January 7, 2015 INCOMPARABLE ω -LIKE MODELS OF SET THEORY 1 5 1 GUNTERFUCHS,VICTORIAGITMAN,ANDJOELDAVIDHAMKINS 0 2 Abstract. We show that the analogues of the Hamkins embedding theo- n rems[Ham13],provedforthecountablemodelsofsettheory,donotholdwhen a extendedtotheuncountablerealmofω1-likemodelsofsettheory. Specifically, J under the ♦ hypothesis and suitable consistency assumptions, we show that 5 thereisafamilyof2ω1 manyω1-likemodelsofZFC,allwiththesameordinals, thatarepairwiseincomparableunderembeddability;therecanbeatransitive ] ω1-likemodelofZFCthatdoesnotembedintoitsownconstructibleuniverse; O andtherecanbeanω1-likemodelofPAwhosestructureofhereditarilyfinite L setsisnotuniversalfortheω1-likemodelsofsettheory. . h t a m 1. Introduction [ We should like to consider the question of whether the embedding theorems of 1 Hamkins [Ham13], recently proved for the countable models of set theory, might v extend to the realm of uncountable models. Specifically, Hamkins proved that (1) 2 any two countable models of set theory are comparable by embeddability; indeed, 2 (2)onecountablemodelofsettheoryembeds into anotherjust incasethe ordinals 0 1 of the first order-embed into the ordinals of the second; consequently, (3) every 0 countable model of set theory embeds into its own constructible universe; and fur- 1. thermore,(4)everycountablemodelofsettheoryembedsintothehereditarilyfinite 0 sets hHF,∈iM of any nonstandard model of arithmetic M |=PA. The question we 5 considerhereis,dotheanalogousresultsholdforuncountablemodels? Ouranswer 1 is that they do not. Indeed, we shall prove that the corresponding statements do : v not hold even in the special case of ω -like models of set theory, which otherwise 1 i amonguncountable modelsoftenexhibit aspecialaffinity withthe countablemod- X els. Specifically, we shall construct large families of pairwise incomparable ω -like 1 r a models of set theory, even though they all have the same ordinals; we shall construct ω -like models of set theory that do not embed into their own L; and we 1 shallconstructω -like models of PAthat are notuniversalfor all ω -like models of 1 1 set theory. TheHamkinsembeddingtheoremsareexpressedcollectivelyintheorem1below. An embedding of one model hM,∈Mi of set theory into another hN,∈Ni is simply a function j : M → N for which x ∈M y ←→ j(x) ∈N j(y), for all x,y ∈ M, and in this case we say that hM,∈Mi embeds into hN,∈Ni; note by extensionality that every embedding is injective. Thus, an embedding is simply an isomorphism of The research of the third author has been supported in part by NSF grant DMS-0800762, PSC-CUNY grant 64732-00-42, CUNY Collaborative Incentive Award 80209-06 20 and Simons Foundationgrant209252. TheauthorsarethankfultoRomanKossakandAliEnayatforconver- sations concerning the ideas in this article. Commentary concerning this paper can be made at http://jdh.hamkins.org/incomparable-omega-one-like-models-of-set-theory. 1 2 FUCHS,GITMAN,ANDHAMKINS N M j x∈M y ←→ j(x)∈N j(y) Figure 1. An embedding j :M →N hM,∈Mi with its range, which is a submodel of hN,∈Ni, as illustrated in figure 1. Although this is the usual model-theoretic embedding concept for relational structures, the reader should note that it is a considerably weaker embedding concept thancommonlyencounteredinsettheory,becausethiskindofembeddingneednot be elementary nor even ∆ -elementary, although clearly every embedding as just 0 definediselementaryatleastforquantifier-freeassertions. Sowecautionthereader not to assume a greater degree of elementarity beyond quantifier-free elementarity for the embeddings appearing in this paper, except where we explicitly remark on it. Theorem 1 (Hamkins [Ham13]). (1) For any two countable models of set theory hM,∈Mi and hN,∈Ni, one of them embeds into the other. (2) Indeed, such an hM,∈Mi embeds into hN,∈Ni if and only if the ordinals of M order-embed into the ordinals of N. (3) Consequently, every countable model hM,∈Mi of set theory embeds into its own constructible universe hLM,∈Mi. LM j :M →LM, x∈y ←→ j(x)∈j(y) M j (4) Furthermore, everycountablemodel ofsettheoryembedsintothehereditary finite sets hHF,∈iM of any nonstandard model of arithmetic M |= PA. Indeed, HFM is universal for all countable acyclic binary relations. One can begin to get an appreciation for the difference in embedding concepts by observing that ZFC proves that there is a nontrivial embedding j : V → V, namely, the embedding recursively defined as follows j(y)=(cid:8) j(x) | x∈y (cid:9)∪(cid:8){∅,y}(cid:9). Weleaveitasafunexercisetoverifythatx∈y ←→j(x)∈j(y)fortheembedding j defined by this recursion.1 Contrast this situation with the well-known Kunen inconsistency[Kun71],whichassertsthattherecanbe nonontrivialΣ -elementary 1 embedding j : V → V. Similarly, the same recursive definition applied in L leads tonontrivialembeddingsj :L→L,regardlessofwhether 0♯ exists. Butagain,the point is that embeddings are not necessarily even ∆ -elementary, and the familiar 0 1See[Ham13];buttogiveahintherefortheimpatient,notethateveryj(y)isnonemptyand also∅∈/j(y);itfollowsthatinsidej(y)wemayidentifythepair{∅,y}∈j(y);itfollowsthatj is injectiveandfurthermore,theonlywaytohavej(x)∈j(y)isfromx∈y. INCOMPARABLE ω1-LIKE MODELS OF SET THEORY 3 equivalenceoftheexistenceof0♯ withanontrivial“embedding”j :L→Lactually requires a ∆ -elementary embedding. 0 We find it interesting to note in contrast to theorem 1 that there is no such embedding phenomenonin the the contextofthe countablemodels ofPeanoarith- metic (where an embedding of models of arithmetic is a function preserving all atomic formulas in the language of arithmetic). Perhaps the main reason for this is that embeddings between models of PA are automatically ∆ -elementary, as a 0 consequence of the MRDP theorem, whereas this is not true for models of set theory, as the example above of the recursively defined embedding j : V → V shows, since this is an embedding, but it is not ∆ -elementary, in light of j(∅) 6= ∅. For 0 countable models of arithmetic M,N |=PA, one can show that there is an embedding j : M → N if and only if N satisfies the Σ -theory of M and the standard 1 system of M is contained in the standard system of N. It follows that there are many instances of incomparability. Meanwhile, it is a consequence of theorem 1 statement (4) that the embedding phenomenon recurs with the countable models of finite set theory ZFC¬∞, that is, with hHF,∈iM for M |=PA, since allnonstan- dard such models are universal for all countable acyclic binary relations, and so in ¬∞ thecontextofcountablemodelsofZFC therearepreciselytwobi-embeddability classes,namely,thestandardmodel,whichisinitial,andthenonstandardcountable models, which are universal. Our main theorems are as follows. Main Theorems. (1) If ♦ holds and ZFC is consistent, then there is a family C of 2ω1 many pairwise incomparable ω -like models of ZFC, meaning that there is no 1 embedding between any two distinct models in C. (2) Themodelsinstatement(1)canbeconstructedsothattheirordinalsorder- embed into each other and indeed, so that the ordinals of each model is a universal ω -like linear order. If ZFC has an ω-model, then the models of 1 statement(1) canbe constructedsoasto haveprecisely the same ordinals. (3) If ♦ holds and ZFC is consistent, then there is an ω -likemodel M |=ZFC 1 andanω -likemodelN |=PAsuchthatM does notembedintohHF,∈iN. 1 (4) If there is a Mahlo cardinal, then in a forcing extension of L, there is a transitive ω -like model M ⊆ ZFC that does not embed into its own 1 constructible universe LM. These results appear later as theorems 4, 5, 6, and 7. Note that the size of the family C in statement (1) is as large as it could possibly be, given that any two elements in a pairwise incomparable family of structures must be non-isomorphic and there are at most 2ω1 many isomorphism types of ω1-like models of set theory or indeed of structures of size ω in any first-order finite language. Statement 1 (2) shows that the models of the family C serve as ω -like counterexamples to the 1 assertion that one model of set theory embeds into another whenever the ordinals of the first order-embed into the ordinals of the second. 2. ω -like models of set theory and other background 1 The ordinal ω is the only uncountable ordinal all of whose proper initial seg- 1 mentsarecountable. Generalizingthis,alinearorderisω -like,ifitisuncountable, 1 4 FUCHS,GITMAN,ANDHAMKINS but all proper initial segments are countable. For example, a model of PA is ω - 1 like, if it is uncountable, but all proper initial segments are countable. Similarly, a model of set theory hM,∈Mi is ω -like, if it is uncountable, but every rank initial 1 segment VM for α ∈ OrdM is countable. For models of ZF, this is equivalent to α saying that M is uncountable, but every object y ∈ M has only countably many ∈M-predecessors, that is, {x ∈ M | x ∈M y} is countable; for models of ZFC, it is also equivalent to asserting that the ordinals OrdM are ω -like as a linear order. 1 Theω -likemodelsconstitute agatewayfromthe realmofcountablemodelstothe 1 uncountable, sharing and blending many of the features of both kinds of models, and they have been extensively studied both in the case of models of arithmetic and of models of set theory [Kau83, Kau77, Kos85, Ena84, MSS]. Oneobviouswaytoconstructanω -likemodelM isastheunionofacontinuous 1 elementary chain of countable models: . M . . .. Mα . M1 M0 ≺M1 ≺···≺Mα ≺···≺M =Sα<ω1Mα, M 0 At each step we should have an elementary top-extension M ≺ M , meaning α t α+1 that the new elements of M have rank exceeding that of any element of M , α+1 α as defined precisely below; and at limit stages λ we take unions Mλ = Sα<λMα. It is a consequence of lemma 2, a result due to Kiesler and Morley [KM68], that every countable model of set theory has such an elementary top-extension. In this way, every proper initial segment of the final model M is contained in some M , α which is countable. Thus, the ω -like model M grows from the bottom out of its 1 countable elementary initial segments. Conversely,however,itis nothardto seethateveryω -likemodel M mustarise 1 exactly in this way as the union of a continuous elementary chain of countable elementary initial segments, because a simple Lo¨wenheim-Skolem argument shows that there will be unboundedly many such countable elementary initial segments. So the obvious construction method is in a sense the only construction method for building ω -like models. Because every ω -like model is thus the union of an ele- 1 1 mentarychainoflengthω ,thesemodelsnaturallyinheritmuchoftheset-theoretic 1 structure and context of ω , such as clubs, stationary sets, and constructions via 1 ♦, and it is by taking advantage of this set-theoretic structure that we shall prove our main theorems. SupposethathM,∈MiandhN,∈Niaremodelsofsettheory. Wesaythatthefirst is a submodel ofthe second, writtenM ⊆N or more properlyhM,∈Mi⊆hN,∈Ni, if M is a subset of N and ∈M is the restriction of ∈N to the domain M, which is to say that the two models agree on whether a∈b for any elements a,b∈M. The extension is a transitive extension, written M ⊑ N, if the larger model adds no new elements to old sets, which is to say, a ∈N b ∈ M implies a ∈ M; this is also sometimes called anend-extension (this is not the same as a top-extension). Thus, INCOMPARABLE ω1-LIKE MODELS OF SET THEORY 5 a transitive extension occurs when the submodel is transitive with respect to the membership relation of the larger model, such as in the case of a forcing extension M ⊆ M[G] or of the inner model LM ⊆ M. A top-extension, in contrast, written M ⊑ N, occurs when the new sets of the larger model all have higher rank in the t vonNeumannhierarchythananyoldset;thatis,ifwhenevera∈N\M andb∈M, thentherankofainN ishigherthantherankofbinN. Forexample,everymodel M |= ZF is a top-extension of its rank initial segments VM ⊑ M. An elementary α t top-extension, written M ≺ N, occurs when a top-extension is also elementary, t meaningthateveryfirst-orderassertionaboutsomeobjectsinM hasthesametruth value in M as it does in N. For models of ZF, an elementary transitive extension (or elementary end-extension) is the same thing as an elementary top-extension, becauseVM is definable inM fromα andsomust by elementaritybe equalto VN α α as defined in N. Meanwhile, the nontrivial forcing extensions M ⊆ M[G] provide examplesoftransitiveextensions(end-extensions)thatarenottop-extensions,and they are never elementary. A model M is topless in a top-extension M ⊑ N, if t there is no least upper bound of OrdM in N; otherwise M is topped in N. For any modelofsettheoryhM,∈Miandanyelementa∈M,letusintroducethefollowing notation aM ={b∈M |M |=b∈a}, to refer to the set of objects in M that M believes to be elements of a. If M ⊆N is a submodel of another model, then aN ∩M is the trace of a on M. Note that another way to say that an extension M ⊆N is transitive is to say that aM =aN for all a∈M. N a M aN∩M Figure 2. The trace of a on M The success of the elementary chain construction in building an ω -like model 1 relies,ofcourse,onthe factthateverycountablemodelofsettheoryindeedhasan elementary top-extension. Lemma 2 (Keisler-Morley[KM68]). Every countable model hM,∈Mi |= ZFC has an elementary top-extension. Let us briefly sketch a folklore proof of this based on definable ultrapowers, as we shall subsequently make use of some of the ideas in the proof. One begins with a countable model hM,∈Mi |= ZFC. The first step is to ensure the global choice axiom, by adding a predicate C ⊆ M such that hM,∈M,Ci satisfies ZFC(C), the version of ZFC that includes instances of the replacement axioms in the expanded language, and also has a C-definable well-ordering of the universe. This can be done by the forcing Add(Ord,1)M to add a Cohen class of ordinals C ⊆ OrdM. Conditions in Add(Ord,1)M are simply the binary ordinal-length sequences s ∈ (

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