Model Theory

Model Theory - Copyright c 19821998 by Stephen G. Simpson...

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Copyright c ± 1982–1998 by Stephen G. Simpson Math 563: Model Theory Stephen G. Simpson May 2, 1998 Department of Mathematics The Pennsylvania State University University Park, State College PA 16802 simpson@math.psu.edu www.math.psu.edu/simpson/courses/math563/ Note: Chapters 12 and 13 are not finished.
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Contents 1 Sentences and models 7 1 . 1 Symbo l s .............................. 7 1 . 2 F o rmu l a s.............................. 8 1 . 3 S t ru c tu r e s............................. 9 1 . 4 T th . .............................. 1 0 1 . 5 M od e l sandth e o r i e s........................ 1 0 2 Complete theories 13 2 . 1 D efin i t i on sandex amp l e s..................... 1 3 2 . 2 V au gh t st e s t ........................... 1 5 2 . 3 App l i c a t i so fV t e s t................... 1 6 3 The compactness theorem 21 3 . 1 P r oo fo fth ec omp a c tn e s sth e o r em. ............... 2 1 3 . 2 S om eapp l i c a t i ofi e ldth e o ry . 2 3 3.3 The L¨ ow enh e im -Sk o l em -T a r sk ith e o r ............ 2 4 4 Decidability 27 4 . 1 R e cu r s iv e lyax i a t i z ab l eth e o r i e s................ 2 7 4 . 2 D e c id l e o r i e s......................... 3 0 4 . 3 D e c l e l 3 1 5E l e n t a r y e x t e n s i o n s 3 5 5 . i t i onandex l e s ..................... 3 5 5 . 2 Ex i s t en c eo fe l t a ryex t s i s ............... 3 8 5 . 3 E l t a rym o rph i sm s................... 4 0 3
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4 CONTENTS 6 Algebraically closed fields 43 6 . 1 S imp l efi e ldex t en s i on s ...................... 4 3 6 . 2 A l g eb r a i cc l o su r e ......................... 4 7 6 . 3 C omp l e t e s sandm od e lc l e t e s s ............. 4 9 6 . 4 H i lb e r t sNu l l s t e l l s a t z ..................... 5 2 7 Saturated models 55 7 . 1 E l em tt yp e s........................... 5 5 7 . 2 S a tu r a t edm e l s......................... 5 8 7 . 3 Ex i s t c eo fs a r a t e l s .................. 6 0 7 . 4 P r e s e rv a t i onth e o r 6 3 8 Elimination of quantifiers 71 8 . 1 Th e l e t i ono fath e o ry. ............... 7 1 8.2 Substructure completeness. ................... 7 3 8 . 3 Th er o l l eex t s i s................... 7 6 9 Real closed ordered fields 79 9 . 1 O rd e r edfi e ld 7 9 9 . 2 Un iqu e s so fr e a l o r e .................... 8 3 9.3 Quantifier elimination for RCOF ................. 8 6 9 . 4 Th es o lu t i fH i e r t s1 7 thp r ob l em . ............ 8 9 10 Prime models (countable case) 93 1 0 . 1Th eom i t t in gt e sth e o r .................. 9 3 1 0 . 2P r im e l s ........................... 9 6 1 0 . 3Th enumb e ro fc oun t ab l e l s ................ 1 0 1 1 0 . 4D e c id l ep r e l s ..................... 1 0 4 11 Differentially closed fields of characteristic 0 109 1 1 . 1S l t s i 1 0 9 1 1 . 2D iff e r t i a l lyc l o s e s .................... 1 1 5 1 1 . 3D e r t i a l o r e( c t l ec a s e )............... 1 1 7 1 1 . 4R i t t l l s t e l l s a t z ....................... 1 2 0 12 Totally transcendental theories 125 12.1 Stability . . ............................ 1 2 5 1 2 . 2R anko fane l e..................... 1 2 5 1 2 . 3Ind i s c e rn ib l e 1 2 5
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CONTENTS 5 1 2 . 4Ex i s t en c eo fs a tu r a t edm od e l s .................. 1 2 5 13 Prime models (uncountable case) 127 1 3 . 1S t r on g lya t om i cm e l s...................... 1 2 7 1 3 . 2N o rm a ls e t s ............................ 1 2 7 1 3 . 3Un iqu e s sandch a r a c t e r i z a t i ono fp r im em e l s ....... 1 2 7
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6 CONTENTS
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Chapter 1 Sentences and models 1.1 Symbols 1. We assume the availability of a large supply of nonlogical symbols of the following kinds: 1. n -ary relation symbols R ( ,..., ), n 1; 2. n -ary operation symbols o ( ), n 1; 3. constant symbols c . These collections of symbols are assumed to be disjoint.
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Model Theory - Copyright c 19821998 by Stephen G. Simpson...

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