(Sakaé Fuchino) fuchino@isc.chubu.ac.jp 2002 8 24 2002 11 11 2002 11 23 2002 11 29 2003 10 30 footnote 0 2 1 3 1.1...................... 3 1.2....................... 5 1.3.......................... 6 1.4.......................... 8 1.5....................... 10 2 11 2.1......................... 11 2.2................ 14 2.3.......................... 19 2.4........................... 21 2.5...................... 24 2.6.......................... 27 3 29 3.1...................... 29 3.2 CH........ 31 3.3....................... 33 36 1
0 2002 7 26 ( ) 7 29 ( ) 2002 2002 meager 1 2 3 1 2 1 (forcing) [12] 1 2.1 2.6 Sabine Koppelberg 1995 [10] 3 1 2 1 2 [15], [2], [3] [15] [2] [3] [11] [7] [11] [2] [7] 1 [14] 2
1 1.1 2 R R R ω N 2.2 N ω ω = {0, 1, 2,...} n ( ) n = {0, 1,..., n 1} 0 = ( ) 1 = {0} = { }, 2 = {0, 1} = {, { }},... ω 0, 1, 2,... 3 X, Y X Y = {f : f X Y } 4 ω 2 2 = {0, 1}! ω 2 0, 1 f : X Y X X f X X f f X = {f(x) : x X } Y Y f 1 Y Y f f 1 Y = {x X : f(x) Y } ω 2 2 = {0, 1} ω 5 2 ω 2 O ( ω 2, O) O ω> 2 = {s : n ω s : n 2} 2 Zermelo-Fraenkel ZFC ZFC [6] [11] 3 2.1 4 f X Y f : X Y 5 X i, i I X i, i I Xi i I X i = {f : f I i I f(i) X i} i I X i X i I Xi = I X 3
6 s ω> 2 [s] = {f ω 2 : s f} 7 [s] ω 2 ω 2 O {[s] : s ω> 2} R ω 2 R R 2 ω 2 f (f(2n), f(2n+1)) ( ω 2) 2 ω 2 ( ω 2) 2 R ω 2 R {, } I = [0, 1] k : ω 2 I ; f f(n) 2 (n+1) n ω W = {f ω 2 : f(n) } Q = {x R : x } s ω> 2 s 0 ω 2 n ω (s s(n) n dom(s) 0)(n) = 0 n dom(s) s 1 ω 2 n ω (s s(n) n dom(s) 1)(n) = 1 n dom(s) 1 (a) k (b) k (c) k W k `W W I \ Q (a): x I 2 0. x 0 x 1 x 2... 0 = 0.000..., 1 = 0.111... f : ω 2 f(n) = x n k(f) = x k 6 ω> 2 <ω 2 7 f, g f g g f 4
(b): Q I O = {(q 0, q 1 ) : 0 q 0 < q 1 1, q 0, q 1 Q } I O k ω 2 (q 0, q 1 ) O k 1 (q 0, q 1 ) = {[s] : s ω> 2, q 0 < k(s 0), k(s 1) < q 1 } ω 2 ω 2 (c): k `W W I \ Q (b) s ω> 2 k [s] \ Q I \ Q q 0 = k(s 0), q 1 = k(s 1) q 0, q 1 Q k [s] = (q 0, q 1 ) (I \ Q ) k [s] I \ Q ( 1) Q ω 2 \ W I ω 2 8 Q ω 2 \ W I ω 2 1 I ω 2 1.2 (X, O) Bor(X) Bor(O) O σ- 9 Bor(X) X = (X, O) Bor(X) = {A : O A P(X), A } Bor(X) Bor(X) = {B(α) : α < ω 1 } B(α), α < ω 1 B(0) = O, B (α) = B(α) {X \ b : b B(α)} B(α + 1) = {y : y B (α) } B(γ) = {B(α) : α < γ}, γ limit 10 F σ - G δ -F σ - G δ - S 1 F σ - G δ - 8 2.4 9 X S X σ- S 10 3.1 5
1.3 (X, O) D X X (dense) O O D O R Q R (X, O) Y X X nowhere dense O O O O O Y = ω 2 Y nowhere dense s ω> 2 t ω> 2 (s t Y [t] = ) Y X (meager set) X nowhere dense Y n X, n ω Y = n ω Y n 2 X = (X, O) (1) Y X nowhere dense Y cl(y ) nowhere dense (2) Y X Y Y X F σ (1): O O, O Y nowhere dense O O O Y = O cl(y ) = cl(y ) nowhere dense (2): Y = n ω Y n Y n nowhere dense Y = n ω cl(y n) Y Y Y F σ (1) cl(y n ) nowehre dense Y ( 2) ω 2 d f, g ω 2 d(f, g) = n ωf(n) g(n) 2 (n+1) ω 2 ω 2 + ω 2 ω 2 3 X X 4 ω 2 ω 2 ω 2 11 11 X ω 2 ω 2 \ X X ω 2 ω 2 \ X 6
M = {X ω 2 : X } 4 ω 2 M! 5 (1) X ω 2 X ω 2. (2) X ω 2 Y X Y (3) X n ω 2, n ω ω 2 n ω X n ω 2 S I P(S) 12 S σ- (0) S I; (1) x S {x} I; (2) X I, Y X Y I; (3) X n I, n ω n ω X n I I S σ- (1) (3) S I (0) S σ- S S σ- I S σ- X I X I S 4 5 6 M ω 2 σ- I P(S) σ- I I I I = {Y P(S) : Y I Y Y } X = (X, O) σ- I P(X) Borel supported X I I I F σ - 2 6 7 M Borel supported ω 2 σ- M I = {X I : X I } 1, Q ω 2 \ W 8 (1) X M I k X M. (2) X M k 1 X M I. 12 P(S) = {X : X S} I P(S) I S 7
1.4 B P(S) σ- µ : B R + B µ (0), (1), (2) (0) µ( ) = 0; (1) µ(s) = 1; (2) A n B, n ω m, n ω A m A n = µ( A n ) = ( k ) µ(a n ) = lim µ(a n ) k n ω n ω (2) m n ω A n A n, n ω (0) (2) (2 ) A n B, n < m µ( A n ) = µ(a n ) n<m n<m (2 ) (3) X Y µ(x) µ(y ) (4) A n B, n ω µ( A n ) µ(a n ) n ω n ω B n B, n ω n ω B n A n n ω A n = n ω B n (2) (3) µ( n ω A n) = n ω µ(b n) n ω µ(a n) (1) (2) 9 (1) O ω 2 B = Bor(O) µ : B R + s n 2 µ([s]) = 2 n (2) O I B = Bor(O) µ : B R + [a, b] I µ([a, b]) = b a µ σ- B P(S) X S µ (null-set) µ(x ) = 0, X X X B S = ω 2 S = I 9 (1) (2) µ N = {X ω 2 : X } (1) 9 (1) µ (2) (1) n=0 8
10 (1) X ω 2 ε > 0 ω 2 O X O, µ(o \ X) < ε (2) X ω 2 G δ X ω 2 X X 11 N ω 2 Borel-supported σ- ( 11) N I = {X I : X } 8 12 (1) X N I k X N. (2) X N k 1 X N I. M N σ- 13 X ω 2 X N ω 2 \ X M ω> 2 ω> 2 = {s n : n ω} 13 n, j ω t n,j ω> 2 (1) s n t n,j ; (2) j < j t n,j t n,j ; (3) dom(t n ) > n + j G j = n ω [t n,j] (3) 8 (4) µ(g j ) µ([t n,j ]) 2 (n+j) = 2 (n 1) n ω n ω (2) j < j G j G j G = G j j ω µ(g) lim j ω µ(g j ) = 0 G F j = ω 2 \ G j F j nowhere dense s ω> 2 s = s i [t i ] [s i ] [t i ] F j = ω 2 \ G = j ω F j ( 13) 13 30 9
1.5 ω ω = {f : f ω ω } ω ω 14 ω (Baire space) ω> ω = {t : t : n ω, n ω} t ω> ω [t] = {f ω ω : t f} O = {[t] : t ω> ω} ω ω ω ω ω 2 I ω 2 I Mω ω, Nω ω ω ω σ- Mω ω Nω ω M, N n, k ω a k n = n {}}{ 1,..., 1, 0 k n {}}{ 1,..., 1, 1 k W I 1 j : ω ω W j(f) = a 0 f(0) a 1 f(1) a 2 f(2) 15 ω ω µ s ω> ω n = dom(s) µ([s]) = 2 (s(i)+1) i<n W 4 ω 2 14 (a) j ω ω W (b) j ω ω W I 14 3 5 15 j(f) a 0 f(0), a1 f(1), a2 f(2),... ω 10
2 2.1 15 X < X X, < (wellordered set well-ordering) 16 (1) X, < 17 (2) X a < min a X, < C X < `C = < (C C) = { x, y : x C, y C x < y} C < `C, < (C C) < C, < X, `X = { x, y X 2 : x y} X, < X, < 15 (2) a X X X, < = N, < X x X X X x y X, x < y z X x < z < y 18 y x X (successor) x X y x = min{z X : z < y}. X z X x X x X (limit) X X x X z X x X x < z x < z x < z x z, z = 16 < X (well-ordering) < X 17 X < (i), (ii), (iii) (i) x X x < x ; (ii) x, y, z X x < y y < z x < z; (iii) x, y X x < y y < x X < 18 {z X : x < z} 11
16 19 N R R {x R : 0 < x} {x R : 2 < x 2 } 17 X, < F : X X 20 x X x F (x) F 0 : X X F 0 F 0 (x) < x x X x 0 = min{x X : F 0 (x) < x} F 0 (x 0 ) < x 0 x 0 F 0 (x 0 ) F 0 (F 0 (x 0 )) F 0 F 0 (x 0 ) < x 0 F 0 F 0 (F 0 (x 0 )) < F 0 (x 0 ) ( 17) 18 X, < X id X X, < 21 id X X, < X, < F : X X F F 1 17 x X x F (x) x = F 1 (F (x)) F (x), x = F (x) F = id X ( 18) 19 X, < Y, < X, < Y, < F : X Y G : X Y X, < Y, < F 1 G X, < X, < 19 F 1 G = id X F G = F ( 19) ( 19) 20 X, < X (initial segment) X C c C, x X, x c x C. C C X (proper initial segment) C X x = min(x \ C). x X < x = {y X : y < x} C = X < x y X < x y < x x 19 N 2.3 20 f X Y f : X Y f : X Y x, y X x < y F (x) < F (y) 21 f X, < Y, < f X Y x, x X x < x f(x) < f(y) 12
y C c C c < x < x c C x C x. c X < x 21 X, < X, < X, < 23 22 F 22 F = F (= f F f) F (compatible ) 23 dom F = f F dom(f) 23 X, < Y, < (a) (b) (c) X, < Y, < X, < Y, < Y, < X, < X, <, Y, < (a), (b), (c) F = {f : X X Y Y f : X, < = Y, <} Claim 23.1 f, g F f g g f f, g F dom(f), dom(g) X dom(f) dom(g) dom(g) dom(f) dom(f) dom(g) 19 f = g ` dom(f) f g dom(g) dom(f) g f (Claim 23.1) Claim 23.1 22 f = F dom(f ) = f F dom(f) Claim 23.2 f F 22 f f X Y = { x, y : x X, y Y } ( ) x X x, y f y X Y x X x, y f f(x) = y f f : X Y X ( ) dom(f) f f X Y f rng(f) 23 F f, g F f g dom(f) dom(g) x dom(f) dom(g) f(x) = g(x) 13
f 22 dom(f ) = {dom(f) : f F}, f dom(f ) = {f dom(f) : f F} f X Y X = dom(f ), Y = f dom(f ) f x, y X, x < y f, g F x dom(f), y dom(g) Claim 23.1 g f x, y dom(f) f F x < y f (x) = f(x) < f(y) = f (y) f F f f F f f f F (Claim 23.2) X X, Y Y x = min(x \ X ), y = min(y \ Y ) f = f { x, y } f F f = f f F ( 23) 2.2 24 X, < E(x) X x x X ( ) y < x E x E x X E X, < X x E(x) E ( ) E(x) x X Y = {x X : E(x) } Y X Y x 0 x 0 y X, y < x 0 E(y) ( ) E(x 0 ) x 0 Y ( 24) < 11 x X X x x X 24 X, < E(x) X x (a), (b), (c) (a) X E 14
(b) (c) x X E x X x E x y < x E x E x X E E 24 ( ) x X E(y) y < x E(x) x X (a) E(x) ( ) x x X x (b) x (c) E(x) ( ) 24 x X E(x) ( 24 ) 25 X, < Y G dom(g) = {f : f X Y } 24 F : X Y x X ( ) F (x) = G(F `X < x ) 25 ( ) F X F F ( ) F F {x X : F (x) F (x)} x 0 y < x 0 x 0 F (y) = F (y) F `X < x0 = F `X < x0 F (x 0 ) = G(F `X < x0 ) = G(F `X < x0 ) = F (x 0 ) x 0 F = F ( ) F F = {f : f X I x I f(x) = G(f `X < x ) } F 22 F = F Claim 25.1 F. 24 G F X ( ) F 25 F `S F S f : X Y f `S = f (S Y ) 15
F 26 (Claim 25.1) Claim 25.2 f F, dom(f) = I x I f `X < x F. F X < x subseteqi (Claim 25.2) Claim 25.3 f, g F f g dom(f) dom(g) f = g ` dom(f) f g (Claim 25.3) Claim 25.4 X = {x X : f F x dom(f)}. {x X : f F x dom(f)} = x 0 f = F Claim 25.3 22 f F x 0 dom(f ) = X < x0 f = f { x 0, G(f ) } f F x 0 dom(f ) x 0 (Claim 25.4) F = F F F Claim 25.4 F X F F ( ) ( 25) 24 25 X F 25 X, < Y H, K dom(h) = rng(h) = rng(k) = Y, dom(k) = {f : f X Y } a Y F : X Y a x X ( ) F (x) = H(F (y)) y X x = y K(F `X < ) x X C y C x y x C x y C x C (transitive) 26 : x 0 X X < x0 = dom(e) 16
26 (a), { }, {, { }} {{ }} {, { }, {{ }}} (b) i I t i i I t i i I t i (c) t t {t} 27 T T (a) x T (b) x, y T, x y y x y = x {x} (c) U T U x T U U = x (a): z y x T z x (b): x y (a) x y x {x} y x {x} = y z y \ (x {x}) z x T z T z x {x} T x z x z y T z y. y x (c): x U U x U x z x \ U u U u z T u = z z u U z U z z U z x x U ( 27) 28 (Mostowski ) X, < T π : X, < = T, T π T T X, < π (Step I): T π : X, < = T, π : X, < = T, T {x X : π(x) π (x)} x 0 x 0 y X x 0 π(y) = π (y) π π(x 0 ) π(y) T 27 (b) π(x 0 ) = π(y) {π(y)} π (x 0 ) = π (y) {π (y)} π(x 0 ) = π (x 0 ) 17
x 0 x 0 π `X < x0 = π `X < x0 π(x 0 ) π X < x0 π X < x0 T 27 (c) π X < x0 = π(x 0 ). π X < x 0 = π (x 0 ) π(x 0 ) = π (x 0 ) (Step II): ( ) x X T x π x π x : X < x, < = T x, x X (IIa): x X X X < x = T x =, π = (IIb): x X T x π x π x : X < x, < = T x, x x 0 x < x 0 x X < x0 T x π x π x : X < x, < = T x, (IIa) x 0 X X < x0 (IIb1): x 0 x T π x0 = π x { x, T x } π x0 : X < x0 = T x {T x }, T x {T x } x 0 (IIb2): x 0 (Step I) x < x 0 T x, π x x < y < x 0 π y `X < x X < x, < π y X < x, π x π y, T x = {z T y π x0 = {π x : x < x 0 } : z π y (x)} π x0 : X < x0, < = {T x : x < x 0 } {T x : x < x 0 } x 0 (Step III): (Step II) (Step I) (Step I) (IIb2) x X T x π x π x : X < x, < = T x, x, y X, x < y π x π y, T x = {z T y : z π y (x)} (IIb1) (IIb2) T x, π x (x X) X, < ( 28) 29 X, < Y X Y, < π X : X, < T, π Y : Y, < (S, ) π Y = π X `Y S T S T π X `Y : Y, < = (π Y, ) 28 π Y = π X `Y S T π X S T ( 29) 18
2.3 On On (1) On 27 28 29 (2) α (3) X, < α, α, α (ordinal number) On = {α : α }. 30 α, β On α < β α β. 30 α α, α, α α α, α, ( 30) 31 (a) (b) (c) (d) (e) (f) β α On β On On α On α = {β On : β < α} α, β On α = β α β β α α On α α α, β On β < α β α M M On < M = sup M (g) On (a): β α On β δ γ β α δ β β α β α β (b): α = {x : x α} (a) 27 28 α On β α β On 29 On On 30 On Ord 19
(c): 23 α β α β β α α β α β α α = β α β γ α γ α α = γ α β β α β α (d): α α α x x x α (e): β < α β α α β α (d) β α\β β α β α ξ α \ β β ξ β ξ η ξ \ β β η < ξ ξ β = γ < α (f): M M M On β M β M (e) M M < M (e) (g): (a) On = On On (f) On On (d) ( 31) 32 (a) On On <On (b) < On C On C On C = min C (c) (d) 0 = < < α On α + 1 = α {α} On α + 1 On, < α (a): 31 (c), (e) (b): C On C On C 31 (e) C C (c): vacantly 31 (e) < On (d): α + 1 On α + 1 31 (e) α + 1 α ( 32) 32 0 =, 1 = 0 + 1, 2 = 1 + 1,... n + 1 = n + 1 31 On n = {0, 1,..., n 1} 32 31 n + 1 = n + 1 n + 1 n + 1 32 (d) +1 32 n 1 n 1! 20
α On α (limit ordinal) α, β On β < α β β α, < N n On n N = {n On : n }. N N On N = N 31 (f) N On N N ω 2.4 Card n n = {0, 1,..., n 1} E n E = n n E (AC Axiom of Choice) F f : F F a F f(a) a 33 33 X X, < X < X X X = min{α On : X α } X, < X < X, < α α On X α {α On : X α } X well-defined On κ 33 f F 21
α < κ α κ κ (cardinal) 35 κ α < κ α κ 2 34 x Cantor 1873 34 x f : x P(x) g : P(x) x 2 34 h : x P(x) f : x P(x); a {a} g : P(x) x 2; y ch y!ch y : x 2 y a x 1 a y ch y (a) = 0 a y h : x P(x) y = {a x : a h(a)} y P(x) h a x h(a ) = y a y y a h(a ) = y a y y a h(a ) = y ( 34) 35 A C g : A C h : C A A C 35 C = g A R C R C \ C < = R R { c, c : c C, c C \ C } < C! C < C π : C, < = α, β = π C α β α π g : A β A β A C 36 h C A A = C j : A A k : C C f = k 1 j A C ( 35) 34 2 = {0, 1} x 2 = {f : f : x 2} 35 36 Card 22
36 (a) (b) (c) ω M sup M (= M) (a) n m < n f : m n m < n n 0 n = n {n } n < n f : m n m 0 m = m {m } m < m f 2 f(m ) = n f : m n f (k) = { f(k) f(k) < m 0 f m n m < n < n n ω n f : n ω l < n f (l) = min{n, f(l)} f : n n + 1 f (b): α ω α α + 1 α f β + 1 f(β) = β α β < ω ω β < α β = α f α + 1 (c): M κ M = κ M M 31 (f) M κ < M f : κ M κ < λ λ M λ M f : κ λ f (α) = { f(α) f(α) < λ 0 f λ M λ 37 x, y x (a) (b) ( 36) x = y f : x y (i) x y (ii) (iii) f : x y g : y x 23
κ µ κ < µ x x = κ x = κ 34 37 κ = x < P(x) κ + = min{µ Card : κ < µ} κ + κ ℵ : On On 37 38 ℵ(0) = ω ℵ(α + 1) = (ℵ(α)) + ℵ(λ) = sup{ℵ(α) : α < λ}, λ. α On ℵ(α) ℵ α ℵ α ω α Card κ ℵ α α On µ = min(card \ ({ℵ α : α On}) ω). ℵ α, α On α ℵ α < µ {ξ On : ξ < µ} {ℵ α : α On} Card \ ω = {ℵ α : α On} X ℵ 0 (countable set) X = ℵ 0 X X > ℵ 0 X 39 α κ = ℵ α α = β +1 β κ = (ℵ β ) + κ (successor cardinal) α κ = ℵ α κ (limit cardinal) 2.5 κ λ κ + λ, κ λ, κ λ A, B A = κ, B = λ A B = 40 κ + λ = A B, κ λ = A B, κ λ = B A 37 ℵ 38 On ZFC 25 39 40 A = κ {0}, B = λ {1} 24
A B A = A, B = B A B = A B A B = A B = A B = A B, B A = B A f λ 2 f 1 {1} P(λ) P(λ) = λ 2 = 2 λ 38 κ, λ, µ (a) κ + λ = λ + κ, κ λ = λ κ (b) (κ + λ) + µ = κ + (λ + µ), (κ λ) µ = κ (λ µ) (c) κ (λ + µ) = κ λ + κ µ (d) (κ λ) µ = κ µ λ µ (e) κ λ+µ = κ λ κ µ (f) (g) (κ λ ) µ = κ λ µ κ κ, λ λ ; (κ, λ ), κ + λ κ + λ, κ λ κ λ, κ λ κ λ α, β On max(α, β) α β On 2 = On On < (α, β) < (γ, δ) max(α, β) < max(γ, δ) ( ) max(α, β) = max(γ, δ) α < γ ( ) max(α, β) = max(γ, δ) α = γ β < δ. 39 < On 2 < X On 2 X < α 1 = min{max{α, β} : (α, β) X} X 1 = {(α, β) X : max{α, β} = α 1 } α 2 = min{α : β (α, β) X 1 } X 2 = {(α, β) X 1 : α = α 2 } α 3 = min{β : (α 2, β) X 2 } < (α 2, α 3 ) X ( 39) (On, ) (On 2, <) 23 K : (On 2, <) (On, ) 40 ν On ν ν On 2 < ν ν = {(α, β) : (α, β) < (0, ν)} 25
(α, β) ν ν (ξ, η) < (α, β) max{ξ, η} max{α, β} < ν (ξ, η) ν ν ν ν On 2 < < (0, ν) On 2 \ ν ν ( 40) K ν ν (On, ) On 41 (a) n, m ω K((m, n)) < ω (b) κ K((0, κ)) = κ (c) κ κ κ = κ (a): k = max{m, n}+1 (m, n) < (0, k) 40 {(α, β) : (α, β) < (0, k)} = k k k k < ω K((m, n)) < K((0, k)) < ω (b) (c): κ Card, κ ℵ 0 κ = ℵ 0 40 (a) K((0, ω)) = {K((m, n)) : m, n ω} = ω (b) K `ω ω ω ω ω ℵ 0 ℵ 0 = ℵ 0 λ < κ (b) (c) κ (b) (c) κ κ κ K((0, κ)) = K κ κ κ α < κ α α = α < κ K((0, α)) < κ (0, κ) (0, α), α < κ K((0, κ)) κ κ = ℵ 0 K `κ κ κ κ κ κ κ = κ ( 41) 42 (a) κ, λ κ ω λ ω κ + λ = max{κ, λ} κ, λ 0 κ λ = max{κ, λ} (b) κ, λ X κ x X x λ X max{κ, λ} (a): κ λ 0 κ 0, λ 0 (κ {0}) (λ {1}) κ λ κ + λ κ λ µ = max{κ, λ} 41 µ κ + λ κ λ µ µ = µ (b): X = X κ X = {x α : α < κ} 41 x α α < κ x α λ x α = {a α,β : β < λ} g : κ λ X g((α, β)) = a α,β g 37 (b) (a) X κ λ = max{κ, λ} ( 42) 41 37 (b) κ X f x α = f(α) 26
43 κ, µ 2 µ 2 κ, ω κ µ κ = 2 κ κ κ = 2 κ 2 κ µ κ (2 κ ) κ = 2 κ κ = 2 κ κ = 2 κ 41 (c) ( 43) κ + λ, κ λ κ λ 3.2 Saharon Shelah Shelah [8] 2.6 α On X α α (cofinal) β < α γ X β γ α cf(α) cf(α) = min{ X : X α, X α } 44 κ Card cf(κ + ) = κ + X κ + κ + β X β κ X κ 42 (b) κ + = κ + = X κ κ = κ ( 44) κ cf(κ) = κ κ (regular) 44 ZFC cf(ℵ ω ) = ω {ℵ n : n ω} ℵ ω 34 45 König κ κ cf(κ) > κ {f α : α < κ} cf(κ) κ f cf(κ) κ \ {f α : α < κ} {α ξ : ξ < cf(κ)} κ κ f : cf(κ) κ f(ξ) = min(κ \ {f α (ξ) : α < α ξ }) {f α (ξ) : α < α ξ } < κ κ \ {f α (ξ) : α < α ξ } = α < κ f f α α < α ξ ξ < cf(κ) f f(ξ) f α (ξ) ( 45) 46 (König) κ cf(2 κ ) > κ 27
κ (2 κ ) = 2 κ X α, α < κ α < κ X α < 2 κ κ (2 κ ) = α<κ X α f : κ 2 f(α) = min(2 κ \ {f(α) : f X α }) α < κ f X α f α<κ X α X α ( 46) 28
3 α < ( = ) 25 25 42 3.1 1.2 ω 2 ω 1 B B(0) = {O : O ω 2 }; B(α + 1) = { Y : Y S Y }, S = B(α) { ω 2 \ x : x B(α)} ; γ < ω 1 B(γ) = α<γ B(α). B = α<ω 1 B(α) 47 (a) F P( ω 2) ω 2 σ- B F (b) α β < ω 1 B(α) B(β) (c) B = Bor( ω 2). (a): α < ω 1 B(α) F α = 0 F ω 2 B(α) F B(α + 1) F B(α + 1) F σ-γ < ω 1 β < γ B(β) F B(γ) = β<γ B(β) F (b): β < ω 1 α β B(α) B(β) β = 0 β = α + 1 B(α + 1) x B(α) x = {x} B(α + 1) B(α) B(α + 1) β < ω 1 B(β) = β <β B(β ) β < β β (c): (a) B ω 2 B B(0) B x B x B(α) α < ω 1 ω 2 \ x = { ω 2 \ x} B(α + 1) B X B x X r(x) = min{α < ω 1 : x B(α)} R = {r(x) : x X} R ω 1 R α α < ω 1 ω 1 = R 42 (b) ω 1 42 3 1.1 29
(b) r( ) X B(α ) X B(α + 1) B ( 47) 48 (a) O ω 2 O = 2 ℵ 0 (b) Bor( ω 2) = 2 ℵ 0 (a): 3 O {[s] : s ω> 2} ω> 2 = ℵ 0 43 ω> 2 = {s n : n ω} ϕ : P(ω) O; x n x [s n] ϕ O P(ω) = 2 ℵ 0 ψ : ω 2 O f ω 2 \ {f} ψ O ω 2 = 2 ℵ 0 (b): B : ω 1 P( ω 2) α < ω 1 B(α) = 2 ℵ 0 47 Bor( ω 2) = α<ω 1 B(α) 42 (b) Bor( ω 2) 2 ℵ 0 (a) 2 ℵ 0 O Bor( ω 2) α = 0 (a) B(0) = O = 2 ℵ 0 B(α) = 2 ℵ 0 B(α) B(α+1) B(α+1) 2 ℵ 0 B = B(α) { ω 2 \ x : x B(α)} η : ω B B(α + 1); f f ω B(α + 1) ω B (2 ℵ 0 + 2 ℵ 0 ) ℵ 0 = 2 ℵ 0 α β < α B(β) = 2 ℵ 0 B(α) = β<α B(β) 42 (b) B(α) 2ℵ 0 B(α) 2 ℵ 0 O B(α) ( 48) 49 X ω 2 (1) X (2) O ω 2 M X = O M 44 (3) O ω 2 N X = O N (1) (2) (1) (3) (2) X = O M B = {O M : O ω 2 M ω 2 } B Bor( ω 2) B σ- X n B, n ω X n = (O n \ M 0 n) M 1 n, n ω M 0 n O n, M 1 n ω 2 \ O n 45 n ω X n B ( X n = O n \ ) ( M 0 ) n Mn 1 n ω n ω n ω n ω 43ω> 2 = { n 2 : n ω} 42 (b) ω> 2 ℵ 0 ω> 2 ℵ 0 44 A, B A B = (A \ B) (B \ A) 45 X n = O n (M 0 n M 1 n) 30
X B X = (O \ M 0 ) M 1 M 0 O, M 1 ω 2 \ O X = ( ( )) cl(o) \ M 0 (cl(o \ O)) M 1 ω 2 \ X = ( ( ω 2 \ cl(o)) \ M 1) ( ) M 0 (cl(o) \ O) ω 2 \ cl(o) M 1 M 0 (cl(o) \ O) ω 2 \ X B ( 49) 3.2 CH 34 κ < 2 κ ℵ 0 < 2 ℵ 0 1.1 R ω 2 R = ω 2 = 2 ℵ 0 (Continuum Hypothesis CH) 2 ℵ 0 = ℵ 1 (ZFC) ZFC 46 R = 2 ℵ 0 R {x α : α < ω 1 } 50 (Sierpiński) (CH) R 2 R 2 = A B, A B = (i) x R A ({x} R) (ii) y R B (R {y}) CH R = {x α : α < ω 1 } A = {(x α, x β ) : α, β ω 1, α > β}, B = {(x α, x β ) : α, β ω 1, α β} R 2 = A B, A B = A, B (i), (ii) : x R x = x α α < ω 1 A ({x} R) = {(x α, x β ) : β < α } A ({x} R) α ℵ 0 B (R {y}) ℵ 0 CH R 2 = A B, A B = A (i) r α R, α < ω 1 C = {x R : α < ω 1 r α, x A} = ( ) A ({r α } R) ℵ 1 R \ C x R \ C {(x α, x ) : α < ω 1 } R 2 \ A = B (R {x }) B 46 [5] 31
( 50) 50 51 (Sierpiński [13]) (CH) f : [0, 1] 2 [0, 1] 1 1 0 0 f(x, y) dx dy 1 1 0 0 f(x, y) dy dx A, B R 2 50 f B y [0, 1] {x [0, 1] : f(x, y) 0} = B ([0, 1] {y}) 1 1 0 0 f(x, y) dx dy = 1 0 0 dy = 0. x [0, 1] {y [0, 1] : f(x, y) 1} = A ({x} [0, 1]) 1 0 1 1 0 0 f(x, y) dy dx = 1 0 f(x, y) dy dx = 1 0 = 1 0 1 0 1 0 1 dy = 1. f(x, y) dx dy ( 51) Tonelli f f dxdy f dydx CH Tonelli f 51 CH 52 (Laczkovich, Friedman, Freiling 47 ) ZFC M M f : [0, 1] 2 [0, 1] M 1 1 0 0 f(x, y) dx dy 1 1 0 0 f(x, y) dy dx M 51 51 47 [3] 32
1. R R = A B (i) s R A ({x} R) < 2 ℵ 0 (ii) y R B (R {y}) < 2 ℵ 0 2. 2 ℵ 0 R 1. ZFC : R = {x α : α < 2 ω } 50 2. non(n ) = 2 ℵ 0 (Martin s Axiom) 50 [1] 53 (Erdős [4]) (CH) analytic functions F z C {f(z) : f F} CH F analytic functions z C {f(z) : f F} f α, α < ω 1 F {α, β} [ω 1 ] 2 S α,β = {z C : f α (z) = f β (z)} S α,β [f, g analytic functions C C 0 {z C : f(z) = g(z)} ]. S = C \ {S α,β : {α, β} [ω 1 ] 2 } S z S f α (z ), α < ω 1 {f(z ) : f F} CH analytic functions F z C {f(z) : f F} C = {v α : α < ω 1 } analytic functions f α, α < ω 1 ( ) β < α f α f β f α (v β ) f β, β < α {v β : β < α} {w n : n ω} α < ω 1 ε n > 0, n ω 0 f α (z) = ε 0 (z w 0 ) + ε 1 (z w 0 )(z w 1 ) + ε 2 (z w 0 )(z w 1 )(z w 2 ) + ( ) f α F = {f α : α < ω 1 } F ( 53) 3.3 σ- σ- f 33
54 (Erdős-Sierpiński Duality Theorem) f : ω 2 ω 2 (1) f f = id ( ω 2) f (2) X P( ω 2) X M f X N. 13 X ω 2 X M ω 2 \ x N M Bor( ω 2) M 7 48 (b) Bor( ω 2) = 2 ℵ 0 2 ℵ 0 = ℵ 1 N {X α : α < ω 1 } X 0 = X ϕ : ω 1 ω 1 ϕ(α) = min{η < ω 1 : ( ξ<η ) ( ) X ξ \ {Xξ : ξ < sup ϕ(β)} 2 ℵ 0 } β<α : ϕ(β), β < α {X ξ : ξ < sup β<α ϕ(β)} {X ξ : ξ < sup β<α ϕ(β)} 4 ω 2 \ {X ξ : ξ < sup β<α ϕ(β)} 2 ℵ 0 2 ℵ 0 X X α, α < ω 1 M X X ξ ξ < ω 1 η = ξ + 1 ϕ(α) {η < ω 1 :...} min{η < ω 1 :...} α < ω 1 ( Y α = ξ<ϕ(α) ) ( ) X ξ \ {Xξ : ξ < sup ϕ(β)} β<α Y 0 = X 0 = X {Y α : α < ω 1 } 2 ℵ 0 ω 2 β < ω 1 α<β Y α = ξ<ϕ(β) X ξ Claim 54.1 X ω 2 β < ω 1 X α<β Y α X ξ < ω 1 X X ξ ξ < ϕ(β) β < ω 1 X X ξ X η = η<ϕ(β) α<β Y α ( 54) ω 2 {Z α : α < ω 1 } Z 0 = ω 2 \ X 0, Z α 2 ℵ 0 Claim 54.2 X ω 2 β < ω 1 X α<β Z α 34
Y 0 =X 0 Y 1 Y 2... {}}{{}}{{}}{{}}{ } {{ }} {{ }... Z 2 }{{} Z 1 } {{ } Z 0 = ω 2\X 0 α < ω 1 f α : Y α Z α f = {f α : α < ω 1 } {(f α ) 1 : α < ω 1 } Claim 54.1 Claim 54.2 f! ( 54) (1), (2) f add(n ), cof(n ) etc. 55 (Erdős-Sierpiński Duality Theorem Bartoszynski-Raisonnier-Stern ) add(n ) = cof(n ) f : ω 2 ω 2 (1) f f = id ( ω 2) f (2) X P( ω 2) X M f X N. add(n ) = cof(n ) add(m) = cof(m) = add(n ) = cof(n ) 54 ( 55) add(n ) = cof(n ) 55 48 N M S. Shelah [16] 48 [9] 35
[1] Proofs from THE BOOK, M. Aigner, G.M. Ziegler, Springer-Verlag (1998). [2] T. Bartoszyński and H. Judah, Set Theory: on the structure of the real line, A K Peters, (1995), i ix,1 546. [3] Krzysztof Ciesielski, Set Theoretic Real Analysis, J. Appl. Anal. 3(2), 143 190 (1997). [4] P. Erdős, An interpolation problem associated with the continuum hypothesis, Michigan Math. J., 11, 9 10 (1964). [5] 23 1, vol.37, no.5, 50 53 (1998). 20 (2000), i iv, 1 235. [6]!, Vol. 41 No. 2, 52 56 (2002). [7] T. Jech, Set Theory, The Third Millennium Edition, revised and expanded 3rd rev. ed., Springer-Verlag (2003). [8] M. Holz, K. Steffens, E. Weitz, Introduction to Cardinal Arithmetic, Birkhäuser (1999). [9] A. Kanamori, The Higher Infinite, Springer Verlag (1994/1997), (1998). [10] S. Koppelberg, Einführung in die Logik und Mengenlehre, Vorlesungsskript an der Freien Universität Berlin, Sommersemester 1995. [11] K. Kunen, Set Theory, North-Holland, i xvi, 1 313 (1980). [12] Azriel Levy, Basic Set Theory, Springer-Verlag (1979). [13] W. Sierpiński, Sur les rapports entre l existence des intégrales 1 0 f(x, y)dx, 1 0 f(x, y)dy et 1 0 dx 1 0 f(x, y)dy, Fund. Math. 1, 142 147 (1920). Reprinted in Oeuvres Choisies, vol. II, 341 345. [14] ;,, (1988). [15] J. C. Oxtoby, Measure and Category, Springer-Verlag, (1971/1980). [16] S. Shelah, Can you take Solovay s inaccessible away? Israel Journal of Mathematics 48, 1 47 (1984). 36