1980年代半ば,米国中西部のモデル 理論,そして未来-モデル理論賛歌

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1 RIMS

2 1 2 3

3 University of Illinois at Chicago (UIC) John T Baldwin

4 University of Illinois at Chicago (UIC) John T Baldwin Y N Moschovakis, Descriptive Set Theory North Holland, 1980

5 University of Illinois at Chicago (UIC) John T Baldwin Y N Moschovakis, Descriptive Set Theory North Holland, 1980 UIC TA

6 1980 UIC UIC Logic Course Herbert B Enderton, A Mathematical Introduction to Logic, Academic Press, 1972

7 1980 UIC UIC Logic Course Herbert B Enderton, A Mathematical Introduction to Logic, Academic Press, 1972 Baldwin John T Baldwin, Fundamentals of Stability Theory, Springer, 1988

8 1980 UIC UIC Logic Course Herbert B Enderton, A Mathematical Introduction to Logic, Academic Press, 1972 Baldwin John T Baldwin, Fundamentals of Stability Theory, Springer, 1988 Pillay Anand Pillay, An introduction to stability theory, Clarendon P, Oxford, 1983

9 Mid-West Model Theory University of Illinois at Chicago John T Baldwin, David Marker University of Illinois Urbana-Champaign Lou van den Dries University of Notre Dame Anand Pillay, Sergei Starchenko (Byunghan Kim)

10 M M φ T T T

11 M M φ T T T 1 Th(M) 2 T

12 L A, B L- L- φ A = φ B = φ A B A B

13 L A, B L- L- φ A = φ B = φ A B A B A B A B L

14 1 T L- A, B T A B = A B

15 1 T L- A, B T A B = A B =

16 T L- T A, B T

17 T L- T A, B T T

18 T I(T, κ) T 1 κ T κ I(T, κ)

19 T I(T, κ) T 1 κ T κ I(T, κ) I(T, κ) = 1 κ- I(T, κ) = 1 T κ

20 2 (Morley, 1965) κ I(T, κ) = 1 κ I(T, κ) = 1 M Morley, Categoricity in power, TAMS, 1965

21 2 (Morley, 1965) κ I(T, κ) = 1 κ I(T, κ) = 1 M Morley, Categoricity in power, TAMS, 1965 ACF 0 0 ACF p p

22 T

23 L ω1 ω 3 (Scott ) L A L- L ω1 ω φ L- B B = φ B A

24 L ω1 ω 3 (Scott ) L A L- L ω1 ω φ L- B B = φ B A L ω ω L ω ω

25 Vaught Martin Martin

26 Vaught 4 Ṭ I(T, ℵ 0 ) > ℵ 0 I(T, ℵ 0 ) = 2 ℵ 0 S Shelah, L Harrington and M Makkai, A proof of Vaught s conjecture for totally transcendental theories, Israel J M 1984

27 Vaught 4 Ṭ I(T, ℵ 0 ) > ℵ 0 I(T, ℵ 0 ) = 2 ℵ 0 S Shelah, L Harrington and M Makkai, A proof of Vaught s conjecture for totally transcendental theories, Israel J M 1984 Vaught T

28 Martin 1 T S(T ) L ω ω { φ p φ : p S(T )} L ω1 ω L 1 (T )

29 Martin 1 T S(T ) L ω ω { φ p φ : p S(T )} L ω1 ω L 1 (T ) 5 I(T, ℵ 0 ) < 2 ℵ 0 T M Th L1 (T )(M) T T L 1 (T )

30 Martin 2 E Bouscalen, Martin s Conjecture for ω-stable theories, Israel J M 1984 C W Wagner, On Martin s Conjecture, Annals of Math Logic, 1982

31 Martin Vaught 6 Ṃartin = Vaught

32 Martin 1 7 ( Martin ) T S(T ) ℵ 0 (1) I(T, ℵ 0 ) < 2 ℵ 0 T M Th L1 (T )(M) (2) T L 1 2 ℵ 0

33 Martin 1 7 ( Martin ) T S(T ) ℵ 0 (1) I(T, ℵ 0 ) < 2 ℵ 0 T M Th L1 (T )(M) (2) T L 1 2 ℵ 0 T T L 1 (T ) T T L 1 -

34 Martin 2 SMC Wagner, 1982 PhD Thesis Baldwin ω-

35 Martin 2 SMC Wagner, 1982 PhD Thesis Baldwin ω- M I, On the strong Martin Conjecture, J S L, 1991 ω- Martin

36 Martin 2 SMC Wagner, 1982 PhD Thesis Baldwin ω- M I, On the strong Martin Conjecture, J S L, 1991 ω- Martin Shelah Vaught L 1 (T )

37 Martin 2 SMC Wagner, 1982 PhD Thesis Baldwin ω- M I, On the strong Martin Conjecture, J S L, 1991 ω- Martin Shelah Vaught L 1 (T ) ω- L 1 (T )-

38 S Shelah, Classification Theory and the Number of Nonisomorphic Models, North-Holland, 1978

39 S Shelah, Classification Theory and the Number of Nonisomorphic Models, North-Holland, 1978 S Shelah, Classification Theory for Abstract Elementary Classes, Studies in Logic vol 18 and 20, Col Pub 2009 J T Baldwin, Categoricity Univ Lect Series, vol 50, Amer Math Soc, 2009

40 1980 Hrushovski

41 (R, +,, 0, 1, <) RCF

42 (R, +,, 0, 1, <) RCF (C, +,, 0, 1) ACF p p 0

43 L M L- M M

44 L M L- M M (C, +,, 0, 1) Zilber Zilber Hrushovski 1980

45 Hrushovski, Zilber 1991

46 Hrushovski, Zilber 1991 Zariski P

47 Hrushovski, Zilber 1991 Zariski P M P M

48 Hrushovski, Zilber 1991 Zariski P M P M Zilber

49 Hrushovski, Zilber 1991 Zariski P M P M Zilber E Hrushovski and B Zilber, Zariski Goemetries, J of AMS, 1996

50 (M, <, ) M < M M

51 (M, <, ) M < M M (R, +,, 0, 1, <) 8 (Pillay, Steinhorn, Knight) Ṃ M N N

52 Wilkie 9 Ṛ exp A J Wilkie, Model completeness results for expansions of the ordered fields of real numbers by restricted Pfaffian functions and the exponential function, J of the AMS, 1996

53 Pila-Wilkie R (R, +,, 0, 1, <) 10 (Pila-Wilkie, 2006) X R n R ε > 0 t 0 = t 0 (ε) t t 0 X trans (Q, t) t ε J Pila and A J Wilkie, The rational points of a definable set, Duke Math J, 133, No 3, 2006,

54 R exp

55 R exp R an

56 R exp R an R an,exp

57 R exp R an R an,exp

58

59 Mordell-Lang Hrushovski 11 ( Mordell-Lang ) k 0 KA X A K Γ A(K ) Stab X (1) (2) (1) X Γ X (2) A B k 0 S S k 0 X 0 B S k0 K h X = a 0 + h 1 ( X 0 k0 K ) E Hrushovski, The Mordell-Lang conjecture for function fields, Jour AMS, 1996

60 Hrushovski X Γ X k 0 k 0 X 0 X 0 h

61 Hrushovski X Γ X k 0 k 0 X 0 X 0 h 0 Buium

62 Hrushovski X Γ X k 0 k 0 X 0 X 0 h 0 Buium Hrushovski Buium 0

63 Hrushovski X Γ X k 0 k 0 X 0 X 0 h 0 Buium Hrushovski Buium 0

64 Pila (2011) J Pila O-minimality and the André-Oort conjecture for C n Ann of Math 173(2011),

65 Pila (2011) J Pila O-minimality and the André-Oort conjecture for C n Ann of Math 173(2011), V C n X V = X X

66 Pila (2011) J Pila O-minimality and the André-Oort conjecture for C n Ann of Math 173(2011), V C n X V = X X André-Oort

67 Pila Pila-Wilkie X

68 Pila Pila-Wilkie X (Siegel ) X

69 Pila Pila-Wilkie X (Siegel ) X X < X

70

71

72

73 B Zilber, Zariski Geometries, Geometry from Logician s Point of View, London Math Soc Lect Note Series, 360, 2010

74 B Zilber, Zariski Geometries, Geometry from Logician s Point of View, London Math Soc Lect Note Series, 360, 2010

75 M I, and Boris Zilber, Notes on a model theory of a quantum 2-torus T 2 q for generic q, arxive: v1, L ω1 ω- L ω ω

76 Logics

77 Logics

78 Logics L ω ω

79 Logics L ω ω

80

Lebesgue可測性に関するSoloayの定理と実数の集合の正則性=1This slide is available on ` `%%%`#`&12_`__~~~ౡ氀猀e

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