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2 page: 2 2 [ 4 ] [11] ( [11] ) Chapter I 0 n ( n ) (2.1) n + 1 n {n} 0, 1, 2, 3, 4,..., { }, {, { }}, {, { }, {, { }}}, {, { }, {, { }}, {, { }, {, { }}}},... n n = {0, 1, 2,..., n 1} n : (2.2) n l, m m n l m l n ; (2.3) n 2) ; (2.4) m n n 1 m ((2.1) +1 ) m + 1 = m m ( n n ) N N ( ) ( (ZF) (AC) (ZFC)) ( ) N N 0, 1, 2 α = N (2.4) (2.2) (2.3) (2.5) (2.6) : (2.5) α β, γ γ α β γ β α (2.6) α α α α, β α β α < β α β α = β α β α α ( ) ( ) N N ω α + 1 = α {α} ( ) ( α) ω On On ( On X On X ) ( ) ( ) 2

3 page: 3 3 ω ω + 1 = ω {ω} : ω ω + 1 ω ω + 1 (2.7) α β α β α ω ω + 1 : ω + ω ω, ω + 1, (ω + 1) + 1,... ω + ω ( ) κ κ + On : (2.8) ℵ 0 = ω ; (2.9) α ℵ α ℵ α+1 (ℵ α ) + ; (2.10) γ ℵ α, α < γ ℵ γ = lim{ℵ α : α < γ} (= {ℵ α : α < γ}) ℵ α, α On {ℵ α : α On} = {κ : κ } x x κ 1 κ x x x x Zorn Zorn ( ) ZFC Zorn Zorn ( Shelah [17] ZFC ) 1 Hamel R Q ( Zorn ) R Q Hamel Hamel : R 1 c R ( ) c 3

4 page: 4 4 r α R, α < c (2.11) r 0 = 1 ; (2.12) r β, β < α (2.12a) {r β : β < α} Q R r α = 1 ; (2.12b) {r β : β < α} Q R r α R {r β : β < α} Q 3) {r α : α < c} R Q Hamel ( ) Hamel (Zorn ) Hamel Hamel R 1 H Hamel a H 1 (2.13) H 0 = {x R : x H ( ) a 0 } H 0 r Q H r = H 0 + ra = {x + ra : x H 0 } R = r Q H r H 0 H r, r Q H 0 r Q H r ( R ) H 0 Steinhaus ( [15], Theorem 4.8) δ (0, ) Q H 0 = H 0 H 0 (= {a b : a, b H 0 }) [ δ, δ] r (0, ) Q H r [ δ, δ] ( x H 1 n 1 n x H 1 [ δ, δ] ) n Hamel (2.11), (2.12) α < c (2.12b) r α C [0, 1] C Hamel H C Hamel H ([14]) ( 4) ) ( Piccard [15], Theorem 4.8 ) 1 : 2 1 R H [19] ZF On : (3.1) V 0 = ; (3.2) V α V α+1 = V α P(V α ) ; (3.3) γ V α, α < γ V γ = {V α : α < γ} V α, α On x x V α α 4

5 page: 5 5 V V = α On V α 5) x P(x) x (x ) V α, α On (V ) L : (3.4) L 0 = ; (3.5) L α L α+1 = L α def(l α ) ; (3.6) γ L α, α < γ L γ = {L α : α < γ} L = α On L α def(l α ) (L α, ) 6) L α a 0,..., a n 1 ZF ϕ {x L α : (L α, ) = ϕ(x, a 0,..., a n 1 )} 7) def(l α ) ( ) ( ) : ZFC ϕ (GCH: κ 2 κ = κ + 8) ) ( ) ϕ (3.7) ZFC (L, ) = ϕ 9) ZFC GCH ( ): ZFC GCH ZFC GCH ZFC ϕ 0,..., ϕ n 1 (3.7) ZFC (L, ) = ϕ 0,..., (L, ) = ϕ n 1 GCH (L, ) = GCH (L, ) = GCH ZFC (3.7) (L, ) = GCH ZFC ZFC L ( x L y x y L ) (3.7) ZF L (3.8) L 0 (R) = tc(r) ; (3.9) L α (R) L α+1 (R) = L α def(l α (R)) ; (3.10) γ L α (R), α < γ L γ (R) = {L α (R) : α < γ} tc(r) R transitive closure, R L(R) = α On L α (R) L(R) ZF (3.7) ( L(R) 1 ) L(R) AC 4 V = L Hamel L L L L L L L = L L 5

6 page: 6 6 L x L α ( V = L ) ZFC ZFC + V = L n R n X X m n R m R n X X ϕ a 0,..., a m 1 R X = { b 0,..., b n 1 R n : (R, +,, Z) = ϕ(b 0,..., b n 1, a 0,..., a m 1 )} ([10], (37.6)) L(R) R n R n ( ω ω = { f : f : ω ω} ω ) 1 ω ω [ 9 ] (R, +,, Z) V = L R R 2 (2.11), (2.12) R Q Hamel H H 3 (ZF + V = L) R R Q Hamel 10) 1, 2 : 4 (ZF + V = L) R 5 ( ) (R R ) ( V = L ) Kanamori [ 9 ] Jech [ 8 ] [ 9 ] [ 7 ], [18] [ 1 ] [ 6 ] κ κ = λ + λ κ κ ℵ 0 κ κ ω 6

7 page: 7 7 ZFC ( ) ZFC ( ) ZFC ZFC ZFC ZFC : κ (5.1) (L κ, ) = ZFC 11) ZFC ZFC ZFC ZFC ZFC ZFC ZFC ZFC ( ) ZFC κ λ 2 λ λ κ 2 λ κ κ 1 κ κ ZFC ZFC κ (V κ, ) ZFC L GCH κ κ κ L ZFC ZFC ZFC ZFC (equiconsistent) ZFC ZFC ( ) ZFC P(κ) κ- (κ ) ( ) κ V = L ([ 9 ], Corollary 5.5) V L ( [ 9 ], Theorem 9.1) ([ 9 ], Theorem 2.8) κ κ κ ([ 9 ], Proposition 6.6 ) κ 0 κ 1 κ 0 κ 1 V κ1 κ 0 ZFC ( MC ) ZFC 7

8 page: 8 8 ( IC ) ZFC + IC MC A B ZFC + A ZFC+ B A B (consistency strength) A 0 ( ) A 1 ( ) A 1 A 0 κ A 1 (κ) κ A 0 (κ) ZFC κ κ κ P(κ) κ- κ λ κ P κ λ = {a P(λ) : a < κ} P(P κ λ) µ λ : (5.2) α λ µ λ ({a P κ λ : α a}) = 1; (5.3) a P κ λ f (a) a f : P κ λ λ α λ µ λ ( f 1 ({α })) = 1 V ( [ 9 ] Theorems 5.4, 5.6, Theorem 22.7 [ 9 ] Theorem 22.7 (5.2), (5.3) ([ 9 ] ) ) (5.2), (5.3) κ {µ < κ : µ } κ ([ 9 ], Proposition 22.1) (Woodin ) SCC ( 1 ) [ 9 ] Theorem 26.11, Corollary 32.14, Theorem 27.9 : 5 (ZFC + SCC) L(R) 5 ZFC + MC : ZFC L ZFC + MC 4 ([ 9 ], Corollary ) 6 Hamel 6 (ZFC + SCC) L(R) Q R Hamel Hamel H L(R) Hamel ( 1 ) H 0 L(R) 2 H 0 5 L(R) R : (2.11), (2.12) 8

9 page: 9 9 R Q Hamel L(R) Q R Hamel L(R) ZFC 7 L(R) Q R Hamel ZFC ZFC ZFC 3 Shelah [16] (Shelah sweetness ), L(R) R ZFC 12) 6 L(R) R R Q Hamel Hamel Zorn Hamel 7 13) Kunen [11] [ 4 ] [11] Jech [ 8 ], Kanamori [ 9 ] [ 1 ] [ 4 ] 1) Handbook of Set Theory [ 2 ] 2) X X Y X Y ( [ 4 ] ) 3) (2.12a) 4) ( ) 9

10 page: Shelah [16] Martin ([15], Chapter 19) 5) V = α On V α ZF 6) ( ) X (X, ) X X 2 = { x, y : x, y X, x y} 7) A ϕ = ϕ(x 0,..., x n 1 ) A a 0,..., a n 1 A A = ϕ(a 0,..., a n 1 ) ϕ x 0,..., x n 1 a 0,..., a n 1 ϕ A 8) κ κ P(κ) 2 κ 2 ℵ0 R c 9) (L, ) = GCH Löwenheim-Skolem Mostowski Condensation Lemma 10) : Sierpiński Burton Jones R Σ 1 1 Hamel ZFC (Miller [12] ) Hamel Hamel 1 2 [12] V = L Π 1 1 Q R Hamel 11) (3.7) (5.1) (3.7) ( ) ZFC + GCH ϕ (L, ) = ϕ (5.1) ZFC ( ( ) ) (L, ) = ZFC ZFC ( ) 12) [16] Shelah L(R) R A. Miller [16] [13] While the second result ( : 7 ) is technically brilliant, important, and likely to be useful for getting other results, the first result is pure magic. 13) [ 3 ] [ 1 ] R. ( ), ( ), [ 2 ] Matthew Foreman and Akihiro Kanamori (Eds.), Handbook of Set Theory, Springer (2010). [ 3 ] Forcing Axioms, Vol.56, No.3 (2004), [ 4 ] 20 4 (2007) I. [ 5 ] unabridged version: fuchino /papers/axiomatic-set-th-unabridged.pdf [ 6 ] [[[ ] ] ] ([ 7 ] [18] ), to appear. [ 7 ] ( ), ( ), ( ), (2006). [ 8 ] Thomas Jech, Set Theory, The Third Millennium Edition, Springer (2002/2006). [ 9 ] Akihiro Kanamori, The Higher Infinite, Springer (2004): : A. ( ) ( ): ( ) (1998). [10] Alexander S. Kechris, Classical Descriptive Set Theory, Grad. Texts in Math., 156, Springer (1995). [11] Kenneth Kunen, Set Theory, An Introduction to Independence Proofs, Stud. Logic Found. Math., 102, North- Holland Publishing (1980). : K. ( ) ( ) (2008). [12] Arnold W. Miller, Infinite combinatorics and definability, Annals of Pure and Applied Logic 41, (1989), [13] Arnold W. Miller, review of [16], The Journal of Symbolic Logic 54(2), (1989), [14] M.G. Nadkarni and V.S. Sunder, Hamel bases and measurability, Mathematics Newsletter Vol.4(3), (2004), 1 3. [15] John. C. Oxtoby, Measure and Category, Grad. Texts in Math., 2, Springer-Verlag, (1980). [16] Saharon Shelah, Can you take Solovay s inaccessible away?, Istael Journal of Mathematics 48, (1984), [17] Saharon Shelah, Cardinal Arithmetic, Vol. 29 of Oxford Logic Guides, Clarendon Press (1994). [18] (2012). [19] Ernst Zermelo, Untersuchungen über die Grundlagen der Mengenlehre. I, Mathematische Annalen 65 (1908), ( ) ( ) 10

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