1 2 X X X X X X X X X X Russel (1) (2) (3) X = {A A A} 1.1.1

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1 G.Cantor ( ) 1874 Unter eines Menge verstehen wir jede Zusammenfassung M von bestimmten wohlunterschiedenen Objekten in unserer Anschauung oder unserers Denkens (welche die Elemente von M genannt werden) zu einem ganzen. (In English) A set is a collection into a whole of definite, distinct objects of our intuition or our thought. The object are called elements (members) of the set. A a A a A a a A a A A = {1, 3, 5, 7} (explicit) A = {x x 1 8 } (implicit) A B x A x B A B B A A B B A A = B A B B.Russel (1902) X = {A A A A}

2 1 2 X X X X X X X X X X Russel (1) (2) (3) X = {A A A} 1.1.1

3 N = {1, 2, 3,...} Z = {0, ±1, ±2, ±3,...} { } q Q = p p, q Z, p 0 R = {x x } C = {x + iy x, y R} A B = {x x A x B} A B = {x x A x B} {A i i I} i I A i = {x i x A i } i I A i = {x i x A i }. A B A c = {x B x A} A B A i B (i I) ( i I A i ) c = i I A c i, ( i IA i ) c = i I A c i. ; x x A A B {A i i I} I = i I A i, i I A i R A A c = sup A A x y A x y x c x c A B sup A sup B sup 1.3 A B f A x B y f : A B y = f(x) f : A B A f B f

4 1 4 f : A B x y f(x) f(y) f f(x) = f(y) x = y f : A B y B y = f(x) x A f f f A B f g : B A y = f(x) x = g(y) g f f 1 f : A B g : B C g f : A C g f(x) = g(f(x)) A id A : A A 1 x A id A (x) = x f : A B f f 1 = id B, f 1 f = id A f 1 : B A A B B A A 2 A f : A B A A f A f A : A B x A f A (x) = f(x) f A f A f : Z N {0} f(x) = x f N {0} f N {0} = id N {0} A A A f : A A (1) f (2) f (3) f A A A A A A A, B, C (A B C) = A + B + C (A B) (B C) (C A) + (A B C) n A = i (i = 1, 2,..., n) A B (B A ) = B A (2 A ) = 2 A 1 id identity

5 A A a a a b b a a b, b c a c A A A = i I C i ( C i C j = (i j)) C i α i C i α i C i S = {α i i I} A A A A/ x A x [x] π : A A/ π(x) = [x] S S A π S A (1) A x y x y (2) A = R x y x y (3) A = Z x y x = y x = 0 y = 1 x = 1 y = 0 (4) A = R x y e x = e y (5) A = C x y R(x 2 ) = R(y 2 ) (6) A x y x y R (1) A q = {x R q x < q + 1} (q Q) (2) B r = {x R x r Q} (r R) (3) C s = {x R x s Z} (r R {0}) (4) D n = {x R n 1 < x 2 n} (n Z)

6 (1) Z x y x y (2) R x y x y Z (3) [0, 1] x y (x = y) (x = 0, y = 1) (x = 1, y = 0) (4) [0, 1] [0, 1] (x, y) (x, y ) ((x, y) = (x, y )) (x = 0, x = 1, y = y ) (5) [0, 1] [0, 1] (x, y) (x, y ) ((x, y) = (x, y )) (x = 0, x = 1, y = 1 y ) (6) L l l l l (7) n M n (C) A B n P P 1 AP = B R V U V x y x y U V/U V/U R

7 A B A B A carda A B carda = cardb ) F A, B F A B A B carda A F/ {1, 2,..., n} A carda = n n N A carda = ℵ 0 ℵ 0 A carda = ℵ ℵ A B carda cardb carda cardb carda cardb carda < cardb A B (1) A N A (2) (3) (1) A N A ( ) A, B, C carda cardb, cardb cardc = carda cardc (Bernstein ) A, B carda cardb, cardb carda = carda = cardb cardq = ℵ 0

8 carda cardb = card(a B) c C card(a c) = card( c C A c) (carda) cardb = card(a B ) A B B A A B = carda + cardb = card(a B) A = c C A c c C card(a c) = carda carda carda = (carda) 2 α α α = α carda = ℵ 0 card(a A) = ℵ 0. ℵ 2 0 = ℵ α 2 α > α Cantor ℵ 0 = ℵ ℵ 0 ℵ cardc = ℵ A = 0 A carda = ℵ (1) ℵ 0 + ℵ 0 = ℵ 0 (2) ℵ + ℵ 0 = ℵ (3) 2 ℵ 0 = ℵ 0 (4) ℵ ℵ = 2 ℵ (5) ℵ 2 = ℵ (6) ℵ ℵ 0 = ℵ A B carda = ℵ 0 cardb > ℵ 0 card(b A) = cardb

9 9 3 one, two, three,.., ℵ 0, ℵ first, second, third, S S [ ] a a [ ] a b, b c a c [ ] a b, b a a = b a, b a b b a (totally ordered set) (partially ordered set) Hasse A P (A) A P (A) Hasse S A a A a A a b for any b A. S A a A a A If b a for b A, then b = a.

10 (sup) (inf) S A B = {b S a b for any a A} A B A B A sup A S a, b a < b a < x < b x S a b b a a S S A = {x S x < a} a sup A a sup A = a S, T f : S T a, b S a b f(a) f(b) S T A = B ( ) S S A S (well ordered set) N Z Q ( ) S a P (a) a S x S, x < a x P (x) P (a) a S P (a) S T f, g a A f(a) = g(a)

11 {1, 2,..., n} n nth N ω S T S T S T S + T S T α, β α + β S + T α β S T n N n + ω = ω ω + n ω n N n ω ω n ω 2 = ω + ω, 2 ω = ω S a S < a >= {x S x < a} ( ) S T (1) S T (2) a S S < a > T (3) b T S T < b > S T α, β (1) (3) α = β, α > β, α < β ( ) α, β α = β, α > β, α < β < 2 < < ω < ω + 1 < < ω 2 < ω < < ω 3 < < ω ω <

12 Zorn ( ) A W = {X a a A} f : A W f(a) X a for any a A f A X a f f(a 1 ) X a1 f(a 2 ) X a2... f pragmatism S, S S A sup A (Zorn ) ( ) S a 1 S S a 1 a 1 < a 2 a 2 S a 1 < a 2 < a 3 a 3 S a 1 < a 2 < a 3 < A = {a 1 < a 2 < a 3 <... } A x A x A x S!? Zorn 3.4 Zorn ( ) ( ) α, β α β β α Zorn

Basic Math. 1 0 [ N Z Q Q c R C] 1, 2, 3,... natural numbers, N Def.(Definition) N (1) 1 N, (2) n N = n +1 N, (3) N (1), (2), n N n N (element). n/ N.

Basic Math. 1 0 [ N Z Q Q c R C] 1, 2, 3,... natural numbers, N Def.(Definition) N (1) 1 N, (2) n N = n +1 N, (3) N (1), (2), n N n N (element). n/ N. Basic Mathematics 16 4 16 3-4 (10:40-12:10) 0 1 1 2 2 2 3 (mapping) 5 4 ε-δ (ε-δ Logic) 6 5 (Potency) 9 6 (Equivalence Relation and Order) 13 7 Zorn (Axiom of Choice, Zorn s Lemma) 14 8 (Set and Topology)