aa r X i v : . [ m a t h . L O ] J a n November 9, 2018
SCOTT’S PROBLEM FOR PROPER SCOTT SETS
VICTORIA GITMAN
Abstract.
Some 40 years ago, Dana Scott proved that every countable Scottset is the standard system of a model of PA. Two decades later, Knight andNadel extended his result to Scott sets of size ω . Here, I show that assumingthe Proper Forcing Axiom (PFA), every proper Scott set is the standard systemof a model of PA. I define that a Scott set X is proper if it is arithmeticallyclosed and the quotient Boolean algebra X / Fin is a proper partial order. I alsoinvestigate the question of the existence of proper Scott sets. Introduction
In this paper, I use the Proper Forcing Axiom (PFA) to make partial progresson a half-century-old question in the folklore of models of Peano Arithmetic aboutwhether every Scott set is the standard system of a model of PA. The ProperForcing Axiom is a generalization of Martin’s Axiom that has found application inmany areas of set theory in recent years. I will show that, assuming PFA, everyarithmetically closed Scott set whose quotient Boolean algebra X / Fin is proper isthe standard system of a model of PA.I will begin with some technical and historical details. We can associate toevery model M of PA a certain collection of subsets of the natural numbers calledits standard system, in short SSy( M ). The natural numbers N form the initialsegment of every model of PA. The standard system consists of sets that arise asintersections of the definable (with parameters) sets of the model with the standard part N . One thinks of the standard system as the traces left by the definable setsof the model on the natural numbers. A different way of characterizing sets in thestandard system uses the notion of coding. Let us say that a set A ⊆ N is coded in amodel M if M has an element a such that ( a ) n = 1 if and only if n ∈ A . Here, ( a ) x can refer to any of the reader’s favorite methods of coding with elements of a modelof PA, e.g., by defining ( a ) x as the x th digit in the binary expansion of a . It is easyto see that for nonstandard models we can equivalently define the standard systemto be the collection of all subsets of the natural numbers coded in the model.What features characterize standard systems? Without reference to models ofarithmetic, a standard system is just a particular collection of subsets of the naturalnumbers. Can we come up with a list of elementary (set theoretic, computabilitytheoretic, etc.) properties that X ⊆ P ( N ) must satisfy in order to be the standardsystem of some model of PA? The notion of a Scott set encapsulates three keyfeatures of standard systems.
Definition 1.1. X ⊆ P ( N ) is a Scott set if(1) X is a Boolean algebra of sets. I would like to thank my Ph.D. advisor Joel David Hamkins, Roman Kossak, and Ali Enayatfor their many helpful suggestions. (2) If A ∈ X and B is Turing computable from A , then B ∈ X . (3) If T is an infinite binary tree coded by a set in X , then X has a set codingsome path through T .It is relatively easy to see that every standard system is a Scott set (see [ ? ], p.175). Conversely, Dana Scott proved in 1962 that every countable Scott set is thestandard system of a model of PA [ ? ]. The proof relies on the fact that Scott setsare powerful enough to carry out internally the Henkin construction used to provethe Completeness Theorem. A crucial fact used in the proof is that if a set in theScott set codes a consistent theory (again using the reader’s favorite coding), thenthe Scott set must contain some completion of that theory as well. This followseasily if one is familiar with the relationship between building completions of atheory and branches through binary trees (see [ ? ] p. 177-182 for a modern versionof Scott’s proof). So as the first step toward characterizing standard systems, wehave: Theorem 1.2 (Scott, 1962) . Every countable Scott set is the standard system of amodel of PA . Thus, countable Scott sets are exactly the countable standard systems of modelsof PA. Scott’s theorem leads naturally to the following folklore question:
Scott’s Problem. Is every Scott set the standard system of a model of PA?In 1982, Knight and Nadel settled the question for Scott sets of size ω . Theorem 1.3 (Knight and Nadel, 1982) . Every Scott set of size ω is the standardsystem of a model of PA . [ ? ]It follows that Scott sets of size ω are exactly the standard systems of size ω of models of PA. I will give a proof of Theorem 1.3 shortly. Corollary 1.4. If CH holds, then Scott’s Problem has a positive answer. Very little is known about Scott’s Problem if CH fails, that is, for Scott setsof size larger than ω . I will use the techniques of forcing together with forcingaxioms to find new conditions under which given a Scott set we can build a modelof PA with that Scott set as the standard system. Given a Scott set X , we obtain apartial order X / Fin consisting of the infinite sets in X under the ordering of almostinclusion . Let us call a Scott set X proper if it is arithmetically closed and theposet X / Fin is proper (see Section 2 for the definition of properness and PFA andSection 3 for the significance of arithmetic closure). My main theorem is:
Main Theorem.
Assuming
PFA , every proper Scott set is the standard system ofa model of PA . I will prove the main theorem by first generalizing a theorem known as Ehren-feucht’s Lemma to uncountable models using PFA. Ehrenfeucht’s Lemma is anunpublished result of Ehrenfeucht from around the 1970’s . One can use Ehren-feucht’s Lemma to give a proof of Theorem 1.3. Similarly, my generalized versionis used to prove the main theorem. It is instructive to see a proof of Ehrenfeucht’s Conditions (1) and (2) together imply that if A , . . . A n ∈ X and B is computable from A ⊕ · · · ⊕ A n , then B ∈ X . Roman Kossak, personal communication.
COTT’S PROBLEM FOR PROPER SCOTT SETS 3
Lemma to understand the difficulties involved in extending it to uncountable models(the proof below follows [ ? ]). Theorem 1.5 (Ehrenfeucht’s Lemma) . If M is a countable model of PA whosestandard system is contained in a Scott set X , then for any A ∈ X there is anelementary extension M ≺ N such that A ∈ SSy( N ) ⊆ X .Proof. First, we consider nonstandard M . Let X be a Scott set such that SSy( M ) ⊆ X and let A ∈ X . Choose a countable Scott set Y ⊆ X containing SSy( M ) and A .Using the truth predicate for Σ n -formulas, we can prove that the Σ n -theory of M is coded by a set in SSy( M ) for every n . Moreover, all computable theories are inSSy( M ) since SSy( M ) is a Scott set. Therefore, the theory T :=“PA + Σ -theoryof M ” is in SSy( M ). The idea that the Henkin construction can be carried outinside a Scott gives more than just Theorem 1.2. From this it follows that for anytheory T ⊇ PA such that T ∩ Σ n ∈ X for every n ∈ N , there exists a model of T with that Scott set as the standard system. In particular, we get a model M ∗ of T with SSy( M ∗ ) = Y . By Friedman’s Embedding Theorem (see [ ? ], p. 161), since M ∗ | = “Σ -theory of M ” and SSy( M ) ⊆ Y , we have M ≺ ∆ M ∗ . Close M underinitial segment and call the resulting submodel N . Then M ≺ N since it is cofinaland ∆ -elementary (by Gaifman’s Embedding Theorem, see [ ? ], p. 87). But alsoSSy( N ) = SSy( M ∗ ) = Y as required since N is an initial segment of M ∗ . Thiscompletes the proof for nonstandard models. Let TA := { ϕ | N | = ϕ } denote TrueArithmetic . It is clear that N ≺ N if and only if N | = TA. The standard system of N is the collection of all arithmetic sets. So suppose that X is a Scott set containingall arithmetic sets and fix A ∈ X . It follows that TA ∩ Σ n is in X for every n ∈ N .Let Y ⊆ X be a countable Scott set containing A and TA ∩ Σ n for every n ∈ N . Bythe remark above, there exists a model N | = TA whose standard system is exactly Y . Thus, N ≺ N and A ∈ SSy( N ) ⊆ X . (cid:3) We are now ready to prove Knight and Nadel’s result. Proof of Theorem 1.3.
Let X be a Scott set of size ω and enumerate X = { A ξ | ξ < ω } . The idea is to build up a model with the Scott set X as the standardsystem in ω steps by successively throwing in one more set at each step and usingEhrenfeucht’s Lemma to stay within X . More precisely, we will define an elementarychain M ≺ M ≺ · · · ≺ M ξ ≺ · · · of length ω of countable models of PA such thatSSy( M ξ ) ⊆ X and A ξ ∈ SSy( M ξ +1 ). Then clearly M = ∪ ξ<ω M ξ will work. Let M be any countable model of PA with SSy( M ) ⊆ X . Such M exists by Scott’stheorem (1.2). Given M ξ , by Ehrenfeucht’s Lemma, there exists M ξ +1 such that M ξ ≺ M ξ +1 , the set A ξ ∈ SSy( M ξ +1 ), and SSy( M ξ +1 ) ⊆ X . At limit stages takeunions. (cid:3) The key ideas in the proof of Theorem 1.3 can be summarized in the followingdefinition and theorem:
Definition 1.6 (The κ -Ehrenfeucht Principle for Γ) . Let κ be a cardinal and Γsome collection of Scott sets. The κ -Ehrenfeucht Principle for Γ states that if M is amodel of PA of size less than κ and X is a Scott set in Γ such that SSy( M ) ⊆ X , thenfor any A ∈ X there is an elementary extension M ≺ N such that A ∈ SSy( N ) ⊆ X . This is not Knight and Nadel’s original proof.
VICTORIA GITMAN
If Γ is the collection of all Scott sets, we will say simply that the κ -EhrenfeuchtPrinciple holds.In view of Definition 1.6, Ehrenfeucht’s Lemma (Theorem 1.5) is the ω -Ehrenfeucht Principle. We can freely assume that the elementary extension N given by the κ -Ehrenfeucht Principle has size less than κ since if this is not thecase, we can always take an elementary submodel N ′ of N such that M ≺ N ′ and A ∈ SSy( N ′ ). A completely straightforward generalization of the proof of Theorem1.3 gives: Theorem 1.7.
If the κ -Ehrenfeucht Principle for Γ holds, then every Scott set in Γ of size κ is the standard system of a model of PA . Thus, one approach to solving Scott’s Problem would be to try to prove the ω -Ehrenfeucht Principle for some collection of Scott sets. However, proofs ofEhrenfeucht’s Lemma hinge precisely on those techniques in the field of modelsof PA that appear to work only with countable models. As an example, Friedman’sfamous Embedding Theorem does not generalize to uncountable models. In whatfollows, I will mainly investigate the Ehrenfeucht principles. The results on Scott’sProblem will follow as a corollary. Under PFA, I will show that the ω -EhrenfeuchtPrinciple for proper Scott sets holds (Theorem 3.4).2. Set Theory and Scott’s Problem
Since the result of Knight and Nadel (Theorem 1.3), very little progress hadbeen made on Scott’s Problem until some recent work of Fredrik Engstr¨om [ ? ]. Itis not difficult to believe that Scott’s Problem past ω might have a set theoreticresolution. Engstr¨om followed a strategy, suggested more than a decade earlierby Joel Hamkins and others, to use forcing axioms to gain new insight into theproblem. We saw that a positive answer to Scott’s Problem follows from CH. It isa standard practice in set theoretic proofs that if a statement follows from CH, wetry to prove it or its negation from ¬ CH + Martin’s Axiom. Martin’s Axiom (MA)is a forcing axiom which asserts that for every c.c.c. poset P and every collection D of less than the continuum many dense subsets of P , there is a filter on P thatmeets all of them. Such filters are often called partially generic filters . Engstr¨omtried to use Martin’s Axiom to find new techniques for building models of PA whosestandard system is a given Scott set.Given a Scott set X , Engstr¨om chose the poset X / Fin, whose elements are infinitesets in X ordered by almost inclusion . That is, for infinite A and B in X , we saythat A ≤ B if and only if A ⊆ Fin B . Observe that X / Fin is forcing equivalent toforcing with the Boolean algebra X modulo the ideal of finite sets. A familiar andthoroughly studied instance of this poset is P ( N ) / Fin. A Scott set is arithmeticallyclosed if whenever A is in it and B is arithmetically definable from A , then B is alsoin it (for a more extensive discussion, see Section 3). For a property of posets P , if X is an arithmetically closed Scott set and X / Fin has P , I will simply say that X has property P . An important point to be noted here is that whenever a Scott set X is discussed as a poset, I will always be assuming that it is arithmetically closed .The significance of arithmetic closure will become apparent in Section 3. Here ω -like models are an obvious counterexample. COTT’S PROBLEM FOR PROPER SCOTT SETS 5
Theorem 2.1 (Engstr¨om, 2004) . Assuming Martin’s Axiom, every c.c.c. Scott setof size less than the continuum is the standard system of a model of PA . [ ? ]To obtain models for Scott sets for which we could not do so before, Engstr¨omneeded that there are uncountable Scott sets that are c.c.c.. Unfortunately: Theorem 2.2.
Every c.c.c. Scott set is countable.Proof.
Let X be a Scott set. If x is a finite subset of N , let p x q denote the code of x using G¨odel’s coding. For every A ∈ X , define an associated A ′ = { p A ∩ n q | n ∈ N } .Clearly A ′ is computable from A , and hence in X . Observe that if A = B , then | A ′ ∩ B ′ | < ω . Hence if A = B , we get that A ′ and B ′ are incompatible in X / Fin.It follows that A = { A ′ | A ∈ X } is an antichain of X / Fin of size | X | . This showsthat X / Fin always has antichains as large as the whole poset. (cid:3)
Thus, the poset X / Fin has the worst possible chain condition, namely | X | + -c.c..Theorem 2.2 implies that no new instances of Scott’s Problem can be obtained fromTheorem 2.1.I will borrow from Engstr¨om’s work the poset X / Fin. But my strategy will bedifferent in two respects. First, instead of MA, I will use the poset together withthe forcing axiom PFA, allowing me to get around the obstacle of Theorem 2.2.In Section 7, I will argue that, unlike the case with c.c.c. Scott sets, uncountableproper Scott sets do exist. However, I will not be able to explicitly obtain any newinstances of Scott’s Problem. Second, my main aim will be to obtain an extension ofEhrenfeucht’s Lemma to uncountable models, while Engstr¨om’s was to directly geta model whose standard system is a given Scott set. This approach will allow me tohandle Scott sets of size continuum, which had not been possible with Engstr¨om’stechniques.Recall that for a cardinal λ , the set H λ is the collection of all sets whose transitiveclosure has size less than λ . Let P be a poset and λ be a cardinal greater than 2 | P | .Since we can always take an isomorphic copy of P on the cardinal | P | , we can assumewithout loss of generality that P and P ( P ) are elements of H λ . In particular, wewant to ensure that all dense subsets of P are in H λ . Let M be a countableelementary submodel of H λ containing P as an element. If G is a filter on P , we saythat G is M - generic if for every maximal antichain A ∈ M of P , the intersection G ∩ A ∩ M = ∅ . It must be explicitly specified what M -generic means in this contextsince the usual notion of generic filters makes sense only for transitive structuresand M is not necessarily transitive. This definition of M -generic is closely relatedto the definition for transitive structures. To see this, let M ∗ be the Mostowskicollapse of M and P ∗ be the image of P under the collapse. Let G ∗ ⊆ P ∗ be thepointwise image of G ∩ M under the collapse. Then G is M -generic if and only if G ∗ is M ∗ -generic for P ∗ in the usual sense. Definition 2.3.
Let P ∈ H λ be a poset and M be an elementary submodel of H λ containing P . Then a condition q ∈ P is M - generic if and only if every V -genericfilter G ⊆ P containing q is M -generic. Definition 2.4.
A poset P is proper if for every λ > | P | and every countable M ≺ H λ containing P , for every p ∈ P ∩ M , there is an M -generic condition below p . VICTORIA GITMAN
It can be shown that it is actually equivalent to consider only some fixed λ > | P | and to show that generic conditions exist only for a club of countable M ≺ H λ [ ? ](p. 102). Definition 2.5.
The Proper Forcing Axiom (PFA) is the assertion that for everyproper poset P and every collection D of at most ω many dense subsets of P , thereis a filter on P that meets all of them.Proper forcing was invented by Shelah, who sought a class of ω -preservingforcing notions that would be preserved under countable support iterations (for anintroduction to proper forcing see [ ? ] (p. 601) or [ ? ]). The two familiar classesof ω -preserving forcing notions, namely the c.c.c. and countably closed forcingnotions, turn out to be proper as well. The Proper Forcing Axiom, introducedby Baumgartner [ ? ], is easily seen to be a generalization of Martin’s Axiom sincec.c.c. posets are proper and PFA decides the size of the continuum is ω . The laterfact is a highly nontrivial result in [ ? ]. In many respects, however, PFA is verymuch unlike MA. Not only does it decide the size of the continuum, the axiom alsohas large cardinal strength. The best known large cardinal upper bound on theconsistency of PFA is a supercompact cardinal [ ? ]. Much fruitful set theoreticalwork in recent years has involved PFA and its consequences.3. Proof of the Main Theorem
I will use PFA to prove the ω -Ehrenfeucht Principle for proper Scott sets . Themain theorem will follow as a corollary.A filter G on the poset X / Fin is easily seen to be a filter on the Boolean algebra X . By extending G to a larger filter if necessary, we can assume without loss ofgenerality that G is an ultrafilter. Recall that to prove the ω -Ehrenfeucht Principle,given a model M of size ω and a Scott set X such SSy(M) ⊆ X , we need to find forevery A ∈ X , an elementary extension N such that A ∈ SSy( N ) ⊆ X . The strategywill be to find ω many dense subsets of X / Fin such that if G is a partially genericultrafilter meeting all of them, then the standard system of the ultrapower of M by G will stay within X . Thus, if X is proper, we will be able to use PFA to obtainsuch an ultrafilter. I will also show that to every A ∈ X , there corresponds a set B ∈ X / Fin such that whenever B is in an ultrafilter G , the set A will end up in theultrapower of M by G .Let S ⊆ P ( N ) and expand the language of arithmetic L A to include unarypredicates for all A ∈ S . Then the structure N S = h N , A i A ∈S is a structure ofthis expanded language with the natural interpretation. Since Scott sets are closedunder relative computability, basic computability theory arguments show that if X is a Scott set, the structure N X = h N , A i A ∈ X is closed under ∆ -definability. Thatis, if B is ∆ -definable in N X , then B ∈ X . Definition 3.1.
A collection
S ⊆ P ( N ) is arithmetically closed if the structure N S = h N , A i A ∈S is closed under definability. That is, if B is definable in N S , then B ∈ S .A Scott set X is arithmetically closed simply when it satisfies Definition 3.1.Observe actually that if S is arithmetically closed, then it is a Scott set. Thus,arithmetic closure subsumes the definition of a Scott set. An easy induction onthe complexity of formulas establishes that if X is a Boolean algebra of sets and COTT’S PROBLEM FOR PROPER SCOTT SETS 7 N X = h N , A i A ∈ X is closed under Σ -definability, then X is arithmetically closed.Hence a Scott set is arithmetically closed if and only if it is closed under the Turingjump operation. Definition 3.2.
Say that h B n | n ∈ N i is coded in X if there is B ∈ X suchthat B n = { m ∈ N | h n, m i ∈ B } . Given h B n | n ∈ N i coded in X and C ∈ X / Fin, say that C decides h B n | n ∈ N i if whenever U is an ultrafilteron X and C ∈ U , then { n ∈ N | B n ∈ U } ∈ X . Call a Scott set X decisive if for every h B n | n ∈ N i coded in X , the set D = { C ∈ X / Fin | C decides h B n | n ∈ N i} is dense in X / Fin.Decisiveness is precisely the property of a Scott set which is required for ourproof of the main theorem. I will show below that decisiveness is equivalent toarithmetic closure.
Lemma 3.3.
The following are equivalent for a Scott set X : (1) X is arithmetically closed. (2) X is decisive. (3) For every sequence h B n | n ∈ N i coded in X , there is C ∈ X / Fin deciding h B n | n ∈ N i .Proof. (1)= ⇒ (2): Assume that X is arithmetically closed. Fix A ∈ X / Fin and a sequence h B n | n ∈ N i coded in X . We need to show that there is an element in X / Finbelow A deciding h B n | n ∈ N i . For every finite binary sequence s , we will define B s by induction on the length of s . Let B ∅ = A . Given B s , where s has length n ,define B s = B s ∩ B n and B s = B s ∩ ( N − B n ). Define the binary tree T = { s ∈ <ω | B s is infinite } . Clearly T is infinite since if we split an infinite set into twopieces one of them must still be infinite. Since X is arithmetically closed and T isarithmetic in A and h B n | n ∈ N i , it follows that T ∈ X . Thus, X contains a cofinalbranch P through T . Define C = { b n | n ∈ N } such that b is least element of B ∅ and b n +1 is least element of B P ↾ n that is greater than b n . Clearly C is infinite and C ⊆ A . Now suppose U is an ultrafilter on X and C ∈ U , then B n ∈ U if and onlyif C ⊆ Fin B n . Thus, { n ∈ N | B n ∈ U } = { n ∈ N | C ⊆ Fin B n } ∈ X since X isarithmetically closed.(2)= ⇒ (3): Clear.(3)= ⇒ (1) : It suffices to show that X is closed under the Turing jump operation.Fix A ∈ X and define the sequence h B n | n ∈ N i by k ∈ B n if and only if theTuring program coded by n with oracle A halts on input n in less than k manysteps. Clearly the sequence is computable from A , and hence coded in X . Let H = { n ∈ N | the program coded by n with oracle A halts on input n } be thehalting problem for A . It should be clear that n ∈ H implies that B n is cofiniteand n / ∈ H implies that B n = ∅ . Let C ∈ X / Fin deciding h B n | n ∈ N i and U be any ultrafilter containing C , then { n ∈ N | B n ∈ U } ∈ X . But this set isexactly H . This shows that H ∈ X , and hence X is closed under the Turing jumpoperation. (cid:3) Theorem 3.4.
Assuming
PFA , the ω -Ehrenfeucht Principle for proper Scott setsholds. That is, if X is a proper Scott set and M is a model of PA of size ω whose Similar arguments have appeared in [ ? ] and other places. I am grateful to Joel Hamkins for pointing out this argument.
VICTORIA GITMAN standard system is contained in X , then for any A ∈ X , there is an elementaryextension M ≺ N such that A ∈ SSy( N ) ⊆ X .Proof. I will build N using a variation on the ultrapower construction introduced byKirby and Paris [ ? ]. Fix a model M of PA and a Scott set X such that SSy( M ) ⊆ X .Let G be some ultrafilter on X . If f : N → M , we say that f is coded in M whenthere is a ∈ M such that ( a ) n = f ( n ) for all n ∈ N . Given f and g coded in M , define f ∼ G g if { n ∈ N | f ( n ) = g ( n ) } ∈ G . The definition makes sensesince clearly { n ∈ N | f ( n ) = g ( n ) } ∈ SSy( M ) ⊆ X . The classical ultrapowerconstruction uses an ultrafilter on P ( N ) and all functions from N to M . Thisconstruction uses only functions coded in M , and therefore needs only an ultrafilteron SSy( M ) ⊆ X . As in the classical construction, we get an equivalence relationand a well-defined L A structure on the equivalence classes. The proof relies on thefact that X is a Boolean algebra. Call Π X M/G the collection of equivalence classes[ f ] G where f is coded in M . Also, as usual, we get: Lemma 3.4.1.
Lo´s Lemma holds. That is, Π X M/G | = ϕ ([ f ] G ) if and only if { n ∈ N | M | = ϕ ( f ( n )) } ∈ G .Proof. Similar to the classical proof of the Lo´s Lemma. (cid:3)
Lemma 3.4.2.
For every A ∈ X , there is B ∈ X / Fin such that if G is any ultrafilteron X containing B , then A ∈ SSy(Π X M/G ) .Proof. Let χ A be the characteristic function of A . For every n ∈ N , define B n = { m ∈ N | ( m ) n = χ A ( n ) } . Then clearly each B n ∈ X and h B n | n ∈ N i is coded in X since the sequence is arithmetic in A . Observe that the intersection of any finitenumber of B n is infinite. Let B = { b n | n ∈ N } where b is least element of B and b n +1 is least element of ∩ m ≤ n +1 B m that is greater than b n . Then B ⊆ Fin B n for all n ∈ N and B ∈ X since it is arithmetic in h B n : n ∈ N i . It follows that if G is any ultrafilter containing B , then G must contain all the B n as well. Let G be an ultrafilter containing B . Let id : N → N be the identity function. I claimthat ([ id ] G ) n = χ A ( n ). It will follow that A ∈ SSy(Π X M/G ). But this is true since([ id ] G ) n = χ A ( n ) if and only if { m ∈ N | ( m ) n = χ A ( n ) } = B n ∈ G . (cid:3) Lemma 3.4.2 tells us that if we want to add some set A to the standard systemof the ultrapower that we are building, we just have to make sure that a correctset gets put into the ultrafilter. It follows that that we can build ultrapowers of M having any given element of X in the standard system.The crucial step of the construction is to find a family of size ω of dense subsetsof X / Fin such that if the ultrafilter meets all members of the family, the stan-dard system of the ultrapower stays within X . It is in this step that we need the decisiveness of X .Recall that a set E is in the standard system of a nonstandard model if and onlyif there is an element e such that E = { n ∈ N | ( e ) n = 1 } , meaning E is coded inthe model. Thus, we have to show that the sets coded by elements of Π X M/G arein X . Lemma 3.4.3.
For every function f : N → M coded in M , there is a dense subset D f of X / Fin such that if G meets D f , then [ f ] G ∈ Π X M/G codes a set in X .Proof. Fix a function f : N → M coded in M and let E f = { n ∈ N | Π X M/G | =([ f ] G ) n = 1 } . By Lo´s Lemma, Π X M/G | = ([ f ] G ) n = 1 if and only if COTT’S PROBLEM FOR PROPER SCOTT SETS 9 { m ∈ N | ( f ( m )) n = 1 } ∈ G . Define B n,f = { m ∈ N | ( f ( m )) n = 1 } andnote that h B n,f | n ∈ N i is coded in SSy( M ). Observe that n ∈ E f if and onlyif B n,f ∈ G . Thus, we have to make sure that { n ∈ N | B n,f ∈ G } ∈ X . Let D f = { C ∈ X / Fin | C decides h B n,f | n ∈ N i} . Since X is decisive, D f is dense.Clearly if G meets D f , the set coded by [ f ] G will be in X . (cid:3) Now we can finish the proof of Theorem 3.4. Let D = { D f | f : N → M iscoded in M } . Since M has size ω , the collection D has size ω also. AssumingPFA guarantees that we can find an ultrafilter G meeting every D f ∈ D . But thisis precisely what forces the standard system of Π X M/G to stay inside X . (cid:3) The main theorem now follows directly from Theorem 3.4.
Proof of Main Theorem.
Since PFA implies 2 ω = ω and Scott sets of size ω arealready handled by Knight and Nadel’s result, we only need to consider Scott sets ofsize ω . But the result for these follows from Theorem 1.7 and the ω -EhrenfeuchtPrinciple established by Theorem 3.4. (cid:3) Extensions of Ehrenfeucht’s Lemma
Below, I will go through some results related to the question of extending Ehren-feucht’s Lemma to models of size ω ( ω -Ehrenfeucht Principle).Theorem 3.4 shows that in a universe satisfying PFA, the ω -Ehrenfeucht Prin-ciple for proper Scott sets holds. Next, I will use the same techniques to show thatthe κ -Ehrenfeucht Principle for arithmetically closed Scott sets holds for all κ if weonly consider models with countable standard systems. For this argument, we donot need to use PFA or properness. Theorem 4.1. If M is a model of PA whose standard system is countable andcontained in an arithmetically closed Scott set X , then for any A ∈ X , there is anelementary extension M ≺ N such that A ∈ SSy( N ) ⊆ X .Proof. Fix an arithmetically closed Scott set X and a model M of PA such thatSSy( M ) is countable and contained in X . To mimic the proof of Theorem 3.4, weneed to find an ultrafilter G on X which meets the dense sets D f = { C ∈ X / Fin | C decides h B n,f | n ∈ N i} . I claim that there are only countably many D f . If thisis the case, then such an ultrafilter exists without any forcing axiom assumption.Given f : N → M , let B f code h B n,f | n ∈ N i . There are possibly as many f aselements of M , but there can be only countably many B f since each B f ∈ SSy( M ).It remains only to observe that D f is determined by B f . So there are as many D f as there are different B f . Thus, there are only countably many D f in spite of thefact that M can be arbitrarily large. (cid:3) The same idea can be used to extend Theorem 3.4 to show that the κ -EhrenfeuchtPrinciple for proper Scott sets holds for all κ if we consider only models whosestandard system has size ω . Theorem 4.2.
Assuming
PFA , if X is a proper Scott set and M is a model of PA whose standard system has size ω and is contained in X , then for any A ∈ X , thereis an elementary extension M ≺ N such that A ∈ SSy( N ) ⊆ X . It is also an easy consequence of an amalgamation result for models of PA thatthe κ -Ehrenfeucht Principle holds for all κ for models with a countable nonstandard elementary initial segment . Neither PFA nor arithmetic closure is required for thisresult. Theorem 4.3.
Suppose M , M , and M are models of PA such that M ≺ cof M and M ≺ end M . Then there is an amalgamation M of M and M over M suchthat M ≺ end M and M ≺ cof M . (See [ ? ], p. 40) Theorem 4.4.
Suppose M is a model of PA with a countable nonstandard elemen-tary initial segment and X is a Scott set such that SSy( M ) ⊆ X . Then for any A ∈ X , there is an elementary extension M ≺ N such that A ∈ SSy( N ) ⊆ X .Proof. Fix a set A ∈ X . Let K be a countable nonstandard elementary initialsegment of M , then SSy( K ) = SSy( M ). By Ehrenfeucht’s Lemma (Theorem 1.5),there is an extension K ≺ cof K ′ such that A ∈ SSy( K ′ ) ⊆ X . By Theorem 4.4,there is a model N , an amalgamation of K ′ and M over K , such that K ′ ≺ end N and M ≺ cof N . It follows that SSy( K ′ ) = SSy( N ). Thus, A ∈ SSy( N ) ⊆ X . (cid:3) Corollary 4.5.
The κ -Ehrenfeucht Principle holds for ω -like models for all car-dinals κ . These observations suggest that if the ω -Ehrenfeucht Principle fails to hold, oneshould look to models with an uncountable standard system for such a counterex-ample. 5. Other Applications of X / FinIt appears that X / Fin is a natural poset to use in several unresolved questions inthe field of models of PA. In the previous sections, I used it to find new conditionsfor extending Ehrenfeucht’s Lemma and Scott’s Problem. Here, I will mention someother instances in which the poset naturally arises.
Definition 5.1.
Let L be some language extending the language of arithmetic L A .We say that a model M of L satisfies PA ∗ if M satisfies induction axioms in theexpanded language. If M | = PA ∗ , then M ⊆ N is a conservative extension if it isa proper extension and every parametrically definable subset of N when restrictedto M is also definable in M .Gaifman showed in [ ? ] that for any countable language L , every M | = PA ∗ in L has a conservative elementary extension. A result of George Mills shows thatthe statement fails for uncountable languages. Mills proved that every countablenonstandard model M | = PA ∗ in a countable language has an expansion to anuncountable language such that M | = PA ∗ in the expanded language, but has noconservative elementary extension (see [ ? ], p. 168). His techniques failed for thestandard model, leaving open the question whether there is an expansion of thestandard model N to some uncountable language that does not have a conservativeelementary extension. This question has recently been answered by Ali Enayat,who demonstrated that there is always an uncountable arithmetically closed Scottset X such that h N , A i A ∈ X has no conservative elementary extension [ ? ]. This raisesthe question of whether we can say something general about Scott sets X for which h N , A i A ∈ X has a conservative elementary extension. Theorem 5.2.
Assuming
PFA , if X is a proper Scott set of size ω , then h N , A i A ∈ X has a conservative elementary extension. COTT’S PROBLEM FOR PROPER SCOTT SETS 11
Proof.
Let L X be the language of arithmetic L A together with unary predicatesfor sets in X . Let G be an ultrafilter on X . We define Π X N /G , the ultrapowerof N by G , to consist of equivalence classes of functions coded in X . We have tomake this modification to the construction of the proof of Theorem 3.4 since theidea of functions coded in the model clearly does not make sense for N . The usualarguments show that we can impose an L X structure on Π X N /G and Lo´s Lemmaholds. I will show, by choosing G carefully, that h Π X N /G, A ′ i A ∈ X is a conservativeextension of h N , A i A ∈ X where A ′ = { [ f ] G ∈ Π X N /G | { n ∈ N | f ( n ) ∈ A } ∈ G } .Fix a set E definable in h Π X N /G, A ′ i A ∈ X by a formula ϕ ( x, [ f ] G ). Observe that n ∈ E ↔ Π X N /G | = ϕ ( n, [ f ] G ) ↔ B ϕ,fn = { m ∈ N | N | = ϕ ( n, f ( m )) } ∈ G . Let D ϕ,f = { C ∈ X / Fin | C decides h B ϕ,fn | n ∈ N i} . The sets D ϕ,f are dense since X isdecisive. Clearly if G meets all the D ϕ,f , the ultrapower h Π X N /G, A ′ i A ∈ X will be aconservative extension of h N , A i A ∈ X . Finally, since X has size ω , there are at most ω many formulas ϕ of L X and functions f coded in X , and hence at most ω manydense sets D ϕ,f . So we can find the desired G by PFA. (cid:3) Another open question in the field of models of PA, for which X / Fin is relevant,involves the existence of minimal cofinal extensions for uncountable models.
Definition 5.3.
Let M be a model of PA, then M ≺ N is a minimal extension ifit is a proper extension and whenever M ≺ K ≺ N , either K = M or K = N . Theorem 5.4.
Every nonstandard countable model of PA has a minimal cofinalextension. (See [ ? ], p. 28)Gaifman showed that every model of PA, regardless of cardinality, has a minimalend extension [ ? ]. Definition 5.5.
Let X ⊆ P ( N ) be a Boolean algebra. If U is an ultrafilter on X ,we say that U is Ramsey if for every function f : N → N coded in X , there is a set A ∈ U such that f is either 1-1 or constant on A . Lemma 5.6. If M is a nonstandard model of PA such that SSy( M ) has a Ramseyultrafilter, then M has a minimal cofinal extension. Proof.
Let U be a Ramsey ultrafilter on SSy( M ). The strategy will be to show thatthe ultrapower Π SSy( M ) M/U is a minimal cofinal extension of M . The meaningof Π SSy( M ) M/U here is identical to the one in the proof of Theorem 3.4. First,observe that for any ultrafilter U, we have Π X M/U = Scl ( M ∪ { [ id ] U } ), the Skolemclosure of the equivalence class of the identity function together with elements of M . This holds since any [ f ] U = t ([ id ] U ) where t is the Skolem term defined by f in M . Next, observe that such ultrapowers are always cofinal. To see this, fix[ f ] U ∈ Π X M/U and let a > f ( n ) for all n ∈ N . Such a exists since f is codedin M . Clearly [ f ] U < [ c a ] U where c a ( n ) = a for all n ∈ N . These observationshold for any Scott set X ⊇ SSy( M ) and, in particular, for X = SSy( M ). To showthat the extension Π SSy( M ) M/U is minimal, we fix M ≺ K ≺ Π X M/U and showthat K = M or K = Π X M/U . It suffices to see that [ id ] U ∈ Scl ( M ∪ { [ f ] U } ) forevery [ f ] U ∈ (Π SSy( M ) M/U ) − M . Fix f : N → M and define g : N → N such that g (0) = 0 and g ( n ) = n if f ( n ) is not equal to f ( m ) for any m < n , or g ( n ) = m where m is least such that f ( m ) = f ( n ). Observe that g ∈ SSy( M ). Also for any The anonymous referee pointed out that similar arguments have appeared in [ ? ]. This was first proved by [ ? ]. A ⊆ N , the function g is 1-1 or constant on A if and only if f is. Since U is Ramsey, g is either constant or 1-1 on some set A ∈ U . Hence f is either constant or 1-1 on A as well. If f is constant on A , then [ f ] U ∈ M . If f is 1-1 on A , let s be the Skolemterm that is the inverse of f on A . Then clearly s ([ f ] U ) = [ id ] U . This completesthe argument that Π SSy( M ) M/U is a minimal cofinal extension of M . (cid:3) The converse to the above theorem does not hold. If M has a minimal cofinalextension, it does not follow that there is a Ramsey ultrafilter on SSy( M ). Theorem 5.7.
Assuming
PFA , Ramsey ultrafilters exist for proper Scott sets ofsize ω . Thus, if M is a model of PA and SSy( M ) is proper of size ω , then M hasa minimal cofinal extension.Proof. The existence of a Ramsey ultrafilter involves being able to meet a family ofdense sets. To see this, fix f : N → N and observe that D f = { A ∈ SSy( M ) / Fin | f is 1-1 on A or f is constant on A } is dense. To see that D f is dense, actuallydoes not require that SSy( M ) is arithmetically closed. (cid:3) The proof of Theorem 5.7 shows that any M with a countable standard systemwill have a minimal cofinal extension since we do not need PFA to construct anultrafilter meeting countably many dense sets.6. Weakening the Hypothesis
There are several ways in which the hypothesis of the main theorem can bemodified. PFA is a very strong set theoretic axiom, and therefore it is importantto see whether this assumption can be weakened to something that is lower inconsistency strength. In fact, there are weaker versions of PFA that still work withthe main theorem. It is also possible to make slightly different assumptions on X .Instead of assuming that X is proper, it is sufficient to assume that X is the unionof a chain of proper Scott sets.The definition of properness refers to countable structures M ≺ H λ and theexistence of M -generic elements for them. If we fix a cardinal κ and modify thedefinition to consider M of size κ instead, we will get the notion of κ -properness .In this extended definition, the notion of properness we considered up to this pointbecomes ℵ -properness . For example, the κ -c.c. and < κ -closed posets are κ -proper.Hamkins and Johnstone [ ? ] recently proposed a new axiom PFA( c -proper) whichstates that for every poset P that is proper and 2 ω -proper and every collection D of ω many dense subsets of P , there is a filter on P that meets all of them.PFA( c -proper) is much weaker in consistency strength than PFA. While the bestlarge cardinal upper bound on the consistency strength of PFA is a supercompactcardinal, an upper bound for PFA( c -proper) is an unfoldable cardinal [ ? ]. Unfold-able cardinals were defined by Villaveces [ ? ] and are much weaker than measurablecardinals. In fact, unfoldable cardinals are consistent with V = L . The axiomPFA( c -proper) also decides the size of the continuum is ω [ ? ]. It is enough for themain theorem to assume that PFA( c -proper) holds: Theorem 6.1.
Assuming
PFA( c -proper ) , every proper Scott set is the standardsystem of a model of PA . I am grateful to Haim Gaifman for pointing this out, see [ ? ] for a detailed argument. COTT’S PROBLEM FOR PROPER SCOTT SETS 13
Proof.
Every κ + -c.c. poset is κ -proper. It is clear that every Scott set X is (2 ω ) + -c.c.. It follows that every Scott set is 2 ω -proper. Thus, PFA( c -proper) applies toproper Scott sets. (cid:3) It is also easy to see that we do not need the whole Scott set X to be proper.For the construction, it would suffice if X was a union of a chain of proper Scottsets. Call a Scott set piecewise proper if it is the union of a chain of proper Scottsets of size ≤ ω . Under this definition, any arithmetically closed Scott set of size ≤ ω is trivially piecewise proper since it is the union of a chain of arithmeticallyclosed countable Scott sets. Also, it is clear that a piecewise proper Scott set isarithmetically closed. The modified construction using piecewise proper Scott setsdoes not require all of PFA but only a much weaker version known as PFA − . Theaxiom PFA − is the assertion that for every proper poset P of size ω and everycollection D of ω many dense subsets of P , there is a filter on P that meets all ofthem. PFA − has no large cardinal strength. The axiom is equiconsistent with ZFC[ ? ] (p. 122). This leads to the following modified version of the main theorem: Theorem 6.2.
Assuming
PFA − , every piecewise proper Scott set of size ≤ ω isthe standard system of a model of PA .Proof. It suffices to show that the ω -Ehrenfeucht Principle holds for piecewiseproper Scott sets of size ω . So suppose M is a model of PA of size ω and X is apiecewise proper Scott set of size ω such that SSy( M ) ⊆ X . Since X is piecewiseproper, it is the union of a chain of proper Scott sets X ξ for ξ < ω . Fix any A ∈ X ,then there is an ordinal α < ω such that SSy( M ) and A are contained in X α . Since X α is proper, the ω -Ehrenfeucht Principle holds for X α by Theorem 3.4. Thus,there is M ≺ N such that A ∈ SSy( N ) ⊆ X α ⊆ X . (cid:3) When is X / Fin
Proper or Piecewise Proper?
Here, I give an overview of what is known about the existence of proper andpiecewise proper Scott sets. Recall that for a property of posets P , if X is arith-metically closed and X / Fin has P , I say that X has property P . Theorem 7.1.
Any arithmetically closed countable Scott set is proper and P ( N ) isproper.Proof. The class of proper posets includes c.c.c. and countably closed posets. Anarithmetically closed countable Scott set is c.c.c. and P ( N ) is countably closed. (cid:3) We are already in a better position than with c.c.c. Scott sets since we havean instance of an uncountable proper Scott set, namely P ( N ). This does not,however, give us a new instance of Scott’s Problem since we already know by theCompactness Theorem that there are models of PA with standard system P ( N ).The easiest way to show that a poset is proper is to show that it is c.c.c. orcountably closed. We already know that if a Scott set is c.c.c., then it is countable(Theorem 2.2). So this condition gives us no new proper Scott sets. It turns outthat neither does the countably closed condition. Theorem 7.2. If X is any Scott set such that X / Fin is countably closed, then X = P ( N ) . Proof.
First, I claim that if X / Fin is countably closed, then X is arithmeticallyclosed. I will show that for every sequence h B n | n ∈ N i coded in X , there is C ∈ X deciding h B n | n ∈ N i . This suffices by Theorem 3.3. Fix h B n | n ∈ N i coded in X .Define a descending sequence B ∗ ≥ B ∗ ≥ · · · ≥ B ∗ n ≥ · · · of elements of X / Fin byinduction on n such that B ∗ = B and B ∗ n +1 is B ∗ n ∩ B n +1 if this intersection isinfinite or B ∗ n ∩ ( N − B n +1 ) otherwise. By countable closure, there is C ∈ X / Finbelow this sequence. Clearly C decides h B n | n ∈ N i . Therefore X is arithmeticallyclosed. Now I will show that every A ⊆ N is in X . Define B n = { m ∈ N | ( m ) n = χ A ( n ) } as before. Let A m = ∩ n ≤ m B n and observe that A ≥ A ≥ · · · ≥ A m ≥ . . . in X / Fin. By countable closure, there exists C ∈ X / Fin such that C ⊆ Fin A m for all m ∈ N . Thus, C ⊆ Fin B n for all n ∈ N . It follows that A = { n ∈ N |∃ m ∀ k ∈ C if k > m, then ( k ) n = 1 } . Thus, A is arithmetic in C , and hence A ∈ X by arithmetic closure. Since A was arbitrary, this concludes the proof that X = P ( N ). (cid:3) The countable closure condition can be weakened slightly. If a poset is juststrategically ω -closed, it is enough to imply properness. Definition 7.3.
Let P be a poset, then G P is the following infinite game betweenplayers I and II: Player I plays an element p ∈ P , and then player II plays p ∈ P such that p ≥ p . Then player I plays p ≥ p and player II plays p ≥ p . PlayerI and II alternate in this fashion for ω steps to end up with the descending sequence p ≥ p ≥ p ≥ . . . ≥ p n ≥ . . . . Player II wins if the sequence has a lower boundin P . Otherwise, player I wins. A poset P is strategically ω -closed if player II has awinning strategy in the game G P .Observe that if X is a Scott set such that X / Fin is strategically ω -closed, then X has to be arithmetically closed. To see this, suppose that X / Fin is strategically ω -closed and h B n | n ∈ N i is a sequence coded in X . We will find C ∈ X / Findeciding the sequence by having player I play either B n or N − B n intersected withthe previous move of player II at the n th step of the game. It is not known whetherthere are Scott sets that are strategically ω -closed but not countably closed.One might wonder at this point whether it is possibly the case that a Scott setis proper only when it is countable or P ( N ) and a Scott set is piecewise proper onlywhen it is of size ≤ ω . In a forthcoming paper [ ? ], I show the following resultsabout the existence of proper and piecewise proper Scott sets.First, I show that one can obtain uncountable proper Scott sets other that P ( N )by considering when the P ( N ) of V remains proper in a generic extension afterforcing to add new reals. Theorem 7.4. If CH holds and P is a c.c.c. poset, then P V ( N ) / Fin remains properin V [ g ] where g ⊆ P is V -generic. In particular, if CH holds in V and we force to add a Cohen real, then the P ( N ) of V will be a proper Scott set in the generic extension.It is also possible to force the existence of many proper Scott sets of size ω andpiecewise proper Scott sets of size ω . Theorem 7.5.
There is a generic extension of V by a c.c.c. poset, which containscontinuum many proper Scott sets of size ω . Theorem 7.6.
There is a generic extension of V by a c.c.c. poset, which containscontinuum many piecewise proper Scott sets of size ω . COTT’S PROBLEM FOR PROPER SCOTT SETS 15
Finally, Enayat showed in [ ? ] that ZFC proves the existence of an arithmeticallyclosed Scott set of size ω which is not proper. Theorem 7.7 (Enayat, 2006) . There is an arithmetically closed Scott set X suchthat X / Fin collapses ω . Hence X is not proper. Clearly X / Fin cannot be proper since proper posets preserve ω .Recall that any arithmetically closed Scott set of size ω is trivially piecewiseproper. It follows that there are piecewise proper Scott sets which are not proper.It is not clear whether every proper Scott has to be piecewise proper.8. Questions
Question 8.1.
Can ZFC or ZFC + PFA prove the existence of an uncountableproper Scott set other than P ( N )? Question 8.2.
Is it consistent with ZFC that there are proper Scott sets of size ω other than P ( N )? Question 8.3.
Are there Scott sets that are strategically ω -closed but not count-ably closed? Question 8.4.
Does the ω -Ehrenfeucht Principle hold or fail (consistently)? Question 8.5.
Does the ω -Ehrenfeucht Principle hold for models with a countablestandard system? That is, can we remove the assumption of arithmetic closure fromTheorem 4.1? New York City College of Technology (CUNY), Mathematics, 300 Jay Street, Brook-lyn, NY 11201 USA
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