed method,which is more precise than the method of Theorem 10. The added precision is necessary to allow the functor to be computed with no more constants than these.In the new method, the functional Φ is given as an oracle the atomic diagram of M in the larger signature, namely ∆ ( M ) ⊕ h p M , c M , d M , d M , . . . i , and searches firstfor the maximum element p of the forcing notion P named by the constant p . The∆ -diagram identifies it, as it is defined by p ∈ P ∧ ( ∀ q ∈ P ) q ≤ P p . Next, for each s in turn, we find the element d Ms and use c to select an extension p s +1 ≤ p s in d s ∩ P . We can do so because the set x = { q ∈ M d s : q ≤ p s } ∈ M is ∆ -definablefrom parameters we have access to. So we can search for the element of M whichsatisfies this defining property and know that the search must terminate. Then wecan search for c ( x ) ∈ M and choose it to be p s +1 . While we use the < relationon the domain ω of M to search for x and then c ( x ), note that this procedure isnevertheless uniform as the search is looking for a uniquely determined object.If f : M → M ∗ is an isomorphism (in the larger signature), then the sameprocedure Φ on ∆ ( M ∗ , p M ∗ , ~d M ∗ , c M ∗ ) will clearly produce a forcing extension M ∗ [ G ∗ ] isomorphic to M [ G ]. The point of this method is that there is a Turingfunctional Φ ∗ which from the oracle∆ ( M, p M , ~d M , c M ) ⊕ f ⊕ ∆ ( M ∗ , p M ∗ , ~d M ∗ , c M ∗ )computes the corresponding isomorphism from M [ G ] onto M ∗ [ G ∗ ]. This map onisomorphisms will respect composition and, if given the identity isomorphism on M ,will produce the identity isomorphism on M [ G ]. We do not consider it necessaryto write out the proof of this result. This Φ ∗ , together with the functional Φthat computes ∆ ( M [ G ]) = Φ ∆ ( M ) , constitutes a program for the forcing functor,showing that F is in fact a computable functor, according to the definition in[MPSS18].We remark additionally that the forcing functor works exactly the same wayin the category with the same objects as above, but in which the morphisms nowinclude all injective homomorphisms from any one forcing structure to any other.Of course, the image of a morphism g now need only be an injective homomorphismitself, not an isomorphism, but functoriality still holds, effectively.None of the foregoing is at all difficult. With the new constant symbols enu-merating the dense sets we get functoriality but, as will be established in the nextsection, if we omit them we do not get functoriality. Meanwhile, though, we canapply Theorem 25 to our computable functor and extract from it an effective in-terpretation, uniformly across all forcing extensions built by the functor. Corollary 27.
There exist fixed computable infinitary Σ -formulae that, for everyground model ( M, ∆ ( M ) , p, c, d , d , . . . ) of ZFC in the larger signature, give an ef-fective interpretation of the forcing extension M [ G ] in ( M, ∆ ( M ) , p, c, d , d , . . . ) ,provided G is built from this presentation as described in this section. (The inter-pretation allows tuples from M of arbitrary finite arity in its domain.) Non-functoriality
In the main theorem, we proved that there is a computable procedure( M, ∈ M , P ) G that takes as input the atomic diagram of a model of set theory h M, ∈ M , P i witha distinguished notion of forcing P ∈ M , and produces an M -generic filter G ⊆ P .We had observed, however, that the particular filter G arising from this process ORCING AS A COMPUTATIONAL PROCESS 21 can depend on how exactly the atomic diagram of M is presented to us. If werearrange the copy of M as it is represented on the natural numbers (to be used asan oracle for the computation), then this can affect the order in which the densesets d s appear and therefore affect which conditions p s arise in the construction.In short, the computational procedure we provided does not respect isomorphismsof presentations, since different isomorphic presentations of the same model canlead to non-isomorphic generic filters. Thus the process of computing the genericfilter is not functorial in the category of presentations of models of set theory underisomorphism.We claimed in Section 7 that in this signature this phenomenon is unavoidable.To justify that claim, we now show that even if one allows the full elementarydiagram of the model M as input, there is no effective procedure that will pick outthe same generic filter in all presentations of the same model. Theorem 28. If ZF is consistent, then there is no computable procedure that takesas input the elementary diagram of a model of set theory h M, ∈ M , P i with a partialorder P and produces an M -generic filter G ⊆ P , such that isomorphic copies ofthe input model result always in the same corresponding isomorphic copy of G . In other words, there is no computable procedure to produce generic filters that isfunctorial in the category of presentations of models of set theory under isomor-phism.
Proof.
Assume toward contradiction that we have a computable procedureΦ : ∆( M, ∈ M , P ) G, where we assume M = N and h M, ∈ M , P i | = ZF and P ∈ M is a partial order in M , such that G ⊆ P is M -generic and this process is functorial, in the sense thatisomorphic presentations of h M, ∈ M i lead always to the same isomorphic copy of G . We assumed that ZF is consistent. It follows by the L´evy–Montague reflectionprinciple and a simple compactness argument that there is a countable model M | =ZF such that M κ ≺ M for some cardinal κ in M , where M κ means the rank-initialsegment ( V κ ) M . Fix any nontrivial forcing notion P ∈ M κ , such as the forcing toadd a Cohen real.The main idea of the proof is to try to run the computational process inside themodel M . This doesn’t make literal sense, of course, since in M we do not havethe elementary diagram of M as a countable set. Nevertheless, in M we do havethe elementary diagram of M κ , which although uncountable, is a set in M andtherefore has its diagram in M . It may be that M is ω -nonstandard and thereforemay also have nonstandard-length formulae in its version of the diagram, which ofcourse cannot be part of the real elementary diagram. But on the standard-lengthformulae, M will agree with us on the elementary diagram of M κ .Inside M , consider all the ways that we might place finitely much informationabout the elementary diagram of M κ . Specifically, we enumerate finitely many In general, M may be ω -nonstandard. But note that the existence of well-founded M with M κ ≺ M is consistent relative an inaccessible cardinal (indeed, less is necessary). However, thisassumption is higher in consistency strength than just asking for a well-founded model of settheory. elements of M κ and then list off finitely many truth judgments about those elements,and use this as a partial oracle in the computational process of Φ.If outside M we were to consider a full presentation of the elementary diagramof M κ , then the process Φ will result in complete judgments about membership inan M κ -generic filter G ⊆ P , and furthermore these judgments will be independentof the presentation of the diagram we consider. It follows that for any particularcondition p ∈ P , there is a finite piece of the (full, actual) presentation that leadsto a judgment by Φ either that p ∈ G or that p / ∈ G .The key observation is that this finite piece of the full actual diagram of M κ will be an element of M and therefore M will be able to see the judgment that Φmakes on whether p ∈ G or not. Furthermore, all minimal-length such pieces of thediagram of M κ in M that decide whether p ∈ G or not will agree on the outcome,since any such minimal-length piece will involve only standard-finite formulae andtherefore in principle can be continued outside M to a presentation of the full actualdiagram of M κ , which always gives the same result about membership in G . Conse-quently, by searching inside M for these minimal-length supporting computationsabout M κ , we conclude that G ∈ M .But this is impossible, since all subsets of P in M are in M κ , since M κ is a rank-initial segment of M , and so G will actually be M -generic, while also an elementof M , which is impossible for nontrivial forcings. (cid:3) It follows that, given suitable consistency assumptions, there also can be no func-torial computable procedure for producing just the atomic diagram of the forcingextension M [ G ], since there are some forcing notions P , such as the self-encodingforcing defined in [FHR15], that produce unique generic filters in their extensions;for such forcing notions over a well-founded ground model, the extensions are iso-morphic if and only if the generic filters are identical.In the argument for the proof of Theorem 28 we produced a specific modelwhich witnessed the failure of a would-be Turing functor Φ. It is natural to wonderwhether given any Φ and given any countable model M of set theory we can witnessthe failure of Φ with an isomorphic copy of M .The following theorem answers this question in the negative. If we restrict to thepointwise definable models then we can produce generics in a functorial manner. Amodel of set theory is pointwise definable if every element of the model is definablewithout parameters. For example, the Shepherdson–Cohen minimum transitivemodel of ZF is pointwise definable. Theorem 29.
There is a computable functor Φ , in which Φ takes as input theelementary diagram of any pointwise definable model h M, ∈ M i | = ZFC and a forcingnotion P ∈ M and returns an M -generic G ⊆ P and the elementary diagram of M [ G ] . That is, if h M ∗ , ∈ M ∗ i and h M † , ∈ M † i are two isomorphic presentations of M then Φ( M ∗ , ∈ M ∗ , P ∗ ) ∼ = Φ( M † , ∈ M † , P † ) .Proof. The key step in getting functoriality is to ensure that isomorphic presenta-tions give rise to corresponding isomorphic generic filters. That is, if π : M ∗ → M † is an isomorphism we want that π " G ∗ = G † , where G ∗ and G † are the genericsproduced by Φ. Given this it is then straightforward that the isomorphism between M ∗ and M † extends to an isomorphism between M ∗ [ G ∗ ] and M † [ G † ].Let us see how to produce the generic. There is a canonical computable listingof the possible definitions in the language of set theory in ordertype ω . From ORCING AS A COMPUTATIONAL PROCESS 23 the elementary diagram of M we can decide which element is defined by whichdefinition. So using the fact that M is pointwise definable we can produce a listing m , m , . . . of the elements of M , starting with the element defined by the zerothdefinition, then the element defined by the first definition, and so on. This listingis canonical, in the sense that isomorphic copies of M will give rise to the samelisting. More formally: if π : M ∗ → M † is an isomorphism then π ( m ∗ i ) = m † i for all i ∈ ω . We then use this listing to list out the dense subsets of P and to define thegeneric, as in the argument for Theorem 8. Because isomorphic copies of M will usethe same listing of their elements, this process will produce the same generic. (cid:3) We claim that the non-functoriality result extends beyond computability to thecase of Borel processes. Note that in the Borel context, from the atomic diagramof a model we can recover its full elementary diagram by a Borel function, and sothe distinction between those cases evaporates.
Theorem 30.
Suppose ZF is consistent. Then there is no Borel function ( M, ∈ M , P ) G mapping codes for countable h M, ∈ M , P i | = ZF with a forcing notion P ∈ M to an M -generic filter G ⊆ P , such that isomorphic models lead always to thesame (isomorphic) filter. Indeed, we cannot even get such a Borel function sothat if h M ∗ , ∈ M ∗ , P ∗ i and h M † , ∈ M † , P † i are elementarily equivalent then so are h M ∗ [ G ∗ ] , ∈ M ∗ [ G ∗ ] i and h M † [ G † ] , ∈ M † [ G † ] i . We first proved this result with a mild extra consistency assumption. PhilippSchlicht privately communicated to us an argument which needs only the minimumassumption that ZF is consistent. With his permission, it is his argument we presenthere.
Proof (Schlicht).
Suppose that there is a Borel function Φ( M, ∈ M , P ) = G , de-fined from some real parameter y , such that if h M ∗ , ∈ M ∗ , P ∗ i and h M † , ∈ M † , P † i are codes for elementarily equivalent countable models of ZF (coded, say, as a bi-nary relation ∈ M on the natural numbers M = N ) equipped with forcing notions P ∗ ∈ M ∗ and P † ∈ M † , then if G ∗ = Φ( M ∗ , ∈ M ∗ , P ∗ ) and G † = Φ( M † , ∈ M † , P † )then h M ∗ [ G ∗ ] , ∈ M ∗ [ G ∗ ] i and h M † [ G † ] , ∈ M † [ G † ] i are elementarily equivalent. In par-ticular, if h M ∗ , ∈ M ∗ i and h M † , ∈ M † i are isomorphic, then Φ produces elementarilyequivalent forcing extensions.The desired counterexample will be a pair of isomorphic presentations of aCohen-generic extension of a pointwise definable model of ZF + V = L . Observethat the existence of such follows from the assumption that ZF is consistent: Afamous result due to G¨odel gives us a model of ZF + V = L and using that model’sdefinable global well-order we get Skolem functions so that the Skolem hull of theempty set is pointwise definable. Call this pointwise definable model h N, ∈ N i .Work in N . Let Q ∈ N denote the Cohen forcing poset to add a single real(in the sense of N ) and let ρ ∈ N be a Q -name for the set of finite variants ofthe generic real. Consider ˙ P a Q -name so that Q forces that ˙ P is the poset whichchooses an element of ρ by lottery and codes that real into the continuum pattern below ℵ ω , via adding Cohen-generic subsets to the ℵ n . Observe that if the genericreal is modified on a finite domain, then this forcing does not change.In V , let H be the transitive collapse of a countable elementary submodel of somelarge enough H θ which contains y and a code for N . Observe that we can thinkof Q as a forcing poset in H . Of course, N may be ω -nonstandard and so Cohenforcing in the sense of N need not be the real Cohen forcing. Nevertheless, H seesthat Q is a poset, as that is absolute even for nonstandard models of set theory.Now let x be H -generic for Q . Then x must also be N -generic for Q . Set M = N [ x ]and P = ˙ P x ∈ M . Then h M, ∈ M , P i will yield the desired counterexample.Because M is countable in H [ x ], it has a real code. Let µ, ε, π ∈ H be Q -names so that h M ∗ , ∈ M ∗ , P ∗ i = h µ x , ε x , π x i is an isomorphic copy of h M, ∈ M , P i on the natural numbers. Moreover, fix a condition q ∈ x which forces this. Then h M ∗ , ∈ M ∗ , P ∗ i is appropriate as input to Φ. Because Φ is Borel as defined usinga real parameter y ∈ H , we have that H [ x ] can compute Φ( M ∗ , ∈ M ∗ , P ∗ ). More-over, in the forcing extension of h M ∗ , ∈ M ∗ , P ∗ i via the generic output by Φ, call it M ∗ [ G ∗ ], we can check the continuum pattern below ℵ ω to determine which finitevariant of x was used in the lottery sum.Take p in this generic G ∗ which forces that the finite variant chosen to codeinto the continuum pattern is x △ a . Now define x ′ so that x ′ ( i ) = x ( i ) for i =max( a ∪ { | p | , | q | } ) + 1 and x ′ agrees with x on all other coordinates. Then x ′ extends q and is H -generic for Q , and thus also N -generic. In particular, in N [ x ′ ]we have that p forces the continuum pattern in the extension to be x ′ △ a . Andsince x ′ and x differ on a single coordinate, M = N [ x ] = N [ x ′ ] and P = ˙ P x = ˙ P x ′ .Thus, h M † , ∈ M † , P † i = h µ x ′ , ε x ′ , π x ′ i is isomorphic to h M ∗ , ∈ M ∗ , P ∗ i . Note nowthat p ∈ G † = Φ( M † , ∈ M † , P † ) because H [ x ] = H [ x ′ ] and x ′ was defined to agreewith x up to the amount of information needed about µ , ε , and π to have Φ put p in the generic. But the models M ∗ [ G ∗ ] and M † [ G † ] have different continuumpatterns, disagreeing at ℵ i .The statements “GCH holds at ℵ n ” or “GCH fails at ℵ n ” show up in the theoryof a model for standard natural numbers n . But if M ∗ [ G ∗ ] and M † [ G † ] are ω -nonstandard and i is nonstandard this may not be enough. This is where we usethe assumption that the models extend a pointwise definable model of V = L ; itfollows that any nonstandard natural number in these models is definable. Thus,the statements show up in their theory, even if n is nonstandard. Thus they cannotbe elementarily equivalent, giving us the desired counterexample. (cid:3) Meanwhile, if we go to the projective level, then there will (consistently) be afunctorial process. For example, if V = L , then in L given any countable oraclecode for a structure, we can find the L -least isomorphic copy in a projective way,specifically at the level ∆ . If we now build M [ G ] using this copy, it will beconstant on the isomorphism class of M , which is a very strong way of respectingisomorphism. Whether we can push this lower down in the projective hierarchyremains open. Given a collection of forcing notions P i , their lottery sum is the poset which contains the P i as suborders, conditions from different P i being incomparable, and with a new maximum elementplaced above each P i . A generic for the lottery sum then chooses one of the P i by lottery andproduces a generic for it. See [Ham00, Section 3] for a precise definition. ORCING AS A COMPUTATIONAL PROCESS 25
Question 31.
Is there an analytic (or co-analytic) functorial method to producegenerics for models of set theory?
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Professor of Logic, Faculty of Philosophy, University of Ox-ford & Mathematics Institute, University of Oxford & Sir Peter Strawson Fellow,University College, High Street, Oxford OX1 4BH, United Kingdom
E-mail address : [email protected] URL : http://jdh.hamkins.org (Russell Miller) The Graduate Center of CUNY, Ph.D. Programs in Mathematics &Computer Science, 365 Fifth Avenue, New York, NY 10016, USA & Queens College ofCUNY, Mathematics Dept., 65-30 Kissena Blvd., Flushing, NY 11367, USA
E-mail address : [email protected]
URL : http://qcpages.qc.cuny.edu/~rmiller (Kameryn J. Williams) University of Hawai ‘ i at M¯anoa, Department of Mathematics,2565 McCarthy Mall, Keller 401A, Honolulu, HI 96822, USA E-mail address : [email protected] URL ::