The Structure of Models of Peano Arithmetic

326 Pages·2006·1.94 MB·English

by  Roman Kossak

OXFORD LOGIC GUIDES
Series Editors
D.M. GABBAY
A.J. MACINTYRE
D. SCOTT
OXFORD LOGIC GUIDES
Available books in the series:
10. Michael Hallett: Cantorian set theory and limitation of size
17. Stewart Shapiro: Foundations without foundationalism
18. John P. Cleave: A study of logics
21. C. McLarty: Elementary categories, elementary toposes
22. R.M. Smullyan: Recursion theory for metamathematics
23. Peter Clote and Jan Kraj´ıcek: Arithmetic, proof theory, and computational
complexity
24. A. Tarski: Introduction to logic and to the methodology of deductive sciences
25. G. Malinowski: Many valued logics
26. Alexandre Borovik and Ali Nesin: Groups of finite Morley rank
27. R.M. Smullyan: Diagonalization and self-reference
28. DovM.Gabbay,IanHodkinson,andMarkReynolds:Temporallogic:Mathematical
foundations and computational aspects, volume 1
29. Saharon Shelah: Cardinal arithmetic
30. Erik Sandewall: Features and fluents: Volume I: A systematic approach to the
representation of knowledge about dynamical systems
31. T.E.Forster:Settheorywithauniversalset:Exploringanuntypeduniverse,second
edition
32. Anand Pillay: Geometric stability theory
33. Dov M. Gabbay: Labelled deductive systems
35. Alexander Chagrov and Michael Zakharyaschev: Modal logic
36. G.SambinandJ.Smith:Twenty-fiveyearsofMartin-L¨ofconstructivetypetheory
37. Mar´ıa Manzano: Model theory
38. Dov M. Gabbay: Fibring Logics
39. Michael Dummett: Elements of intuitionism, second edition
40. D.M. Gabbay, M.A. Reynolds and M. Finger: Temporal Logic: Mathematical
foundations and computational aspects, volume 2
41. J.M. Dunn and G. Hardegree: Algebraic methods in philosophical logic
42. H.Rott:Change, choiceandinference: Astudyofbeliefrevisionandnonmonotoic
reasoning
43. Johnstone: Sketches of an elephant: A topos theory compendium, volume 1
44. Johnstone: Sketches of an elephant: A topos theory compendium, volume 2
45. David J. Pym and Eike Ritter: Reductive logic and proof search: Proof theory,
semantics and control
46. D.M. Gabbay and L. Maksimova: Interpolation and definability: Modal and
intuitionistic logics
47. John L. Bell: Set Theory: Boolean-valued models and independence proofs, third
edition
48. Laura Crosilla and Peter Schuster: From sets and types to topology and analysis:
Towards practicable foundations for constructive mathematics
49. Steve Awodey: Category theory
50. Roman Kossak and James Schmerl: The structure of models of Peano Arithmetic
The Structure of Models of
Peano Arithmetic
ROMAN KOSSAK
City University of New York
JAMES H. SCHMERL
University of Connecticut, Storrs
CLARENDON PRESS • OXFORD
2006
3
GreatClarendonStreet,OxfordOX26DP
OxfordUniversityPressisadepartmentoftheUniversityofOxford.
ItfurtherstheUniversity’sobjectiveofexcellenceinresearch,scholarship,
andeducationbypublishingworldwidein
Oxford NewYork
Auckland CapeTown DaresSalaam HongKong Karachi
KualaLumpur Madrid Melbourne MexicoCity Nairobi
NewDelhi Shanghai Taipei Toronto
Withofficesin
Argentina Austria Brazil Chile CzechRepublic France Greece
Guatemala Hungary Italy Japan Poland Portugal Singapore
SouthKorea Switzerland Thailand Turkey Ukraine Vietnam
OxfordisaregisteredtrademarkofOxfordUniversityPress
intheUKandincertainothercountries
PublishedintheUnitedStates
byOxfordUniversityPressInc.,NewYork
(cid:1)c OxfordUniversityPress,2006
Themoralrightsoftheauthorshavebeenasserted
DatabaserightOxfordUniversityPress(maker)
Firstpublished2006
Allrightsreserved.Nopartofthispublicationmaybereproduced,
storedinaretrievalsystem,ortransmitted,inanyformorbyanymeans,
withoutthepriorpermissioninwritingofOxfordUniversityPress,
orasexpresslypermittedbylaw,orundertermsagreedwiththeappropriate
reprographicsrightsorganization.Enquiriesconcerningreproduction
outsidethescopeoftheaboveshouldbesenttotheRightsDepartment,
OxfordUniversityPress,attheaddressabove
Youmustnotcirculatethisbookinanyotherbindingorcover
andyoumustimposethesameconditiononanyacquirer
BritishLibraryCataloguinginPublicationData
Dataavailable
LibraryofCongressCataloginginPublicationData
Dataavailable
TypesetbyNewgenImagingSystems(P)Ltd.,Chennai,India
PrintedinGreatBritain
onacid-freepaperby
BiddlesLtd.,King’sLynn,Norfolk
ISBN 0–19–856827–4 978–0–19–856827–8
1 3 5 7 9 10 8 6 4 2
PREFACE
In the 1930’s, nonstandard models of arithmetic were introduced into
mathematics by Thoralf Skolem in two papers [188] and [189]. Even though
the logic community was slow in recognizing the importance of Skolem’s contri-
bution of the method of definable ultrapowers (see [207]), it now seems almost
obligatory to include nonstandard models in introductory courses on mathem-
atical logic. The point that they help to emphasize is the limited expressive
power of first order logic: there are mathematical structures (one of them being
themostclassicalofmathematicalstructures—thestandardmodelofarithmetic)
whichareindistinguishablewithrespecttotheirfirst-orderpropertiesbutwhose
isomorphism types are dramatically different. The drama is personified by non-
standard elements. However, the discussion at the introductory level usually
ends here, leaving out the complex picture obtained by a closer scrutiny of the
spectrumofisomorphismtypesofnonstandardmodels.Onceweknowthatnon-
standard models exist, it is very natural to ask how different they are from the
standard one and also from each other. In other words, we would like to know
to what extent the first-order theory of a model of arithmetic determines prop-
erties that are not first-order expressible. A priori, there is no guarantee that
the possible answers would be relevant to other developments in model theory.
Even a quick initial glance at a nonstandard model reveals a very rich struc-
ture. It could be that the diversity among nonstandard models is so vast that
no coherent picture in the form of a relative classification can emerge. In fact,
this might be the the state of affairs for the spectrum of all nonstandard mod-
els. Fortunately, when we consider some well defined and important subclasses
of nonstandard models, a more attractive picture can be painted. This is the
subject of this book.
TherewaslittleprogressinthemodeltheoryofarithmeticbetweenSkolem’s
discovery and two important developments that took place at the end of 1950’s.
Inthefirstofthese, StanleyTennenbaumprovedinafamousunpublishedpaper
[205] that in no nonstandard model can either addition or multiplication have a
recursivepresentation. Thisresultpointedtoanessentialdifficultyinconstruct-
ing nonstandard models (there are no such objects in the world of constructive
mathematics!). The second of these was the fundamental theorem of Robert
MacDowell & Ernst Specker [123]. Skolem had proved that the standard model
has an elementary end extension. The MacDowell–Specker Theorem involves a
refinement of Skolem’s method to show that every model of Peano Arithmetic
(PA) has an elementary end extension. Refining this further, Haim Gaifman (in
several papers, but most importantly in [45]) developed a technology of iter-
ating elementary end extensions along linear orders to obtain models having
vi PREFACE
additional interesting features. Thus, Tennenbaum says: there are no effective
constructions; buttheresponsefromMacDowell, Specker, andGaifmanis: some
interesting set-theoretic constructions are easily available.
A very important concept that emerged from the proof of Tennenbaum’s
theorem is that of the standard system of a nonstandard model. This is the col-
lectionofsetsofnaturalnumberscodedinthemodel. Thestandardsystemsare
preciselytheω-modelsofthefragmentsecond-orderarithmeticknownasWKL .
0
Having been studied by Dana Scott [180] such models are also called Scott sets.
Thereare2ℵ0 countableScottsets, andeachofthemisthestandardsystemofa
model of PA. Under the Continuum Hypothesis, every Scott set is the standard
system of a model of PA. Every countable nonstandard model has 2ℵ0 distinct
initial segments which are themselves models of PA. The standard system of the
model puts some restrictions on the possible complete theories of these initial
segments (including a restriction on their theories). Still, each countable Scott
set X is the standard system of 2ℵ0 elementarily inequivalent models of PA, and
also for each model M having a standard system of X, there are 2ℵ0 pairwise
nonisomorphiccountablemodelselementarilyequivalenttoM havingXastheir
standardsystems.Consideringthecomplexityofbothclassesofobjectsinvolved,
this is a mess! However, for us this is only a point of departure. Suppose the
completetheoryofamodelanditsstandardsystemaregiven. Whatelsecanbe
said about the model? The theorem of MacDowell and Specker suggests that it
might be fruitful to consider elementary submodels. For any model M, the fam-
ily of elementary submodels of M forms a lattice Lt(M), with naturally defined
operations ∧ and ∨. What are the lattices which can be represented as Lt(M)
for some model of arithmetic M? This is one of the main questions we will con-
sider. Some answers are given in Chapter 3 and much of the material from the
previous chapters, especially from Chapter 3 on types, is developed with an eye
on applications to the lattice problem. While, all distributive lattices satisfying
an obvious necessary condition (they must be algebraic) can be represented as
Lt(M), many questions concerning nondistributive lattices are open. In partic-
ular, we do not know if there is a finite lattice which cannot be represented as
Lt(M).
The MacDowell–Specker Theorem has an interesting feature. It is almost
independent of the language in which arithmetic is formalized. Let L be any
countable language extending the language of PA, and let T be a theory in L
thatextendsPAprovestheschemeofinductionforallformulasofL.Itturnsout
that most arguments concerning models of PA apply without modifications to
models of T. We address this by formulating our results forPA∗ rather than PA,
where PA∗ is any T as above. Actually, many results carry over to uncountable
languagesaswell,withonenotableexception,theMacDowell–SpeckerTheorem.
There are models of PA∗ in a language of cardinality ℵ with no elementary end
1
extensions. This result is due to George Mills, and it uses forcing in models of
arithmetic. The purpose Chapter 6 is to give a proof of Mills’ theorem and to
show how forcing can be applied to construct interesting models of PA∗.
PREFACE vii
Whilewedonotknowexactlywhichlatticescanberepresentedbylatticesof
elementary submodels of models of PA, the analogous question concerning auto-
morphism groups has a complete answer, which we give in Chapter 5. For every
infinite linearly ordered structure (A,<,...), there is a model M of PA such
that the automorphism groups of (A,<,...) and M are isomorphic. Moreover,
one can obtain such an M as an elementary end extension of any model of PA.
Conclusion: nothing special here about arithmetic (but, it should be noted that
the proof uses the full power of arithmetic and involves a formalized Ramsey
style theorem of Neˇsetˇril and Ro¨dl). So we learn that in general there is no
connection between the standard system of a model or its theory and its auto-
morphism group. But this is not the end of the story. In Chapters 8 and 9, we
discuss countable recursively saturated models of PA and show that something
special is happening in this class of models. Most of the results there are for-
mulated in terms of automorphisms and automorphism groups. In particular,
many properties characterizing the important class of arithmetically saturated
models involve automorphisms; for example, a countable recursively saturated
model M of PA is arithmetically saturated iff it has an automorphism moving
all undefinable elements, and this happens iff the automorphism group of M
has uncountable cofinality. The automorphism group of a countable arithmetic-
ally saturated model, considered as an abstract group, determines the standard
system of the model.
SomeimportantresultsonautomorphismgroupsofmodelsofPAareincluded
inKayeandMacpherson’svolumeonautomorphismsoffirst-orderstructures[77].
Here we concentrate on the results obtained after the volume was published,
although we do include the complete proof of the theorem of Daniel Lascar on
the small index property of countable arithmetically saturated models of PA.
The aim of Chapter 10 is to present some exotic species of models with
propertiesdramaticallycontrastingthoseofthecountableones. Inparticularwe
construct a recursively saturated rather classless model using ♦ (a result due to
MattKaufmann[67])andthenagainusingweak♦.Wedoitdespitethefactthat
SaharonShelahhasalreadyprovedthattheexistenceofsuchmodelsisatheorem
ofZFC(andweexplainwhy).Othertopicsinthischapterincludenonisomorphic,
but still very similar, models. Previous constructions of such models used extra
set-theoretic assumptions. Here it is done in ZFC. Rigid recursively saturated
modelsarealsoconstructedinthischapterandasaremodelsofPeanoArithmetic
in the language with Ramsey quantifier and with the stationary quantifier.
One of the topics we neglect in this book is reducts. Any classification of
modelsofPAmustincludesubclassificationsoftheirnaturalreducts.Foramodel
M, let (M,+), (M,×), and (M,<) denote, respectively, the reducts of M to,
respectively,+,×,and<.Itturnsoutthatinthecountablecaseallthesereducts
are nicely classifiable. All nonstandard countable models share the same order
type (ω∗+ω)ρ, where ρ is the order type of the rationals. For every countable
the isomorphism type if its reducts to + and × is determined uniquely by the
standard system of M. Consequently, for any countable models M and N of
viii PREFACE
PA, (M,+) ∼= (N,+) iff (M,×) ∼= (N,×). In Chapter 10 we prove that this is
not the case for models of cardinality ℵ . Very little is known about order types
1
of uncountable models. We honor this important and rather unexplored topic
in Chapter 11 where we give proofs of two striking theorems concerning order
types, oneduetoJean-Franc¸oisPabion[142]onκ-saturatedreducts(M,<)and
one due to Shelah [185] on the existence of (κ,κ)-gaps.
There is extensive literature on models of PA, but essentially there are only
two books: Kaye’s [71] and Ha´jek & Pudla´k’s [50]. It has been often noted that
bookwriting in this area is not easy. Here is what Laurie Kirby wrote in his
review of Kaye’s book [82]
Our vocabulary lacks a term to denote a person whose calling is the study of mod-
els of arithmetic. Model theorists, topologists, even functional analysts can identify
themselves succinctly, but we have to resort to such locutions as “I’m in models of
arithmetic.” And the name of the field itself—“models of arithmetic”—also seems to
bespeakaninsecurityaboutwhetheritisafieldatall: theobjectsofstudyarebaldly
named without any pretensions to a grand theory or -ology. Models of arithmetic cer-
tainlyisabonafidefield.Ithasitsownmeetings,folklore,andstars.Ithasbuiltupa
coherent body of knowledge relevant to some of the central problems of modern logic.
But it has never sat comfortably within the traditional fourfold division of logic, it is
sparsely populated and has been known to lie dormant for decades, and it has never
had a “bible.” Access to this difficult terrain has been daunting to outsiders.
These remarks were appropriate in 1992 and are still appropriate today.
Kaye’s book is still the “bible,” while Ha´jek and Pudla´k’s book has become
a standard reference in the model theory of fragments of arithmetic. Before
1991, the only, and not easily available, source was the excellent notes of Craig
Smoryn´ski [191] from his lectures on nonstandard models at the University of
Utrechtin1978.Thenoteswereaninspirationformanyofuswhostudiedmodels
of PA in the 1980s.
While writing this book we assumed that the reader is familiar with Kaye’s
book,andwetriedtoavoidrepetitions.ThereissomeoverlapintheChapter7on
cuts. In this chapter we discuss strong cuts and their various characterizations.
Strong cuts play an important role in arithmetic saturation—one of the main
themes in this book.
Mostoftheresultspresentedherehavebeenpreviouslypublished, butmany
oftheproofsarenew.Someproofsaresimpler,oratleastmuchshorter,thanthe
original ones. This is a great advantage of having the whole range of techniques
presented in a unified way in one place. Some results appear here for the first
time.IntheRemarksandReferenceswemadeanefforttoincludeaccuratecred-
its and references. Anyone who has worked in this or any area understands well
that this was not an easy job, and certainly there will be errors and omissions.
Exercises are an integral part of the book. To measure the difficulty of exer-
cises, we have designed a suitable ranking system. Exercises marked with ♣ are
theeasiest. Thesearetraditionalexercisesforpractice. Usuallytheyjustinvolve
PREFACE ix
going through the definitions of the concepts involved. The highest rank is ♠.
We have used it sparingly, as it is reserved for those exercises we could not do.
Exercisesmarkedwith♦and♥areinbetween. Includedinthe♦and♥categor-
ies are many lemmas, propositions, and theorems we took from the literature.
Werecommendthatthereaderatleastreadthem, astheyhelptoprovidearel-
atively complete account of what is known about the subject. One should note
thattherankingsofalltheseexercisesisquitesubjective; othersmayrankthem
differently, and we probably would too if we were to do it again. The rankings
are also relative, as they are based on the material developed in the book. As
stand alone exercises, many would have to be ranked higher.
Throughout the book, the reader will find numerous instances of the (Do
it!)command.Asinmostmathematicaltexts,itisassumedthatthereaderwill
fill in the more routine parts of the arguments. There will be many statements
starting with “Clearly, ...,” or “One can verify ...,” or “A similar argument
shows....”Thesearetreacherouspoints.Itisalwaysgoodforthereadertopause
for a moment and to verify whether she or he really believes the author(s). The
role of the (Do it!)’s is to provide alert stopmarks. They remind you not to
read too fast.