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1 2007 I II III 1, 2, 3, 4, 5, 6, (!) ?

2 2 1! 4, 5 1? ?? 2 1 1, 2 1, 2 1, 2, 3,...

3 3 1, 2 1, 3? , 2 2, 3? k,l m, n k,l m, n kn > ml...? 2 m, n n m

4

5 5 3 4 Q 4 Dedekind Schnitt

6 6 Q A A A, A ) 1. A A = Q 2. a, a ( a A, a A ) a<a 3. A 5 2 A A = A A A A Q A, A A, A A A A = { } x Q x A B = {x Q x<0 x 2 2} B B B A A () 1. x x a A a x A x a A a x A 5 A

7 7. A A a A x A a<x x a a A x A,, A B A B A B B A A B A B A B B A A B A = B A B A B 1. A B A = {x Q x 1} B = {x Q x 0 x 2 2} A B

8 8 1. A A A. A A A A 2. A B A B A B A = B 7. A B A B A B A B A = B A = B 3. A, B, C A B B C A C. A B A B B C B C A C A C 4. A, B A B A B. A B A B A B A B B A b 0 A A b 0 A A a a<b 0 1 A B A B 8 7 A = B? = A = B

9 A B C C = {a + b a A,b B } C C A B = C A B C A B 8 9 A B. C A, B C a A b B a A b B a + b<a + b a + b C C C C C C C = Q c C c C c c C c = a + b a A b B c c c b a c b A c C c C c C c<c C c 1 c 1 = a 1 + b 1 A B a 2 A b 2 B a 2 <a 1 b 2 <b 1 a 2 + b 2 C c 1 C 2. 1 A, B A B = C C = {x Q x>1 (x 1) 2 > 2} well-defined

10 10 1. [] A, B A B = B A. A B = C B A = D C = {a + b a A,b B } D = {b + a b B,a A } C = D C = D 2. [] A, B, C (A B) C = A (B C) [] A A O = A A O. O O = {x Q x>0} A A O = B B b A a O o b = a + o o > 0 a + o >a a + o A B A B A B b A a O o b = a + o o 0 a + o a a + o A B A B A 2 B = A 3 1. O P A O = A = A P A O = P. A A P P O = P 1 P O = O P = O O = P

11 11 (zero) 3 O 4. [ ] A A B = O B A. A B B = {o a a A, o O } B B B A B A a A a a<a a B B B B B B = Q B b = o a A a o B B B b a + b 0 b b =(b + a) (a + b) > 0 b<b B b b = o a o o 2 o > o 2 b 1 = o 2 a b 1 B b 1 <b B B B A B = O A B = C B C c A a A a o c = a a + o a >a o > 0 c > 0 C O C O o A a A a a+a 2 A A a+a 2 A a = a+a 2 a+a 2 A a = a+a 2 a A, a A a a a a< o 2 C c c = a a + o 2 o >c 1 o C o O C C O 2 A B = O 4 A 2. A B C A B = O = A C

12 12 B = C. A B = O (A B) C = O C 1, 2 A C = O B 3 C B = C 11 O A A O A A 3. A ( A) =A. A ( A) =O A B A B = A ( B) A B A B C A C B C. A C = E, B C = F A B A B E F E F 11!

13 13 4. A B A B C A C B C. 5.3 A C B C A C B C A C = B C C A = B A B A A O A O O 5. A A O A A O A. A O A 4 A A O A O A B C = {a b a A,b B }, C C A B = C (( A) B) A O B A B = (A ( B)) A O B ( A) ( B) A, B O well-defined

14 14. A B A B A B = C A, B C A, B A B C C C C C C = Q C c c x C c = a b A a 0 y = x a b y y B x = a y C C c c = a b A B a 1 <a b 1 <b a 1 a 2 c 1 = a 1b 1 c 1 C c 1 <c C C 3. A B A B = C C A = {x Q x>0 x 2 > 2} B = {x Q x>0 x 2 > 3} C = {x Q x>0 x 2 > 6} [] A, B A B = B A. A, B A B = C, B A = D C = {a b a A,b B } D = {b a b B,a A } C = D C = D A B A ( A) B = E B ( A) =F E = {(o a)b o O a A, b B } F = {b (o a) o O a A, b B } ( A) B = B ( A) (( A) B) = (B ( A))

15 15 2. [] A, B, C (A B) C = A (B C) [] A A I = A A I. I I = {x Q x>1} I 3 A A I = B A a I i i > 1 a i >a B A A B A a a >a 1 A 0 a 1 a a 1 > 1 a a 1 I a = a 1 a a 1 B A B A B A A I = A A A I = (( A) I = ( A) =A 3 6. I J A I = A = A J A I = J. I, J 3 I 3 J I = J 1 J 3 J I = I J = I

16 16 3 I 4. [ ] O A A B = I B A. A A O B { } i B = a a A, a > 0 i > 1 A B = I A B = C C c I i A a A a c = a i a 0 <a<a i <c C I C I 4 A a A a a a A a a a 0 I i I i 1 a, a a i 1 a <i C I C I C = I A A ( A) D = I D D A ( D) =( A) ( ( D))=( A) D = I B = D A 4 A A

17 17 A B A B = A ( B) A B A B. A B A B A B = C C C [] A, B, C A (B C) =(A B) (A C). D = A (B C) E =(A B) (A C) D ={a (b + c ) a A,b B,c C } E ={a b + a c a A,b B,c C } D = E 2. O I. O I 1 O I

18 18 O 7 D 15 Dedekind D 3 D a A = {x Q x a} A A a )a( = )a(, )a( 16 Q )a( ={x Q x a} Q = {A A } = {)a( a Q} a

19 19 D Q a )a( 7. Q A, B A B Q A Q A B Q A Q A O 17. A B = C, A = D, A B = E, A = F A B Q a, b A = )a(, B = )b( C = {x Q x a + b}, D = {x Q x a}, E = {x Q x ab}, F = {x Q x a 1 } C a + b C a + b C a + b c δ = c a b a + δ A b + δ B c =(a + δ)+(b + δ) 2 c C a + b C A B = {x Q x> a} a B 8. a, b )a + b( = )a( )b(, )ab( = )a( )b( a b )a( )b(. 7 )a( )b( a + b )a( )b( ab )a + b( = )a( )b( )ab( = )a( )b( )a( ={x Q x a} )b( ={x Q x b} a b )a( )b( )a( )b( D A a α, β 18,,,,,, <, +,, ( ) 1 17 Q 18

20 α<β α<c<β c. α = A, β = B α <β A B A B c B c A c c C = {x Q x c} C A C B A C B α <c<β !!

21 X X α α ρ ρ α ρ X X X 3 X 1 π 3 X. X X α A α A A = α X well-defined well-definedness A A A A = Q A α A α A A A X α A α A A α A α A A A X A α A A X A X B A B A B A A a B A a A α B A α B A α B X D? : A α 21 22

22 22 D. A A A A 9 D D X Y φ: X Y : φ X x 1, x 2 φ(x 1 + x 2 )=φ(x 1 )+φ(x 2 ) X x 1, x 2 φ(x 1 x 2 )=φ(x 1 ) φ(x 2 ) X x 1, x 2 x 1 x 2 φ(x 1 ) φ(x 2 )

23 23

2 2 1?? 2 1 1, 2 1, 2 1, 2, 3,... 1, 2 1, 3? , 2 2, 3? k, l m, n k, l m, n kn > ml...? 2 m, n n m

2 2 1?? 2 1 1, 2 1, 2 1, 2, 3,... 1, 2 1, 3? , 2 2, 3? k, l m, n k, l m, n kn > ml...? 2 m, n n m 2009 IA I 22, 23, 24, 25, 26, 27 4 21 1 1 2 1! 4, 5 1? 50 1 2 1 1 2 1 4 2 2 2 1?? 2 1 1, 2 1, 2 1, 2, 3,... 1, 2 1, 3? 2 1 3 1 2 1 1, 2 2, 3? 2 1 3 2 3 2 k, l m, n k, l m, n kn > ml...? 2 m, n n m 3 2

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