Simmons - An introduction to Good old fashioned model theory (2004).pdf

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[ Heldin120../A000-wholething..LastchangedJuly26,2004 ]
Anintroductionto
Goodoldfashionedmodeltheory
[ Heldin120../Preamble..LastchangedJuly26,2004 ]
Thecoursewillbetaught
EachTuesdayfrom2–5pminroomJ
withsomelecturesandsometutorialactivities.
Themainbodyofcoursewillcoverthefollowingtopics.
•Basicideasoflanguage,satisfation,andcompactness
•Someexamplesofeliminationofquantifiers
•Thediagramtechnique,thecharacterizationof8 1 -and8 2 -axiomatizabletheories,
andsimilarresults
•Modelcompletetheories,companiontheories,existentiallyclosedstructures,and
variousrefinements
•Atomicstructuresandsucientlysaturatedstructures
•Thesplittingtechniqueand@ 0 -categoricity
Dependingonthetimeavailable,someofthefollowingtopicswillbelookedat.
•Henkinconstructions
•Omitingtypes
•Theback-and-forthtechnique
•Saturationmethods.
Fullcoursenoteswillbeprovidedofwhichthisisthefirstpart.
Alookatthecontentspagewillindiactehowthesestandatthemoment.Thesenotes
willbemodifiedasthecourseprogresses.Themainbodyofthecourseiscontainedin
PartI.TheextratopicsarecontainedinPartII.
Remarksandcomments inthiskindoftype indicatethatsomethingneedstobe
donebeforethefinalversionisproduced.
1
2
Contents
Introduction............................................... 7
0.1 Historicalaccount– tobedone ............................. 7
0.2 Asurveyoftheliterature– tobere-done ....................... 7
IThedevelopment 9
1 Syntaxandsemantics ......................................11
1.1 Signatureandlanguage.................................11
Exercises .........................................15
1.2 Basicnotions.......................................15
Exercises .........................................18
1.3 Satisfaction........................................19
Exercises .........................................22
1.4 Consequence.......................................23
Exercises .........................................27
1.5 Compactness.......................................29
Exercises .........................................32
2 Theeectiveeliminationofquantifiers.............................33
2.1 Thegeneralitiesofquantifierelimination........................33
Exercises .........................................35
2.2 Thenaturalnumbers...................................35
Exercises .........................................40
2.3 Lines............................................41
Exercises .........................................43
2.4 Someotherexamples– tobere-done ........................44
Exercises– needed .....................................44
3 Basicmethods ..........................................45
3.1 Somesemanticrelations.................................45
Exercises .........................................47
3.2 Thediagramtechnique .................................47
Exercises .........................................52
3.3 Restrictedaxiomatization................................52
Exercises .........................................55
3.4 Directedfamiliesofstructures..............................55
Exercises .........................................60
3.5 Theupanddowntechniques..............................60
Theuptechnique.....................................61
Thedowntechnique...................................62
Exercises .........................................63
4Modelcompleteandsubmodelcompletetheories.......................65
4.1 Modelcompletetheories.................................65
Exercises .........................................67
4.2 Theamalgamationproperty...............................67
Exercises .........................................70
4.3 Submodelcompletetheories...............................71
Exercises– needed .....................................72
5 Companiontheoriesandexistentiallyclosedstructures....................73
5.1 Modelcompanions....................................73
Exercises .........................................76
5.2 Companionoperators ..................................77
Exercises .........................................79
5.3 Existentiallyclosedstructures..............................79
Exercises .........................................82
5.4 Existenceandcharacterization.............................82
Exercises .........................................87
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5.5 Theorieswhichareweaklycomplete..........................87
Exercises .........................................90
6 PertandBuxomstructures...................................91
6.1 Atomicity.........................................91
Exercises .........................................97
6.2 Existentiallyuniversalstructures............................97
Exercises– needed .....................................100
6.3 Acompanionoperator..................................100
Exercises .........................................103
6.4 Existenceofe.u.structures...............................103
Exercises– needed .....................................108
7 Ahierarchyofproperties.....................................109
7.1 SplittingwithGoodformulas..............................110
Exercises .........................................113
7.2 Splittingwithnot-Badformulas.............................113
Exercises .........................................115
7.3 Countableexistentiallyuniversalstructures......................116
Exercises– needed .....................................117
7.4 Categoricityproperties..................................117
Exercises .........................................120
7.5 Someparticularexamples................................121
IIConstructiontechniques 123
8 Theconstructionofcanonicalmodels..............................125
8.1 Helpfulsetanditscanonicalmodel...........................126
Exercises .........................................132
8.2 Consistencyproperty...................................132
Exercises .........................................137
9 Omittingtypes..........................................139
Exercises .........................................142
10Thebackandforthtechnique..................................143
Exercises .........................................146
11Homogeneous-universalmodels.................................147
12Saturation– seeearlier ....................................148
13Forcingtechniques........................................149
14Ultraproducts– tobedone ...................................150
IIISomesolutionstotheexercises 151
AForsection1...........................................153
A.1 For§1.1..........................................153
A.2 For§1.2– notyetdone .................................153
A.3 For§1.3..........................................153
A.4 For§1.4– mostnotdone ................................153
A.5 For§1.5..........................................156
BForsection2...........................................159
B.1 For§2.1..........................................159
B.2 For§2.2..........................................159
B.3 For§2.3..........................................163
B.4 For§2.4– notyetdone .................................164
CForsection3...........................................165
C.1 For§3.1..........................................165
C.2 For§3.2..........................................165
C.3 For§3.3..........................................166
C.4 For§3.4..........................................166
C.5 For§3.5..........................................167
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DForsection4...........................................169
D.1 For§4.1- tobedone ..................................169
D.2 For§4.2..........................................169
D.3 For§4.3- noexercisesyet ..............................171
EForsection5...........................................173
E.1 For§5.1..........................................173
E.2 For§5.2..........................................174
E.3 For§5.3..........................................174
E.4 For§5.4- tobedone ..................................175
E.5 For§5.5..........................................175
FForsection6...........................................177
F.1 For§6.1..........................................177
F.2 For§6.2- noexercisesyet ..............................178
F.3 For§6.3- tobedone ..................................178
F.4 For§6.4- noexercisesyet ..............................178
GForsection7...........................................179
G.1 For§7.1..........................................179
G.2 For§7.2- tobedone ..................................180
G.3 For§7.3- noexercisesyet ..............................180
G.4 For§7.4- tobedone ..................................180
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