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Large cardinals and pcf theory in topology and infinite combinatorics Lajos Soukup Alfrd Rnyi Institute of Mathematics Hungarian Academy of Sciences RIMS Set Theory Workshop 2010 Soukup, L (HAS) RIMS 2010 1 / 23 Content Pcf theory and

  1. Cofinality spectrum of groups Sym ( ω ) the group of all permutation of natural numbers Define the cofinality spectrum of Sym ( ω ) as follows: λ ∈ CF ( Sym ( ω )) iff Sym ( ω ) is the union of an increasing chain of λ proper subgroups. � ω increasing � Shelah and Thomas: (1) if { κ n : n < ω } ∈ ( Sym ( ω )) then pcf ( { κ n : n < ω } ) ⊂ CF ( Sym ( ω )) (2) IF GCH holds and K ⊂ Reg s.t Soukup, L (HAS) RIMS 2010 8 / 23

  2. Cofinality spectrum of groups Sym ( ω ) the group of all permutation of natural numbers Define the cofinality spectrum of Sym ( ω ) as follows: λ ∈ CF ( Sym ( ω )) iff Sym ( ω ) is the union of an increasing chain of λ proper subgroups. � ω increasing � Shelah and Thomas: (1) if { κ n : n < ω } ∈ ( Sym ( ω )) then pcf ( { κ n : n < ω } ) ⊂ CF ( Sym ( ω )) (2) IF GCH holds and K ⊂ Reg s.t (i) K has maximal elements, Soukup, L (HAS) RIMS 2010 8 / 23

  3. Cofinality spectrum of groups Sym ( ω ) the group of all permutation of natural numbers Define the cofinality spectrum of Sym ( ω ) as follows: λ ∈ CF ( Sym ( ω )) iff Sym ( ω ) is the union of an increasing chain of λ proper subgroups. � ω increasing � Shelah and Thomas: (1) if { κ n : n < ω } ∈ ( Sym ( ω )) then pcf ( { κ n : n < ω } ) ⊂ CF ( Sym ( ω )) (2) IF GCH holds and K ⊂ Reg s.t (i) K has maximal elements, (ii) if µ is singular, sup ( K ∩ µ ) = µ then µ + ∈ K Soukup, L (HAS) RIMS 2010 8 / 23

  4. Cofinality spectrum of groups Sym ( ω ) the group of all permutation of natural numbers Define the cofinality spectrum of Sym ( ω ) as follows: λ ∈ CF ( Sym ( ω )) iff Sym ( ω ) is the union of an increasing chain of λ proper subgroups. � ω increasing � Shelah and Thomas: (1) if { κ n : n < ω } ∈ ( Sym ( ω )) then pcf ( { κ n : n < ω } ) ⊂ CF ( Sym ( ω )) (2) IF GCH holds and K ⊂ Reg s.t (i) K has maximal elements, (ii) if µ is singular, sup ( K ∩ µ ) = µ then µ + ∈ K (iii) if µ is inaccessible, sup ( K ∩ µ ) = µ then µ ∈ K , Soukup, L (HAS) RIMS 2010 8 / 23

  5. Cofinality spectrum of groups Sym ( ω ) the group of all permutation of natural numbers Define the cofinality spectrum of Sym ( ω ) as follows: λ ∈ CF ( Sym ( ω )) iff Sym ( ω ) is the union of an increasing chain of λ proper subgroups. � ω increasing � Shelah and Thomas: (1) if { κ n : n < ω } ∈ ( Sym ( ω )) then pcf ( { κ n : n < ω } ) ⊂ CF ( Sym ( ω )) (2) IF GCH holds and K ⊂ Reg s.t (i) K has maximal elements, (ii) if µ is singular, sup ( K ∩ µ ) = µ then µ + ∈ K (iii) if µ is inaccessible, sup ( K ∩ µ ) = µ then µ ∈ K , THEN CF ( Sym ( ω )) = K in some c.c.c generic extension Soukup, L (HAS) RIMS 2010 8 / 23

  6. Cofinality spectrum of groups Sym ( ω ) the group of all permutation of natural numbers Define the cofinality spectrum of Sym ( ω ) as follows: λ ∈ CF ( Sym ( ω )) iff Sym ( ω ) is the union of an increasing chain of λ proper subgroups. � ω increasing � Shelah and Thomas: (1) if { κ n : n < ω } ∈ ( Sym ( ω )) then pcf ( { κ n : n < ω } ) ⊂ CF ( Sym ( ω )) (2) IF GCH holds and K ⊂ Reg s.t (i) K has maximal elements, (ii) if µ is singular, sup ( K ∩ µ ) = µ then µ + ∈ K (iii) if µ is inaccessible, sup ( K ∩ µ ) = µ then µ ∈ K , THEN CF ( Sym ( ω )) = K in some c.c.c generic extension Problem : Full characterization of CF ( Sym ( ω )) Soukup, L (HAS) RIMS 2010 8 / 23

  7. Additivity spectrum of ideals Soukup, L (HAS) RIMS 2010 9 / 23

  8. Additivity spectrum of ideals I ideal Soukup, L (HAS) RIMS 2010 9 / 23

  9. Additivity spectrum of ideals I ideal ADD ( I ) : the additivity spectrum of I . Soukup, L (HAS) RIMS 2010 9 / 23

  10. Additivity spectrum of ideals I ideal ADD ( I ) : the additivity spectrum of I . κ ∈ ADD ( I ) iff there is an increasing chain { A α : α < κ } ⊂ I with ∪ α<κ A α / ∈ I . Soukup, L (HAS) RIMS 2010 9 / 23

  11. Additivity spectrum of ideals I ideal ADD ( I ) : the additivity spectrum of I . κ ∈ ADD ( I ) iff there is an increasing chain { A α : α < κ } ⊂ I with ∪ α<κ A α / ∈ I . M meager ideal; N null ideal Soukup, L (HAS) RIMS 2010 9 / 23

  12. Additivity spectrum of ideals I ideal ADD ( I ) : the additivity spectrum of I . κ ∈ ADD ( I ) iff there is an increasing chain { A α : α < κ } ⊂ I with ∪ α<κ A α / ∈ I . M meager ideal; N null ideal add ( M ) = min ( ADD ( M )) , add ( N ) = min ( ADD ( N )) Soukup, L (HAS) RIMS 2010 9 / 23

  13. Additivity spectrum of ideals I ideal ADD ( I ) : the additivity spectrum of I . κ ∈ ADD ( I ) iff there is an increasing chain { A α : α < κ } ⊂ I with ∪ α<κ A α / ∈ I . M meager ideal; N null ideal add ( M ) = min ( ADD ( M )) , add ( N ) = min ( ADD ( N )) B the σ -ideal generated by the compact subsets of the irrationals. Soukup, L (HAS) RIMS 2010 9 / 23

  14. Additivity spectrum of ideals I ideal ADD ( I ) : the additivity spectrum of I . κ ∈ ADD ( I ) iff there is an increasing chain { A α : α < κ } ⊂ I with ∪ α<κ A α / ∈ I . M meager ideal; N null ideal add ( M ) = min ( ADD ( M )) , add ( N ) = min ( ADD ( N )) B the σ -ideal generated by the compact subsets of the irrationals. ( R \ Q ) ≈ ω ω Soukup, L (HAS) RIMS 2010 9 / 23

  15. Additivity spectrum of ideals I ideal ADD ( I ) : the additivity spectrum of I . κ ∈ ADD ( I ) iff there is an increasing chain { A α : α < κ } ⊂ I with ∪ α<κ A α / ∈ I . M meager ideal; N null ideal add ( M ) = min ( ADD ( M )) , add ( N ) = min ( ADD ( N )) B the σ -ideal generated by the compact subsets of the irrationals. ( R \ Q ) ≈ ω ω A ⊂ R \ Q is compact iff A is ≤ -bounded in � ω ω , ≤� . Soukup, L (HAS) RIMS 2010 9 / 23

  16. Additivity spectrum of ideals I ideal ADD ( I ) : the additivity spectrum of I . κ ∈ ADD ( I ) iff there is an increasing chain { A α : α < κ } ⊂ I with ∪ α<κ A α / ∈ I . M meager ideal; N null ideal add ( M ) = min ( ADD ( M )) , add ( N ) = min ( ADD ( N )) B the σ -ideal generated by the compact subsets of the irrationals. ( R \ Q ) ≈ ω ω A ⊂ R \ Q is compact iff A is ≤ -bounded in � ω ω , ≤� . B = { B ⊂ ω ω : B is ≤ ∗ -bounded in � ω ω , ≤ ∗ �} Soukup, L (HAS) RIMS 2010 9 / 23

  17. Additivity spectrum of ideals Soukup, L (HAS) RIMS 2010 10 / 23

  18. Additivity spectrum of ideals ADD ( I , A )= { κ ∈ Reg : ∃ increasing { A α : α < κ } ⊂ I s.t. ∪ α<κ A α = A } Soukup, L (HAS) RIMS 2010 10 / 23

  19. Additivity spectrum of ideals ADD ( I , A )= { κ ∈ Reg : ∃ increasing { A α : α < κ } ⊂ I s.t. ∪ α<κ A α = A } ADD ( I )= ∪{ ADD ( I , A ) : A ∈ I + } . Soukup, L (HAS) RIMS 2010 10 / 23

  20. Additivity spectrum of ideals ADD ( I , A )= { κ ∈ Reg : ∃ increasing { A α : α < κ } ⊂ I s.t. ∪ α<κ A α = A } ADD ( I )= ∪{ ADD ( I , A ) : A ∈ I + } . Theorem Assume that I ⊂ P ( I ) is a σ -complete ideal, Y ∈ I + , and A ⊂ ADD ( I , Y ) is countable. Then pcf ( A ) ⊂ ADD ( I , Y ) . Soukup, L (HAS) RIMS 2010 10 / 23

  21. pcf ( A ) ⊂ ADD ( I , Y ) for countable A ⊂ ADD ( I , Y ) α : α < a } ⊂ I increasing � F a = Y . For a ∈ A let F a = { F a Soukup, L (HAS) RIMS 2010 11 / 23

  22. pcf ( A ) ⊂ ADD ( I , Y ) for countable A ⊂ ADD ( I , Y ) α : α < a } ⊂ I increasing � F a = Y . For a ∈ A let F a = { F a Let κ ∈ pcf ( A ) . Fix an ultrafilter U on A such that cf ( � A / U ) = κ Soukup, L (HAS) RIMS 2010 11 / 23

  23. pcf ( A ) ⊂ ADD ( I , Y ) for countable A ⊂ ADD ( I , Y ) α : α < a } ⊂ I increasing � F a = Y . For a ∈ A let F a = { F a Let κ ∈ pcf ( A ) . Fix an ultrafilter U on A such that cf ( � A / U ) = κ Let { g α : α < κ } ⊂ � A be ≤ U -increasing, ≤ U -cofinal sequence. Soukup, L (HAS) RIMS 2010 11 / 23

  24. pcf ( A ) ⊂ ADD ( I , Y ) for countable A ⊂ ADD ( I , Y ) α : α < a } ⊂ I increasing � F a = Y . For a ∈ A let F a = { F a Let κ ∈ pcf ( A ) . Fix an ultrafilter U on A such that cf ( � A / U ) = κ Let { g α : α < κ } ⊂ � A be ≤ U -increasing, ≤ U -cofinal sequence. For g ∈ � A let U ( g )= � x ∈ I : { a ∈ A : x ∈ F a � g ( a ) } ∈ U . Soukup, L (HAS) RIMS 2010 11 / 23

  25. pcf ( A ) ⊂ ADD ( I , Y ) for countable A ⊂ ADD ( I , Y ) α : α < a } ⊂ I increasing � F a = Y . For a ∈ A let F a = { F a Let κ ∈ pcf ( A ) . Fix an ultrafilter U on A such that cf ( � A / U ) = κ Let { g α : α < κ } ⊂ � A be ≤ U -increasing, ≤ U -cofinal sequence. For g ∈ � A let U ( g )= � x ∈ I : { a ∈ A : x ∈ F a � g ( a ) } ∈ U . � U ( g α ) : α < κ � witnesses that κ ∈ ADD ( I , Y ) Soukup, L (HAS) RIMS 2010 11 / 23

  26. pcf ( A ) ⊂ ADD ( I , Y ) for countable A ⊂ ADD ( I , Y ) α : α < a } ⊂ I increasing � F a = Y . For a ∈ A let F a = { F a Let κ ∈ pcf ( A ) . Fix an ultrafilter U on A such that cf ( � A / U ) = κ Let { g α : α < κ } ⊂ � A be ≤ U -increasing, ≤ U -cofinal sequence. For g ∈ � A let U ( g )= � x ∈ I : { a ∈ A : x ∈ F a � g ( a ) } ∈ U . � U ( g α ) : α < κ � witnesses that κ ∈ ADD ( I , Y ) (1) U ( g ) ∈ I for each g ∈ � A Soukup, L (HAS) RIMS 2010 11 / 23

  27. pcf ( A ) ⊂ ADD ( I , Y ) for countable A ⊂ ADD ( I , Y ) α : α < a } ⊂ I increasing � F a = Y . For a ∈ A let F a = { F a Let κ ∈ pcf ( A ) . Fix an ultrafilter U on A such that cf ( � A / U ) = κ Let { g α : α < κ } ⊂ � A be ≤ U -increasing, ≤ U -cofinal sequence. For g ∈ � A let U ( g )= � x ∈ I : { a ∈ A : x ∈ F a � g ( a ) } ∈ U . � U ( g α ) : α < κ � witnesses that κ ∈ ADD ( I , Y ) (1) U ( g ) ∈ I for each g ∈ � A (2) If g 1 ≤ I g 2 then U ( g 1 ) ⊂ U ( g 2 ) . Soukup, L (HAS) RIMS 2010 11 / 23

  28. pcf ( A ) ⊂ ADD ( I , Y ) for countable A ⊂ ADD ( I , Y ) α : α < a } ⊂ I increasing � F a = Y . For a ∈ A let F a = { F a Let κ ∈ pcf ( A ) . Fix an ultrafilter U on A such that cf ( � A / U ) = κ Let { g α : α < κ } ⊂ � A be ≤ U -increasing, ≤ U -cofinal sequence. For g ∈ � A let U ( g )= � x ∈ I : { a ∈ A : x ∈ F a � g ( a ) } ∈ U . � U ( g α ) : α < κ � witnesses that κ ∈ ADD ( I , Y ) (1) U ( g ) ∈ I for each g ∈ � A (2) If g 1 ≤ I g 2 then U ( g 1 ) ⊂ U ( g 2 ) . (3) � { U ( g α ) : α < κ } = Y . Soukup, L (HAS) RIMS 2010 11 / 23

  29. pcf ( A ) ⊂ ADD ( I , Y ) for countable A ⊂ ADD ( I , Y ) α : α < a } ⊂ I increasing � F a = Y . For a ∈ A let F a = { F a Let κ ∈ pcf ( A ) . Fix an ultrafilter U on A such that cf ( � A / U ) = κ Let { g α : α < κ } ⊂ � A be ≤ U -increasing, ≤ U -cofinal sequence. For g ∈ � A let U ( g )= � x ∈ I : { a ∈ A : x ∈ F a � g ( a ) } ∈ U . � U ( g α ) : α < κ � witnesses that κ ∈ ADD ( I , Y ) (1) U ( g ) ∈ I for each g ∈ � A (2) If g 1 ≤ I g 2 then U ( g 1 ) ⊂ U ( g 2 ) . (3) � { U ( g α ) : α < κ } = Y . Does A ⊂ ADD ( I ) imply pcf ( A ) ⊂ ADD ( I ) ? What happens if | A | = ω ? Soukup, L (HAS) RIMS 2010 11 / 23

  30. The ideals B and N : restrictions Soukup, L (HAS) RIMS 2010 12 / 23

  31. The ideals B and N : restrictions Theorem If A ⊂ ADD ( N ) is countable, then pcf ( A ) ⊂ ADD ( N ) . Soukup, L (HAS) RIMS 2010 12 / 23

  32. The ideals B and N : restrictions Theorem If A ⊂ ADD ( N ) is countable, then pcf ( A ) ⊂ ADD ( N ) . Problem: What about M ? Soukup, L (HAS) RIMS 2010 12 / 23

  33. The ideals B and N : restrictions Theorem If A ⊂ ADD ( N ) is countable, then pcf ( A ) ⊂ ADD ( N ) . Problem: What about M ? If κ n ∈ ADD ( N , Y n ) for n ∈ ω then { κ n : n < ω } ⊂ ADD ( N , Y ) for some Y ∈ N + . Soukup, L (HAS) RIMS 2010 12 / 23

  34. The ideals B and N : restrictions Theorem If A ⊂ ADD ( N ) is countable, then pcf ( A ) ⊂ ADD ( N ) . Problem: What about M ? If κ n ∈ ADD ( N , Y n ) for n ∈ ω then { κ n : n < ω } ⊂ ADD ( N , Y ) for some Y ∈ N + . Problem: Given κ n ∈ ADD ( M , Y n ) for n ∈ ω then find Y ∈ M + s.t. { κ n : n < ω } ⊂ ADD ( M , Y ) . Soukup, L (HAS) RIMS 2010 12 / 23

  35. The ideals B and N : restrictions Theorem If A ⊂ ADD ( N ) is countable, then pcf ( A ) ⊂ ADD ( N ) . Problem: What about M ? If κ n ∈ ADD ( N , Y n ) for n ∈ ω then { κ n : n < ω } ⊂ ADD ( N , Y ) for some Y ∈ N + . Problem: Given κ n ∈ ADD ( M , Y n ) for n ∈ ω then find Y ∈ M + s.t. { κ n : n < ω } ⊂ ADD ( M , Y ) . B is the σ -ideal generated by the compact subsets of the irrationals. Soukup, L (HAS) RIMS 2010 12 / 23

  36. The ideals B and N : restrictions Theorem If A ⊂ ADD ( N ) is countable, then pcf ( A ) ⊂ ADD ( N ) . Problem: What about M ? If κ n ∈ ADD ( N , Y n ) for n ∈ ω then { κ n : n < ω } ⊂ ADD ( N , Y ) for some Y ∈ N + . Problem: Given κ n ∈ ADD ( M , Y n ) for n ∈ ω then find Y ∈ M + s.t. { κ n : n < ω } ⊂ ADD ( M , Y ) . B is the σ -ideal generated by the compact subsets of the irrationals. Theorem If A ⊂ ADD ( B ) is progressive and | A | < h , then pcf ( A ) ⊂ ADD ( B ) . Soukup, L (HAS) RIMS 2010 12 / 23

  37. The ideals B and N : construction Soukup, L (HAS) RIMS 2010 13 / 23

  38. The ideals B and N : construction Theorem Assume that I is one of the ideals B , M and N . Soukup, L (HAS) RIMS 2010 13 / 23

  39. The ideals B and N : construction Theorem Assume that I is one of the ideals B , M and N . If A = pcf ( A ) is a non-empty set of uncountable regular cardinals, | A | < min ( A ) + n for some n ∈ ω , then A = ADD ( I ) in some c.c.c generic extension V P . Soukup, L (HAS) RIMS 2010 13 / 23

  40. The ideals B and N : construction Theorem Assume that I is one of the ideals B , M and N . If A = pcf ( A ) is a non-empty set of uncountable regular cardinals, | A | < min ( A ) + n for some n ∈ ω , then A = ADD ( I ) in some c.c.c generic extension V P . If ∅ � = Y ⊂ pcf ( {ℵ n : 1 ≤ n < ω } ) then pcf ( Y ) = ADD ( I ) in some c.c.c generic extension V P . Soukup, L (HAS) RIMS 2010 13 / 23

  41. The ideals B and N : construction Theorem Assume that I is one of the ideals B , M and N . If A = pcf ( A ) is a non-empty set of uncountable regular cardinals, | A | < min ( A ) + n for some n ∈ ω , then A = ADD ( I ) in some c.c.c generic extension V P . If ∅ � = Y ⊂ pcf ( {ℵ n : 1 ≤ n < ω } ) then pcf ( Y ) = ADD ( I ) in some c.c.c generic extension V P . If ℵ ω + 1 < max pcf ( {ℵ n : 1 ≤ n < ω } ) then there is an infinite Y ⊂ {ℵ n : 1 ≤ n < ω } such that ADD ( I ) = Y ∪ {ℵ ω + 2 } in some c.c.c generic extension V P . Soukup, L (HAS) RIMS 2010 13 / 23

  42. The ideals B and N : construction Theorem: Assume that I is one of the ideals B , M and N . If A = pcf ( A ) is a non-empty set of uncountable regular cardinals, | A | < min ( A ) + n for some n ∈ ω , then A = ADD ( I ) in some c.c.c generic extension V P . Soukup, L (HAS) RIMS 2010 14 / 23

  43. The ideals B and N : construction Theorem: Assume that I is one of the ideals B , M and N . If A = pcf ( A ) is a non-empty set of uncountable regular cardinals, | A | < min ( A ) + n for some n ∈ ω , then A = ADD ( I ) in some c.c.c generic extension V P . The ideals B , M and N have the Hechler property Soukup, L (HAS) RIMS 2010 14 / 23

  44. The ideals B and N : construction Theorem: Assume that I is one of the ideals B , M and N . If A = pcf ( A ) is a non-empty set of uncountable regular cardinals, | A | < min ( A ) + n for some n ∈ ω , then A = ADD ( I ) in some c.c.c generic extension V P . The ideals B , M and N have the Hechler property I has the Hechler property iff given any σ -directed poset Q there is a c.c.c poset P such that V P | = a cofinal subset { I q : q ∈ Q } of �I , ⊂� is isomorphic to Q . Soukup, L (HAS) RIMS 2010 14 / 23

  45. Hechler property Hechler: B has the Hechler property, Soukup, L (HAS) RIMS 2010 15 / 23

  46. Hechler property Hechler: B has the Hechler property, Bartoszynski and Kada: M has the Hechler property, Soukup, L (HAS) RIMS 2010 15 / 23

  47. Hechler property Hechler: B has the Hechler property, Bartoszynski and Kada: M has the Hechler property, Burke and Kada: N has the Hechler property. Soukup, L (HAS) RIMS 2010 15 / 23

  48. Hechler property Hechler: B has the Hechler property, Bartoszynski and Kada: M has the Hechler property, Burke and Kada: N has the Hechler property. Hechler: � ω ω , ≤ ∗ � has the Hechler property, Soukup, L (HAS) RIMS 2010 15 / 23

  49. Hechler property Hechler: B has the Hechler property, Bartoszynski and Kada: M has the Hechler property, Burke and Kada: N has the Hechler property. Hechler: � ω ω , ≤ ∗ � has the Hechler property, Define map Φ : � ω ω , ≤ ∗ � → B by the formula Φ( b ) = { x : x ≤ ∗ b } Soukup, L (HAS) RIMS 2010 15 / 23

  50. Hechler property Hechler: B has the Hechler property, Bartoszynski and Kada: M has the Hechler property, Burke and Kada: N has the Hechler property. Hechler: � ω ω , ≤ ∗ � has the Hechler property, Define map Φ : � ω ω , ≤ ∗ � → B by the formula Φ( b ) = { x : x ≤ ∗ b } Φ is a natural, cofinal, order preserving embedding. Soukup, L (HAS) RIMS 2010 15 / 23

  51. How to obtain a model of A = ADD ( I ) ? Soukup, L (HAS) RIMS 2010 16 / 23

  52. How to obtain a model of A = ADD ( I ) ? A = pcf ( A ) , | A | < min ( A ) + n Soukup, L (HAS) RIMS 2010 16 / 23

  53. How to obtain a model of A = ADD ( I ) ? A = pcf ( A ) , | A | < min ( A ) + n Q = � � A , ≤� . Soukup, L (HAS) RIMS 2010 16 / 23

  54. How to obtain a model of A = ADD ( I ) ? A = pcf ( A ) , | A | < min ( A ) + n Q = � � A , ≤� . I has the Hechler property: f : � Q , ≤� ֒ → �I , ⊂� Soukup, L (HAS) RIMS 2010 16 / 23

  55. How to obtain a model of A = ADD ( I ) ? A = pcf ( A ) , | A | < min ( A ) + n Q = � � A , ≤� . I has the Hechler property: f : � Q , ≤� ֒ → �I , ⊂� A ⊂ ADD ( I ) is easy Soukup, L (HAS) RIMS 2010 16 / 23

  56. How to obtain a model of A = ADD ( I ) ? A = pcf ( A ) , | A | < min ( A ) + n Q = � � A , ≤� . I has the Hechler property: f : � Q , ≤� ֒ → �I , ⊂� A ⊂ ADD ( I ) is easy Need: λ / ∈ A then λ / ∈ ADD ( I ) Soukup, L (HAS) RIMS 2010 16 / 23

  57. How to obtain a model of A = ADD ( I ) ? A = pcf ( A ) , | A | < min ( A ) + n Q = � � A , ≤� . I has the Hechler property: f : � Q , ≤� ֒ → �I , ⊂� A ⊂ ADD ( I ) is easy Need: λ / ∈ A then λ / ∈ ADD ( I ) Key observation: If B = pcf ( B ) is a progressive set of regular cardinals , λ / ∈ B , then for each { f i : i < λ } ⊂ � B there is g ∈ � B such that |{ i : f i ≤ g }| = λ . Soukup, L (HAS) RIMS 2010 16 / 23

  58. The ideals B and N Soukup, L (HAS) RIMS 2010 17 / 23

  59. The ideals B and N Theorem Assume that I = B or I = N . Given a nonempty, countable subset A of uncountable regular cardinals, T.F .A.E Soukup, L (HAS) RIMS 2010 17 / 23

  60. The ideals B and N Theorem Assume that I = B or I = N . Given a nonempty, countable subset A of uncountable regular cardinals, T.F .A.E A = pcf ( A ) Soukup, L (HAS) RIMS 2010 17 / 23

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