page: 1 1 Club guessing sequence , 1). M. Foreman ([ 7 ], [ 8 ] ). 1960,,,, ZFC,,.,,., ZFC,, 20.. G. Cantor.., Zermelo ZFC 2).,., ZFC,., ZFC

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1 page: 1 1 Club guessing sequence , 1). M. Foreman ([ 7 ], [ 8 ] ). 1960,,,, ZFC,,.,,., ZFC,, 20.. G. Cantor.., Zermelo ZFC 2).,., ZFC,., ZFC ZFC, ZFC., K. Gödel [19] L P. Cohen [ 4 ], [ 5 ]., ZFC,,. K. Gödel, ZFC., Cohen ZFC., ZFC., 1970 ZFC., 1980., S. Shelah PCF 3) S. Todorcevic minimal walk. ZFC,., 1

2 page: 2 2. ZFC,.,. club guessing sequence, PCF,.,. 1.2 S. Shelah,.,,., [34] 4.2., S. Shelah [42]. 1 G. Cantor,.,,., 2 ℵ 0 = ℵ1. 2 ℵ 0 = ℵ1 (CH)., κ 2 κ = κ + (GCH)., K. Gödel [19] 2 ℵ 0 = ℵ1 P. Cohen [ 4 ] 2 ℵ 0 > ℵ1, ZFC 4). ℵ 0, (1) κ < λ, 2 κ 2 λ. (2) (König 5) ) cf(2 κ ) > κ. δ cf(δ), δ., δ, otp(x) cf(δ) δ X 6). cf(κ) = κ κ... κ κ +. ℵ ω {ℵ n : n < ω} ℵ ω, ℵ 0., ℵ ω., ZFC. W. B. Easton [12], 1.1 ZFC., J. Silver [45]. 1.2 (J. Silver [45]) κ ℵ 0 < cf(κ) < κ., λ < κ 2 λ = λ + 2 κ = κ +. 7) 2

3 page: 3 Club guessing sequence 3,, 1.1., M. Magidor (M. Magidor [33]), : n < ω 2 ℵ n = ℵn+1 2 ℵ ω = ℵω+2., ℵ ω GCH. ZFC., 2 ℵ ω., S. Shelah [39], PCF,. ℵ 1.4 ℵ 0 ω < max{ℵ ω4, (2 ℵ 0 ) + }. L. Harrington 1986, S. Shelah [42] :. 8),.., S. Shelah ZFC 9). club guessing sequence. J. Cummings [ 6 ].. (1) V K. Gödel L. (2) V, L, canonical,. (3) PCF,. (4). canonical., (1) (3)..,., [16] K. Kunen [31] ( ) T. Jech [24].,.,.,., Skolem hull.,,. 3

4 page: club guessing sequence 2.1,,. α X α β < α γ β γ X.. X, X δ δ, δ X. δ {γ : γ < δ}, X δ = {γ X : γ < δ}. lim(x) X. X,. sup(x) X, max(x) ( )X. X Y,, X = X \ ζ Y ζ < sup(x), X Y, X Y. X., X \ ζ = {α X : α ζ}, X Y X Y. X Y X Y. 2.1 δ. D δ, δ : (1) D δ. (2) γ < δ γ D, γ D., closed unbounded, club. 10) (2) D., δ, δ X, δ lim(x) δ δ cf(δ). {D α : α < µ} δ µ < cf(δ), α<µ D α δ.. α<µ D α,., ζ < δ, γ ζ γ α<µ D α. δ γ n : n < ω. γ 0 = ζ. γ n. α < µ, D α δ ξ n,α > γ ξ n,α D α. {ξ n,α : α < µ} δ µ < cf(δ), cf(δ)., α < µ ξ n,α < γ n+1 γ n+1 < δ. γ = sup n<ω γ n. cf(δ), γ < δ. α < µ D α γ γ. D α, γ D α., γ α<µ D α., γ γ 0 = ζ,. cf(δ) = ω., δ = ω, D 1 = {2n : n < ω} D 2 = {2n + 1 : n < ω} ω. 2.3 X, X F, F X. 4

5 page: 5 Club guessing sequence 5 (1) F X F. (2) A B X A F B F. (3) A F B F A B F., (X, E, µ), F = {Y E : µ(y ) = 1} X., φ(x) x X F, φ., F X. δ., F = {Y δ : Y δ }, F δ ). club X I, I X : (1) I X I. (2) A B X B I A I. (3) A I B I, A B I., (X, E, µ), I = {Y E : µ(y ) = 0} X. I, X. F X, I = {Y X : X \ Y F } X. F. I X F = {Y X : X \ Y I} X, I., 1 0,. club., δ, δ X δ D X D =. δ, δ., δ X, δ D X D. 2.2, δ S D S D., club, κ club NS κ..,. 2.5 κ, X α : α < κ κ. (1) X α : α < κ diagonal intersection κ. α<κ X α = {γ < κ : α < γ(γ X α )}. 5

6 page: 6 6 (2) X α : α < κ diagonal union κ. α<κ X α = {γ < κ : α < γ(γ X α )}.,. 2.6 κ \ α<κ X α = α<κ (κ \ X α ). κ., κ D α : α < κ diagonal intersection α<κ D α.. D = α<κ D α., D, ζ < κ, ζ D. κ γ n : n < ω. γ 0 = ζ. γ n γ n , α γ n D α., γ n α γ n D α γ n+1. γ = sup n<ω γ n. γ D. diagonal intersection, α < γ γ D α. α < γ. γ = sup n<ω γ n, α < γ m m < ω. γ n, m n < ω γ n D α., γ D α. D α γ D α. D δ D. α < δ δ D α., D \ (α + 1) D α 12)., δ D α. δ D α. α < κ D α = κ \ α, α<κ D α =., α < κ D D α κ., 2.2 κ., 2.6, κ κ,. 2.7 I X., I κ : λ < κ {Y α : α < λ} I α<λ Y α I. κ I, I : I X α : α < κ α<κ X α I ,. 2.8 κ, NS κ κ. NS κ κ κ. 2.9 κ, I κ κ κ., NS κ I. 2.2 Club guessing sequences Lim. κ, Cof(κ) κ. λ < κ, κ Cof(λ) κ 6

7 page: 7 Club guessing sequence 7. Cof( κ) κ (S. Shelah [39]) κ, S κ., C δ : δ S, S club guessing sequence (fully club guessing sequence) : (1) δ S, C δ δ, (2) κ D, C δ D δ S. C δ : δ S, (1) (2) (2), S club guessing sequence (tail club guessing sequence). (2) κ D, C δ D δ S., club guessing sequence FCG-, club guessing sequence TCG-., A δ : δ S κ (S), S S δ A δ δ, A δ : δ S FCG-. ( κ (S) 2.3 ). (2) C δ D δ S., C FCG-, κ, S = {δ S : C δ D}., κ D S D =., D D κ, C FCG- C δ D D δ S., C δ D, δ S., δ D, δ D. S D =. TCG-. FCG- TCG-,. C δ : δ S FCG-, C δ {0} : δ S TCG- FCG-., TCG- FCG-, ( [23]) κ, S κ Lim., C δ : δ S TCG-, C δ \ ζ : δ S \ (ζ + 1) FCG- ζ < κ., S TCG- S FCG-., FCG-, ZFC (S. Shelah [39]) θ κ θ + < κ., κ Cof(θ) S, S FCG-., θ.,., X, acc(x) X α X α α., acc(x) = X lim(x). nacc(x) X \ acc(x). δ, D, acc(d) δ. 7

8 page: θ κ θ + < κ. S κ Cof(θ), C δ : δ S δ S C δ δ C δ = θ., C δ E : δ S acc(e) FCG- κ E.. E. κ D α : α θ +. D 0 = κ. β < α D β D α. α, D α = β<α D β. 2.2, D α κ. α, α = β + 1 β., δ S acc(d β ), C δ D β δ. cf(δ) > ℵ 0, C δ D β., C δ D β : δ S acc(d β ) FCG-., δ S acc(d β ) C δ D β D β+1 κ D β+1. D β+1 D β. D α : α θ +, δ S acc(d θ +)., C δ D α : α < θ +. α < θ +, D α+1 C δ D α+1 C δ D α. γ α C δ (D α \ D α+1 )., γ α : α < θ + C δ., C δ = θ.,. D β+1.,.,.,. M. Foreman ( [ 7 ] [14] ). θ = ℵ 0, cf(δ) = ℵ 0 δ,.,., S. Shelah [39], M. Kojman [28] 13) (S. Shelah [39]) θ, λ, κ θ < λ < κ. S κ Cof(θ), δ S Cof(λ)., S FCG- C δ : δ S δ S C δ Cof( λ).. δ κ Lim, otp(e δ ) = cf(δ) δ e δ. D κ, δ S., n < ω δ f δ (D, n) f δ(d, n). f δ(d, 0) = e δ. f δ (D, n) = {sup(d ξ) : ξ f δ(d, n) ξ > min(d)}. f δ(d, n + 1) = {e γ : γ f δ (D, n) Cof(<λ)}. 8

9 page: 9 Club guessing sequence 9, f δ (D) = n<ω f δ(d, n). D, n < ω, f δ (D, n) D,, f δ (D) D δ S acc(d), f δ (D, 0) δ., f δ (D) δ. δ S acc(d), ζ < δ. δ acc(d), D δ, γ D ζ < γ < δ. e δ δ, ξ e δ ξ > γ., ξ e δ = f δ (D, 0), ξ > γ min(d)., sup(d ξ) f δ (D, 0)., sup(d ξ) γ > ζ., f δ (D, 0) δ δ S n < ω, f δ (D, n) < λ f δ(d, n) < λ. δ S, n < ω., f δ (D, 0) = e δ = cf(δ) = θ < λ. f δ (D, n) < λ, f δ(d, n) f δ (D, n) < λ. f δ(d, n) < λ f δ (D, n + 1) < λ. γ f δ(d, n) Cof(<λ), e γ = cf(γ) < λ., f δ (D, n + 1) λ f δ(d, n). f δ (D, n) < λ λ, f δ (D, n + 1) < λ κ D, f δ (D) : δ S acc(d) FCG- D. κ D α : α λ. D 0 κ Cof(λ) κ.. α λ, β < α D β, D α = β<α D β. D α, D α+1., f δ (D α ) : δ S acc(d α ) FCG-. δ S acc(d α ), f δ (D α ) δ., κ D α+1 δ S acc(d α ) f δ (D α ) D α+1. D α+1 D α. δ S acc(d λ )., λ α n : n < ω n : n < ω, α n α < λ f δ (D α, n) = f δ (D αn, n). m < n α m., α n α < λ f δ (D α, n) = f n δ (D α, n) α n < λ. n = 0 α 0 = 0. f δ (D, 0) D. n > 0 α n = α n 1., α n α < λ, f δ (D α n, n 1) = f δ (D α, n 1). f, f δ (D α, n) = f n δ (D α, n). α n α < λ f δ (D α, n) = f δ (D αn, n) α n < λ., β < λ β α < λ α f δ (D α, n) f δ (D β, n)., β 0 = α n λ β ν : ν < λ, ν < λ f δ (D βν+1, n) f δ (D βν, n). ν < λ, f δ 9

10 page: f δ (D β ν+1, n) = f δ (D β ν, n) = f δ (D α, n), n (1) min(d βν ) < ξ ν min(d βν+1 ) (2) min(d βν+1 ) < ξ ν sup(d βν+1 ξ ν ) < sup(d βν ξ ν ) ξ ν f δ (D β ν, n). f δ (D β ν, n) < λ, ξ f δ (D β ν, n) {ν < λ : ξ ν = ξ} λ. X = {ν < λ : ξ ν = ξ}., min(d βν ) < ξ ν min(d βν+1 ) ν X., ν < µ < λ µ X., D βµ D βν+1, ξ µ = ξ min(d βµ )., ξ µ., ν X min(d βν+1 ) < ξ sup(d βν+1 ξ) < sup(d βν ξ)., ν < µ X, sup(d βµ ξ) sup(d βν+1 ξ) < sup(d βν ξ)., sup(d βν ξ) : ξ X.., α n. α ω = sup n<ω α n. λ, α ω < λ., β < λ n < ω, α n, β α ω f δ (D β, n) = f δ (D αω, n). f δ (D β ) = f δ (D αω ).,, f δ (D αω +1) = f δ (D αω ) D αω. D αω +1. S f δ (D) acc(d) δ S acc(d)., δ S, C δ = nacc(f δ (D)). 2.17, S, C δ : δ S FCG-., δ S, C δ Cof( λ). δ S γ C δ. cf(γ) < λ. γ f δ (D)., γ nacc(f δ (D)). ζ < γ, ζ < ξ < γ ξ f δ (D). f δ (D) = n<ω f δ(d, n), γ f δ (D, n) n < ω., γ f δ (D, n) Cof(<λ), e γ f δ (D, n). S, f δ (D) acc(d), γ acc(d)., D γ γ. ζ < ζ < γ D ζ. e γ γ, ξ > ζ ξ e γ. e γ f δ (D, n), ξ f δ (D, n)., sup(d ξ) f δ(d, n + 1) f δ (D). ζ D ξ, sup(d ξ) ζ > ζ., γ f δ (D) θ + < κ, A. Ros lanowski S. Shelah [37]., κ κ ℵ 2, κ Lim FCG-. 10

11 page: 11 Club guessing sequence 11 κ ℵ 2 S. Shelah [41]. 2.3 FCG- θ κ θ + < κ, κ Cof(θ), club guessing sequence,., diamond κ, S κ., κ (S) A δ : δ S : (1) δ S, A δ δ. (2) X κ, {δ S : X δ = A δ } κ., κ(s) A δ : δ S : (1) δ S, A δ P(δ) A δ δ. (2) X κ, κ D, δ D S X δ A δ. S, S = κ κ S, κ(s) κ (S) (D. Jensen [26]) V = L, κ, κ κ ineffable 14)., κ κ. κ S κ (S). V = L,. κ 2 <κ = κ., S. Shelah, κ ℵ (S. Shelah [43]) κ = λ + = 2 λ, S κ. α S cf(α) cf(λ) κ (S) (S. Shelah [43]) κ = λ +, 2 λ = λ + κ.. λ J. Gregory, λ S. Shelah (J. Gregory [21], S. Shelah [40]) GCH. κ = λ +, T = {α < κ : cf(α) cf(λ)}., κ(t ). square λ. λ, C α : α λ + Lim. (1) α λ + Lim, C α α. (2) α λ + Lim, cf(α) < λ otp(c α ) < λ. (3) β < α C α, C β = C α β. C α : α λ + Lim λ. (2) (3), α λ + Cof(λ) otp(c α ) = λ. λ + 11

12 page: 12 12, α λ + Lim, cf(α) λ, α λ. λ, (3) α λ + Lim. V = L (D. Jensen [26]) V = L, λ λ. S. Shelah λ θ < λ, S λ + Cof(θ). C α : α S, S square : (1) α S, C α α otp(c α ) = θ. (2) α, β S, γ C α C β, C α γ = C β γ. square. λ λ, λ θ λ λ + Cof(θ) square., S. Shelah ZFC (S. Shelah [39]) θ < λ., λ + Cof(θ) λ square λ, 2.29 λ. θ κ, κ Cof(θ).,. 3 ℵ 2 club guessing sequence, M. Gitik S. Shelah., club guessing sequence,.,. I κ. 3.1 P(κ) I. X I Y (X \ Y ) (Y \ X) I. P(κ)/I = {[X] I : X P(κ) \ I}. P(κ)/I I. [X] I X I. [X] I I [Y ] I X \ Y I. well-defined. 12

13 page: 13 Club guessing sequence P(κ)/I [X] I [Y ] I, [X] I [Y ] I compatible : [Z] I I [X] I [Z] I I [Y ] I [Z] I P(κ)/I. [X] I [Y ] I compatible, incompatible. [X] I [Y ] I imcompatible, X Y I. 3.3 A P(κ)/I A P(κ)/I : A A B incompatible. P(κ)/I A, A P(κ)/I A A, A = A. A, [X] I P(κ)/I [X] I compatible [A] I A. 3.4 P(κ)/I A A κ, I 15)., generic embedding,., R. Solovay [46]. generic embedding, M. Foreman [13]. NS ω1, J. Steel R. Van Wesep [47] ZF + DC + AD R + Θ, M. Foreman, M. Magidor S. Shelah [15] supercompact. 16), ω 1.,. 3.5 X I S I S. I S = {Y X : Y S I}. 3.6 (M. Gitik and S. Shelah [17]) (1) κ ℵ 2 κ, NS κ. (2) θ κ θ + < κ, NS κ (κ Cof(θ)). κ ℵ 3 (1), M. Gitik S. Shelah [17] TCG-. NS κ, NS κ (κ Cof(θ)). 3.7 (S. Shelah [39]) C = Cδ : δ S κ S TCG-. κ TCG( C) X κ : κ D, C δ D δ δ X. TCG( C) TCG( C). κ X TCG( C), κ D δ X C δ D. 3.8 C = C δ : δ S κ S TCG-. TCG( C) κ κ.. X κ, X TCG( C). X, X ζ ζ < κ. κ \ ζ κ, {δ S : C δ κ \ ζ} TCG( C). X {δ S : C δ κ \ ζ}, X TCG( C). 13

14 page: κ, {X α : α < µ} TCG( C) µ < κ., α < µ κ D α δ X α C δ D α. D = α<µ D α., µ < κ 2.2 D. X = α<µ X α TCG( C) δ X C δ D. δ X, δ X α α < µ. C δ D α., D D α, C δ D. X α : α < κ TCG( C). α < κ, δ X α C δ D α κ D α. D = α<κ D α. 2.6 D. X = α<κ X α TCG( C), δ X C δ D. diagonal union, δ X α α < δ. D α C δ D α., 2.6, D \ (α + 1) D α., C δ D. 3.9 (J. Baumgartner, A. Taylor, and S. Wagon [ 2 ]) I J κ. I, I J., J = I X X P(κ) \ I. NS κ, κ κ I, κ S, I = NS κ S. S κ., {[X α ] NSκ S : α < µ} P(κ)/NS κ S, {[X α S] NSκ : α < µ} P(κ)/NS κ., NS κ NS κ S (folklore) I κ κ. {[X α ] I : α < κ} P(κ)/I., P(κ) \ I X α : α < κ, (1) α < κ [X α ] = [X α] (2) β < α < κ X β X α =. disjointing property. (P, P ) D, D I κ. D P(κ)/I : [X] I P(κ)/I, [D] I D [D] I I [X] I., A D P(κ)/I A., κ ℵ 3 3.6(1). ( 3.6). κ ℵ 3, NS κ. S κ Cof(ω) Cof( ℵ 2 ). 2.14, S S S TCG- C = C δ : δ S δ S C δ Cof( ℵ 2 ). 3.8 TCG( C) κ 14

15 page: 15 Club guessing sequence 15., 2.9 NS κ TCG( C). NS κ κ, 3.9 S T, TCG( C) = NS κ T. 3.11, S S α : α < κ. (1) {[S α ] NSκ S : α < κ} P(κ)/NS κ S. (2) α < κ TCG- C α δ : δ S α TCG( C α δ : δ S α ) = NS κ S α. δ S α otp(c α δ ) = ω. NS κ, NS κ S. 3.10, κ S α : α < κ α < κ [S α ] NSκ S = [S α] NSκ S β < α < κ S β S α =. α < κ S α S α., [S α ] NSκ S = [S α] NSκ S S α S α, TCG( C δ α : δ S α ) = NS κ S α, TCG( C δ α : δ S α ) = NS κ S α. C δ : δ S : δ S α α < κ, C δ = C α δ. α < κ, C δ δ Cof( ℵ 2 ) ω., δ S otp(c δ ) = ω. C δ : δ S TCG κ D, κ E : δ S E C δ D. D κ. S = {δ S : C δ D}. S (κ \ S), E., κ \ (S (κ \ S)). κ \ (S (κ \ S)) = (κ \ S) S = S \ S. S \ S S, {[S α ] NSκ S : α < κ}, S α ( S \ S) α < κ. S α S, S α ( S \ S) = S α \ S. TCG( C δ α : δ S α ), S α \ S = {δ S α : Cδ α D} TCG( C δ α : δ S α )., TCG( C δ α : δ S α ) = NS κ S α, S α \ S NS κ S α. S α \ S S α, S α \ S... κ D α : α ω 1. D 0 Cof( ℵ 2 ) κ. α ω 1 D β : β < α D α = β<α D β. D α D α+1. D α κ, 3.12 κ D α+1 δ D α+1 S C δ D α. D α+1 acc(d α ). δ D ω1 ω., otp(d ω1 δ) = ω D ω1. cf(δ) = ω, δ D 0, D 0 δ Cof( ω 2 )., δ S. α < ω 1 δ D α+1 S, C δ D α. C δ \ ζ α D α ζ α < δ. ζ α C δ. C δ = ℵ 0 15

16 page: 16 16, {α < ω 1 : ζ α = ζ} ζ C δ., β < ω 1, ζ α = ζ β α < ω 1 C δ \ ζ D α D β., C δ \ ζ D ω1. γ C δ \ ζ., α < ω 1 γ D α+1 acc(d α )., D α γ γ. cf(γ) ω 2 > ω 1, α<ω 1 (D α γ) γ. α<ω 1 (D α γ) = α<ω 1 D α γ = D ω1 γ., otp(d ω1 δ) otp(d ω1 γ) ω 2., δ D ω1 ω. 4 PCF club guessing sequence S. Shelah [39] PCF,. club guessing sequence. ℵ, PCF ℵ 0 ω < max{ℵ ω4, (2 ℵ 0 ) + }, club guessing sequence X, X U :, F X U F, F = U., Y X Y U X \ Y U. A. ΠA, dom(f) = A α A f(α) < α f. U A, ΠA = U U. f = U g {α A : f(α) = g(α)} U. f U g {α A : f(α) g(α)} U., = U, U. X, X X. Y X, X x, x X y Y y, X Y (X, X ) (cofinal). X (X, X ) cf(x, X ). X cf(x, X ). 4.2 A, pcf(a). pcf(a) = {cf(πa, U ) : U A }. non-principal, principal 17). pcf possible cofinality, PCF. S. Shelah, pcf,. 18) 16

17 page: 17 Club guessing sequence 17 pcf. 4.3 (1) A, A (+) = {κ + : κ A}. (2) A A < min A, progressive. (3), A : µ κ A, µ < λ < κ λ A., A = {ℵ n : n < ω or n = ω + 1}. ℵ ω ℵ 1 ℵ ω+1 A, ℵ ω. 4.4 A progressive. (1) A pcf(a). (2) A A, pcf(a ) pcf(a). (3) pcf(a) max pcf(a). (4) B = A + β B max pcf(b β) < β pcf(a) B. 19) 4.5 X, κ, [X] κ X κ. 4.6 A progressive. (1) pcf(a). (2) cf([sup A] A, ) = max pcf(a)., S. Shelah scale. 4.4(1)(3) 4.6(1), A = {ℵ n : 0 < n < ω} ℵ ω+1 pcf(a)., cf(πa, U) = ℵ ω+1 A U., S. Shelah [39]. f g ΠA, f < fin g {κ A : f(κ) g(κ)} < ℵ 0., κ A f(κ) < g(κ). 4.7 A = {ℵ n : 0 < n < ω}. B A ΠB f α : α < ω ω+1. (1) f α : α < ω ω+1 < fin -. (2) {f α : α < ω ω+1 } (ΠB, < fin )., f ΠB, f < fin f α α < ω ω+1. ΠA/fin ω ω+1 scale. pcf. 4.8 µ., µ D µ + = max pcf(d (+) )., [ 1 ]. 4.9 A = {ℵ n : 0 < n < ω}., pcf(a) < ℵ 4.. pcf(a) ℵ 4. A, 4.6(1) pcf(a). 4.4(1), ℵ 1 A pcf(a)., ℵ ω4 17

18 page: pcf(a) ω 3 Cof(ω 1 ) FCG- C δ : δ ω 3 Cof(ω 1 ). δ ω 3 Cof(ω 1 ) C δ otp(c δ ) = ℵ 1., ℵ ω4 κ α : α < ω 3. κ 0 = ℵ 1. α, β < α κ β, κ α = sup β<α κ β. β α κ β, κ α+1. δ ω 3 Cof(ω 1 ), B α,δ = {κ + β : β nacc(c δ ) β α}, λ α,δ = max pcf(b α,δ ). λ α = sup{λ α,δ : δ ω 3 Cof(ω 1 ) λ α,δ < ℵ ω4 }. λ α ℵ ω4 ℵ 3, ℵ ω4. κ α+1 = (max{κ α, λ α }) +. D 0 = {κ α : α < ω 3 }, η = sup D 0., D 0 η, cf(η) = ω 3 ω η D 1, max pcf(d (+) 1 ) = η +. D 2 = D 0 D 1, D 2 η, D 2 D 0, 4.4(2) max pcf(d (+) 2 ) η +. D (+) 2 η max pcf(d (+) 2 ) η., max pcf(d (+) 2 ) η., max pcf(d (+) 2 ) η +., max pcf(d (+) 2 ) = η +. E = {α < ω 3 : κ α D 2 }., E ω 3. C δ : δ ω 3 Cof(ω 1 ) FCG-, C δ E δ ω 3 Cof(ω 1 ). B = {κ + γ : γ nacc(c δ )}. B 4.4(4). B = ℵ 1 = A +. pcf(a) ℵ ω4, B pcf(a)., κ B max pcf(b κ) < κ. κ B, B κ = κ + γ γ nacc(c δ ). γ nacc(c δ ), α = max(c δ γ) α < γ 20)., B κ = {κ + β : β nacc(c δ ) β α} = B α,δ., λ α,δ = max pcf(b α,δ ) = max pcf(b κ) max pcf(b) max pcf(d (+) 2 ) = η + < ℵ ω4., λ α,δ λ α < κ α+1 κ γ < κ., max pcf(b κ) < κ. 4.4(4). ℵ 4.10 ℵ 0 ω < max{ℵ ω4, (2 ℵ 0 ) + }.., 2 ℵ 0 ℵω, ℵ ℵ 0 ω (2 ℵ 0 ) ℵ 0 = 2 ℵ 0. 2 ℵ 0 < ℵω. A = {ℵ n : 0 < n < ω} (2), cf([ℵ ω ] ℵ 0, ) = max pcf(a) < ℵω4. λ = cf([ℵ ω ] ℵ 0, ), Xα : α < λ ([ℵ ω ] ℵ 0, )., X [ℵω ] ℵ 0 X X α α < λ., ℵ ω ℵ 0 λ 2 ℵ 0 = λ < ℵ ω4 18

19 page: 19 Club guessing sequence 19. pcf FCG-,., pcf,., ZFC, S. Shelah. PCF, M. Burke M. Magidor [ 3 ] U. Abraham M. Magidor [ 1 ]. 5 Universality M. Džamonja, M. Kojman, S. Shelah universality, club guessing sequence.., A. A, B A, A B A B., A B A B., A., U A A universal family, A A A B B U. universal family A universality., B A basis, A A B A B B., A. A A A Q G. Cantor., {Q} A universal family, A universality 1. singleton universal family, universal., A A ω A ( ω) A., ω ω reverse, ω n ( ω) m m n., {ω, ω} basis., A., A, B A A B A B., A B., universal family basis. A., universal family basis,. universality, club guessing sequence. 5.1 (M. Kojman, S. Shelah [29]) θ κ θ < κ < 2 θ.,. (1) θ = ℵ 0, κ = ℵ 1, ω 1 TCG-. (2) θ + < κ, κ universality 2 θ. 19

20 page: 20 20, 2 <κ = κ universality 1 21)., ZFC universality., universality. 5.1, (1) ω 1 TCG- universality.., S = κ Cof(θ). (1) (2), S FCG- C δ : δ S δ S otp(c δ ) = θ. (2), TCG- truly tight, P ξ : ξ < κ. (1) ξ < κ P ξ P(ξ) P ξ < κ. (2) δ S γ nacc(c δ ) C δ γ ξ<γ P ξ. θ + < κ κ Cof(θ) truly tight FCG-, S. Shelah [38]. δ S i < θ, otp(c δ γ) = i γ C δ C δ (i)., C δ (0) C δ, C δ (1) C δ. L, L κ., L = L γ : γ < κ. (1) γ < κ L γ < κ. (2) γ < γ < κ L γ L γ, L γ : γ < κ. (3) γ < κ L γ = ξ<γ L ξ ( ). (4) γ<κ L γ = L. L filtration 22). δ S, inv L, C (δ, x) i < θ : {y L Cδ (i) : y < L x } = {y L Cδ (i) : y < L x} x L Cδ (i+1) \ L Cδ (i). filtration., L = L γ : γ < κ L = L γ : γ < κ L filtration, {γ < κ : L γ = L γ}., filtration A θ, κ L, L L filtration L : {δ κ Lim : x L(inv L, C (δ, x) = A)}. 5.3 L, L L, L κ, L L filtration, f : L L,., κ E : δ S x L, C δ E inv L, C (δ, x) = inv L, C (δ, f(x))., C truly tight FCG construction lemma., A θ (club ) δ x A. 5.3 preservation lemma 20

21 page: 21 Club guessing sequence 21., L L, (TCG( C) ) L L., 5.1. ( 5.1). universality 2 θ µ. µ universal family U. {L α : α < µ} U. α < µ L α filtration L α., B = {inv Lα, C (δ, x) : δ S, x L α α < µ}., B max{κ, µ} B P(θ)., max{κ, µ} < 2 θ, B A θ. 5.2, κ L, L, filtration L κ D : δ S D inv L, C (δ, x) = A x L. U universal family, α < µ f : L L α. 5.3 κ E. E D. C FCG-, Cδ E δ S. δ E D., inv L, C (δ, x) = A x L., inv Lα, C (δ, f(x)) = inv L, C (δ, x) = A., A B. A.,., universality,., M. Džamonja [10]. club guessing sequence, basis. S. Todorcevic minimal walk, J. T. Moore. 5.4 (J. T. Moore [36]) basis ℵ 1. L,. S. Shelah PCF, minimal walk, ZFC. minimal walk S. Todorcevic [48]., J. T. Moore. 5.5 Proper Forcing Axiom(PFA) 23)., {R, ω 1, ω 1, C, C} basis., (1) R R ℵ 1 ℵ 1-24), (2) ω 1 ω 1, (3) ω 1 ω 1 reverse, (4) C Countryman line 25), (5) C C reverse., ZFC, 5 21

22 page: CH, basis 2 ℵ 1 B. Dushnik E. W. Miller [ 9 ]., basis 5. universality,. 6 precipitous, ZFC.,,.,,., κ precipitous.. [16] K. Kunen [31]. I κ. P(κ)/I, V [G]., G P(κ)/I. V κ V G (V κ V )/G., I precipitous. 26), precipitous, R. Solovay [46]., I precipitous, G P(κ)/I, j : V M (V κ V )/G 27).. I, critical point 28) κ. κ κ precipitous., T. Jech, M. Magidor, W. Mitchell, K. Prikry. 6.1 (T. Jech, M. Magidor, W. Mitchell, K. Prikry [25]). (1). (2) ω 1 precipitous. (3) NS ω1 precipitous., critical point. precipitous. precipitous., j., κ TCG- C = C δ : δ κ Lim TCG( C) precipitous. j : V M V [G] TCG( C)., M κ C, : V κ D, C D., outside 22

23 page: 23 Club guessing sequence 23 club guessing, M. Džamonja S. Shelah [11]., precipitous., M. Foreman, M. Magidor, S. Shelah [15]. 6.2 (M. Foreman, M. Magidor, and S. Shelah [15]) κ, λ > κ 29)., λ κ + Levy, NS κ precipitous. precipitous,,., 6.2,.,. 6.3 ( [23]) κ, λ > κ Woodin. λ κ + Levy, κ TCG- C, TCG( C) precipitous., κ + TCG-., κ + precipitous 30). K. Kunen J. Paris [32], κ, 2 2κ κ. precipitous, precipitous., precipitous, 6.3.,,. 7 S. Shelah J. Steel 60 Very Informal Gathering 31) UCLA.,.,,., 4 S. Shelah PCF 5 J. T. Moore ZFC L,.,. ( ),. 8 :,, 23

24 page: 24 24,,,,,,, (, ). ( ). 32) 1) natural structure, canonical structure. 2), ZFC. ZFC, ZFC. 3) pcf, PCF. 4) K. Gödel [18], ZFC. W. H. Woodin.,. 5) Julius König Dénes König, König. 6) X, X otp(x) f : X otp(x). 7), 2 λ = λ + λ < κ κ, 8) Cardinal arithmetic? Yes, it had been a great problem, but now... 9) [39], S. Shelah.,. 10) Kunen [31] c.u.b.. 11) δ,., 12) α + 1 α, D \ (α + 1) = {γ D : γ > α}. 13) [22],. 14) κ ineffable, f : [κ] 2 2 κ X {f({a, b}) : a b X} = 1. [X] 2 X. ineffable. 15), P(κ)/I κ ),, [34] A. Kanamori [27]. 17) X F principal X 0 X F = {Y X : X 0 Y }. 18), pcf, pcf(a)., scattered. 19) localization. A progressive, B progressive pcf(a)., λ pcf(b), B B 0, B 0 A λ pcf(b 0 ) 20), α C δ γ. 21) 2 <θ = θ, T θ T universality 1. 22) γ < κ L γ L.,. 23) PFA T. Jech [24]. 24) X L, L., X, X ℵ 1 - : x y X x < L y, {z X : x < L z < L y} ℵ 1. 25) C, C Countryman line : C 2 chain., X C 2 chain, C 2 coordinate-wise, X, (c 1, c 2 ), (d 1, d 2 ) X c 1 d 1 c 2 d 2, d 1 c 1 d 2 c 2. 26) precipitous, [34]. 27) j : V M, ( ) φ(v 1, v 2,..., v n ) a 1, a 2,..., a n V, V φ(a 1, a 2,..., a n ) M φ(j(a 1 ), j(a 2 ),..., j(a n )). 28) j critical point j(κ) > κ κ. 29) Woodin, N. Goldring [20] 30) I κ precipitous I κ + precipitous., I I,. κ + precipitous 24

25 page: 25 Club guessing sequence 25,, precipitous κ +. 31) Very Informal Gathering(VIG) UCLA 1976, ) This material is based upon work supported by the National Science Foundation under Grant No [ 1 ] U. Abraham and M. Magidor, Cardinal Arithmetic, to appear in the Handbook of Set Theory, [ 2 ] J. E. Baumgartner, A. D. Taylor, and S. Wagon, On Splitting Stationary Subsets of Large Cardinals, J. Symbolic Logic, 42(1977)(2), [ 3 ] M. Burke and M. Magidor, Shelah s pcf theory and its applications, Ann. Pure Appl. Logic, 50(1990)(3), [ 4 ] P. Cohen, The independence of the continuum hypothesis, Proc. Nat. Acad. Sci. U.S.A., 50(1963), [ 5 ] P. Cohen, The independence of the continuum hypothesis. II, Proc. Nat. Acad. Sci. U.S.A., 51(1964), [ 6 ] J. Cummings, Notes on singular cardinal combinatorics, Notre Dame J. Formal Logic, 46(2005)(3), [ 7 ] J. Cummings, M. Foreman, and M. Magidor, Canonical structure in the universe of set theory. I, Ann. Pure Appl. Logic, 129(2004), [ 8 ] J. Cummings, M. Foreman, and M. Magidor, Canonical structure in the universe of set theory. II, Ann. Pure Appl. Logic, 142(2006), [ 9 ] B. Dushnik, and E. Miller, Partially ordered sets, Amer. J. Math., 63(1941), [10] M. Džamonja, Club guessing and the universal models, Notre Dame J. Formal Logic 46(2005)(3), [11] M. Džamonja and S. Shelah, On squares, outside guessing of clubs and I <f [λ], Fund. Math, 148(1995)(2), [12] W. B. Easton, Powers of regular cardinals, Ann. Math. Logic, 1(1970), [13] M. Foreman, Ideals and generic embeddings, to appear in the Handbook of Set Theory [14] M. Foreman, Introduction to the Special Issue on Singular Cardinals Combinatorics, Notre Dame J. Formal Logic 46(2005)(3), 249 [15] M. Foreman, M. Magidor, and S. Shelah, Martin s maximum, saturated ideals, and nonregular ultrafilters. I, Ann. of Math., 127(1988)(1), 1 47 [16],, 20 4 I, ,, 2007 [17] M. Gitik and S. Shelah, Less saturated ideals, Proc. Amer. Math. Soc., 125(1997)(5), [18] K. Gödel, What is Cantor s continuum problem?, Amer. Math. Monthly, 54(1947), [19] K. Gödel, The Consistency of the Continuum Hypothesis, Annals of Mathematics Studies, no. 3, Princeton University Press, Princeton, N. J., 1940 [20] Woodin cardinals and presaturated ideals, N. Goldring, Ann. Pure Appl. Logic, 55(1992)(3), [21] J. Gregory, Higher Souslin trees and the generalized continuum hypothesis, J. Symbolic Logic, 41(1976)(3), [22] Y. Hirata, Nonsaturation of the club filter on P k λ, Master s Thesis at University of Tsukuba, [23] T. Ishiu, Club guessing sequences and filters, J. Symbolic Logic, 70(2005)(4), [24] T. Jech, Set Theory, Springer Monographs in Mathematics, The third millennium edition, revised and expanded, Springer-Verlag, Berlin, 2003 [25] T. Jech, M. Magidor, W. Mitchell, and K. Prikry, Precipitous ideals, J. Symbolic Logic, 45(1980)(1), 1 8 [26] R. B. Jensen, The fine structure of the constructible hierarchy, With a section by Jack Silver, Ann. Math. Logic 4(1972), ; erratum ibid. 4(1972), 443 [27] A. Kanamori, The Higher Infinite: Large Cardinals in Set Theory from Their Beginnings, 2nd edition, Springer-Verlag(2003). :,, (1998) [28] M. Kojman, The A, B, C of PCF: A companion to pcf theory, preprint. [29] M. Kojman and S. Shelah, Nonexistence of universal orders in many cardinals, J. Symbolic Logic, 57(1992)(3), [30] B. König, P. Larson, J. T. Moore, and B. Veličković, Bounding the consistency strength of a five element linear basis, Israel J. Math., 164(2008), 1 18 [31] K. Kunen, Set Theory, An introduction to independence proofs, Reprint of the 1980 original, Studies in Logic and the Foundations of Mathematics 102, North-Holland Publishing Co., Amsterdam, 1983, :,,, 2008 [32] K. Kunen and J. B. Paris, Boolean extensions and measurable cardinals, Ann. Math. Logic, 2(1970/1971)(4), [33] M. Magidor, On the singular cardinals problem. II., Ann. Math. (2), 106(1977)(3), [34],, 20 4 II, ,,

26 page: [35] J. T. Moore, A five element basis for the uncountable linear orders, Ann. of Math., 163(2006)(2), [36] J. T. Moore, A solution to the L space problem, J. Amer. Math. Soc., 19(2006)(3), [37] A. Ros lanowski and S. Shelah, Iteration of λ- complete forcing notions not collapsing λ +, Int. J. Math. Math. Sci., 28(2001)(2), [38] S. Shelah, Advances in cardinal arithmetic, In: Finite and infinite combinatorics in sets and logic (Banff, AB, 1991), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 411, Kluwer Acad. Publ., Dordrecht, 1993 pp [39] S. Shelah, Cardinal Arithmetic, Oxford Logic Guides 29, The Clarendon Press Oxford University Press, New York, 1994 [40] S. Shelah, On successors of singular cardinals, Logic Colloquium 78 (Mons, 1978), Stud. Logic Foundations Math. 97, [41] S. Shelah, Proper and improper forcing, Perspectives in Mathematical Logic, 2nd edition, Perspectives in Mathematical Logic, Springer-Verlag, Berline, 1998 [42] S. Shelah, You can enter Cantor s paradise!, Paul Erdős and his mathematics, II (Budapest, 1999), Bolyai Soc. Math. Stud. 11, , János Bolyai Math. Soc., Budapest, 2002 [43] S. Shelah, Diamonds, preprint, 2008, [44] M. Shioya, Splitting P κ λ into maximally many stationary sets, Israel J. Math., 114(1999), [45] J. Silver, On the singular cardinals problem, Proceedings of the International Congress of Mathematicians (Vancouver, B. C., 1974), Vol. 1, , Canad. Math. Congress, Montreal, Que., 1975 [46] R. Solovay, Real-valued measurable cardinals, In Axiomatic Set Theory (edited by D. Scott), , American Mathematical Society, 1971 [47] J. Steel and R. Van Wesep, Two consequences of determinacy consistent with choice. Trans. Amer. Math. Soc. 272(1982)(1), [48] S. Todorcevic, Coherent Sequences, to appear in the Handbook of Set Theory ( ) ( Miami University) 26