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2 page: 2 2 [ 4 ]  (  ) 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  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 ( , 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 () ( 4) ) ( Piccard , Theorem 4.8 ) 1 : 2 1 R H  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 )} (, (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 ],  [ 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  (Shelah sweetness ), L(R) R ZFC 12) 6 L(R) R R Q Hamel Hamel Zorn Hamel 7 13) Kunen  [ 4 ]  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  Martin (, 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  ) Hamel Hamel 1 2  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)  Shelah L(R) R A. Miller   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 ]  ), 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).  Alexander S. Kechris, Classical Descriptive Set Theory, Grad. Texts in Math., 156, Springer (1995).  Kenneth Kunen, Set Theory, An Introduction to Independence Proofs, Stud. Logic Found. Math., 102, North- Holland Publishing (1980). : K. ( ) ( ) (2008).  Arnold W. Miller, Infinite combinatorics and definability, Annals of Pure and Applied Logic 41, (1989),  Arnold W. Miller, review of , The Journal of Symbolic Logic 54(2), (1989),  M.G. Nadkarni and V.S. Sunder, Hamel bases and measurability, Mathematics Newsletter Vol.4(3), (2004), 1 3.  John. C. Oxtoby, Measure and Category, Grad. Texts in Math., 2, Springer-Verlag, (1980).  Saharon Shelah, Can you take Solovay s inaccessible away?, Istael Journal of Mathematics 48, (1984),  Saharon Shelah, Cardinal Arithmetic, Vol. 29 of Oxford Logic Guides, Clarendon Press (1994).  (2012).  Ernst Zermelo, Untersuchungen über die Grundlagen der Mengenlehre. I, Mathematische Annalen 65 (1908), ( ) ( ) 10