Banach-Tarski Hausdorff May 17, 2014

3 Contents 1 Hausdorff 5 1.1 ( Unlösbarkeit des Inhaltproblems) 5

5 1 Hausdorff Banach-Tarski Hausdorff [H1, H2] Hausdorff Grundzüge der Mangenlehre [H1] Inhalte von Punktmengen Lebesugue Jordan (pp.399-). [H1]pp.469 Hausdorff [H2] [H1, H2] Hausdorff 1. Hausdorff 2. Hasdorff 3. + 1.1 ( Unlösbarkeit des Inhaltproblems) 1.1.1 Hausdorff K (α) (β) f(k) = 1 (γ) f(a + B) = f(a) + f(b)

6 1 Hausdorff K (α),(β),(γ) f(k) K Q A, B, C A, B, C A B + C ( Q Q 0 f f(q) 1 nf(k) n = 1, 2, 3... F (A) > 0 K = Q + A + B + C F (A) = 1 2 f(k) = 1 3f(K) 1.1.2 φ π φ 2 3π ψ : (G) 1 φ, ψ, ψ 2 φψ, φψ 2, ψφ, ψ 2 φ φ 2 = ψ 3 = 1 [ ]φ ψ φ, ψ, ψ 2 α, β, γ, δ m 1, m 2,..., m n 1 2 α = φψ m 1 φψ m2 φψ m n β = ψ m 1 φψ m2 φψ m n φ γ = φψ m1 φψ m2 φψ mn φ δ = ψ m1 φψ m2 φψ mn α, β, γ, δ 1 φ, ψ α 1 β = 1 φβφ = α = 1 γ = 1 φγφ = δ = 1 δ = 1 ψ m1 δψ m1 = α = 1 [ ] α 1 1.1.3 α 0.ψ z

1.1. ( Unlösbarkeit des Inhaltproblems) 7 (ψ) (φ) (φψ) λ = cos 2 3 π = 1 2, µ = sin 2 3 π = 3 2, x = xλ yµ y = xµ + yλ z = z x = x cos ϑ + z sin ϑ y = y z = x sin ϑ + z cos ϑ x = xλ cos ϑ + yµ + xλ sin ϑ y = xµ cos ϑ yλ + zµ sin ϑ z = x sin ϑ + z cos ϑ (φψ 2 ) (φψ) µ, µ ϑ/2 α φψ φψ 2 n α = αφψ α = αφψ 2 φψ φψ 2 n + 1. 0, 0, 1 α x, y, z α (φψ) (φψ 2 ) x, y, z α = αφψ 2 α = αφψ µ µ x sin ϑ, y sin ϑ (n 1) cos ϑ z cos ϑ n x = sin ϑ(a cos ϑ n 1 +...) y = sin ϑ(b cos ϑ n 1 +...) z = c cos ϑ n +... n = 1 0, 0, 1 φψ φψ 2 λ sin ϑ, ±µ sin ϑ, cos ϑ n n + 1 x, y, z φψ φψ 2 x = sin ϑ(a cos ϑ n +...) y = sin ϑ(b cos ϑ n +...) z = c cos ϑ n+1 +... a = λ(c a), b = ±µ(c a), c = c a, c a = (1 λ)(c a) = 3 (c a), 2 c a = ( 3 2) n.

8 1 Hausdorff α 0, 0, 1 z ( ) n 1 3 z = cos ϑ n + 2 ϑ 1 1 = ( ) n 1 3 cos ϑ n + 2 cos ϑ (n 1) cos ϑ k cos ϑ 1 ( ) 3 n 1 cos ϑ n + 2 1.1.4 (G) A, B, C (G) A, B, C 1. A 1. 2. ρ A ρφ A + B 3. ρ A ρψ, ρψ 2 B, C 1 A φ, ψ B ψ 2 C ψ n ψ ψ 2 φ, ψ, ψ 2 n φ n φ φ, ψ, ψ 2 n φ n, ψ n n + 1 ψ n A, B, C ψ n φ B, A, A φ n A, B, C φ n ψ B, C, A φ n ψ 2 C, A, B (G) A 1 ψφ, ψ 2 φ, φψ 2 φψφ B φ, ψ φψ 2 φ, ψφψ, ψ 2 φψ C ψ 2 φψ ψφψ 2, ψ 2 φψ 2

1.1. ( Unlösbarkeit des Inhaltproblems) 9 1.1.5 ) ) Q 0 K = P + Q P x (G) x, xφ, xψ, xψ 2, P x P x,p y P x x M = {x, y,...} P = M + Mφ + Mψ + Mψ 2 +... P A, B, C P = A + B + C, A = M + Mψφ + Mψ 2 φ + Mφψ 2 + B = Mφ + Mψ + C = Mψ 2 + Mφψ + Aφ = B + C, Aψ = B, Aψ 2 = C. A, B, C, B + C

11 Bibliography [B1] S.Banach Sur Plobléme de measure,findamenta Mathematicae 4,pp.7-33(1923), Banach pp.66-89 http://matwbn.icm. edu.pl/ksiazki/fm/fm4/fm412.pdf [B2] S.Banach Un Théorème sur les translations biuniques,findamenta Mathematicae 6, pp.236-239(1924) Banach pp.115-117 http://matwbn.icm.edu.pl/ksiazki/fm/fm6/fm6125.pdf [BT] S.Banach et A.Tarski, Sur la décomposition des ensembles de points en parties respectivement congruentes,findamenta Mathematicae 6 pp.244-277(1924), Banach pp.118-148 http://matwbn. icm.edu.pl/ksiazki/fm/fm6/fm6127.pdf [H1] Felix Hausdorff Grundzüge der Mangenlehre Leipzig 1914 [H2] Felix Hausdorff Bemerkung über den Inhalt von Punktmengen. Mathematische Annalen 75 (1914), pp.428-434 http://gdz.sub.uni-goettingen.de/index.php?id=11&ppn= PPN235181684_0075&DMDID=DMDLOG_0033&L=1 [S1] 1988 [S2] 2009 [W] Stan Wagon, The Banach-Tarski Paradox,Cambridge Univ. Press 1985 Paper back 1993