Volume 12, 2000

University of Bialystok

Copyright (c) 2000 Association of Mizar Users

### The abstract of the Mizar article:

### Basic Facts about Inaccessible and Measurable Cardinals

**by****Josef Urban**- Received April 14, 2000
- MML identifier: CARD_FIL

- [ Mizar article, MML identifier index ]

environ vocabulary ORDINAL1, ORDINAL2, BOOLE, FINSET_1, CARD_1, CARD_5, SETFAM_1, FILTER_0, SUBSET_1, FILTER_2, TARSKI, WAYBEL_0, ZFMISC_1, CARD_2, FUNCT_1, RELAT_1, WAYBEL11, FUNCT_2, CARD_3, PRALG_1, FUNCT_5, CARD_FIL, HAHNBAN; notation TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, RELAT_1, FUNCT_1, PRALG_1, BINOP_1, FUNCT_5, SETFAM_1, FINSET_1, ORDINAL1, FUNCT_2, ORDINAL2, CARD_1, CARD_2, CARD_3, CARD_5; constructors XREAL_0, WELLORD2, BINOP_1, PRALG_1, CARD_2, CARD_5, ZFREFLE1, XCMPLX_0, MEMBERED; clusters FUNCT_1, ORDINAL1, CARD_1, CARD_5, RELSET_1, FINSET_1, SUBSET_1, PRALG_1, SETFAM_1, MEMBERED, NUMBERS, ORDINAL2; requirements NUMERALS, SUBSET, BOOLE; begin :: this should be in ordinal2 definition cluster being_limit_ordinal Ordinal; end; definition let X,Y be set; redefine func X \ Y -> Subset of X; end; theorem :: CARD_FIL:1 for x being set for X being infinite set holds Card {x} <` Card X; definition let X be infinite set; cluster Card X -> infinite; end; scheme ElemProp{D()-> non empty set,x()->set,P[set]}: P[x()] provided x() in {y where y is Element of D(): P[y]}; :::::::::::::::::::::: Initial reservations ::::::::::::::::::::: reserve N for Cardinal; reserve M for Aleph; reserve X for non empty set; reserve Y,Z,Z1,Z2,Y1,Y2,Y3,Y4 for Subset of X; reserve S for Subset-Family of X; reserve x for set; ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :: :: :: Necessary facts about filters and ideals on sets :: :: :: ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: theorem :: CARD_FIL:2 { X } is non empty Subset-Family of X & not {} in { X } & for Y1,Y2 holds (Y1 in { X } & Y2 in { X } implies Y1 /\ Y2 in { X }) & ((Y1 in { X } & Y1 c= Y2) implies Y2 in { X }); definition let X; mode Filter of X -> non empty Subset-Family of X means :: CARD_FIL:def 1 (not {} in it) & for Y1,Y2 holds (Y1 in it & Y2 in it implies Y1 /\ Y2 in it) & ((Y1 in it & Y1 c= Y2) implies Y2 in it); end; theorem :: CARD_FIL:3 for F being set holds F is Filter of X iff (F is non empty Subset-Family of X & not {} in F & for Y1,Y2 holds (Y1 in F & Y2 in F implies Y1 /\ Y2 in F) & ((Y1 in F & Y1 c= Y2) implies Y2 in F)); theorem :: CARD_FIL:4 { X } is Filter of X; reserve F,Uf for Filter of X; theorem :: CARD_FIL:5 X in F; theorem :: CARD_FIL:6 Y in F implies not (X \ Y) in F; theorem :: CARD_FIL:7 for I being non empty Subset of bool X st (for Y holds Y in I iff Y` in F) holds (not X in I) & for Y1,Y2 holds (Y1 in I & Y2 in I implies Y1 \/ Y2 in I) & ((Y1 in I & Y2 c= Y1) implies Y2 in I); definition let X,S; redefine func COMPLEMENT S; synonym dual S; end; reserve S for non empty Subset-Family of X; definition let X,S; cluster COMPLEMENT S -> non empty; end; theorem :: CARD_FIL:8 dual S = {Y:Y` in S}; theorem :: CARD_FIL:9 dual S = {Y`:Y in S}; definition let X; mode Ideal of X -> non empty Subset-Family of X means :: CARD_FIL:def 2 (not X in it) & for Y1,Y2 holds (Y1 in it & Y2 in it implies Y1 \/ Y2 in it) & ((Y1 in it & Y2 c= Y1) implies Y2 in it); end; definition let X,F; redefine func dual F -> Ideal of X; end; reserve I for Ideal of X; theorem :: CARD_FIL:10 (for Y holds not (Y in F & Y in dual F)) & (for Y holds not (Y in I & Y in dual I)); theorem :: CARD_FIL:11 {} in I; definition let X,N,S; pred S is_multiplicative_with N means :: CARD_FIL:def 3 for S1 being non empty set st (S1 c= S & Card S1 <` N) holds meet S1 in S; end; definition let X,N,S; pred S is_additive_with N means :: CARD_FIL:def 4 for S1 being non empty set st (S1 c= S & Card S1 <` N) holds union S1 in S; end; definition let X,N,F; redefine pred F is_multiplicative_with N; synonym F is_complete_with N; end; definition let X,N,I; redefine pred I is_additive_with N; synonym I is_complete_with N; end; theorem :: CARD_FIL:12 S is_multiplicative_with N implies dual S is_additive_with N; definition let X,F; attr F is uniform means :: CARD_FIL:def 5 for Y holds Y in F implies Card Y = Card X; attr F is principal means :: CARD_FIL:def 6 ex Y st Y in F & ( for Z holds Z in F implies Y c= Z); attr F is being_ultrafilter means :: CARD_FIL:def 7 for Y holds Y in F or (X \ Y) in F; end; definition let X,F,Z; func Extend_Filter(F,Z) -> non empty Subset-Family of X equals :: CARD_FIL:def 8 {Y: ex Y2 st (Y2 in {Y1 /\ Z : Y1 in F} & Y2 c= Y)}; end; theorem :: CARD_FIL:13 for Z1 holds ( Z1 in Extend_Filter(F,Z) iff ex Z2 st Z2 in F & Z2 /\ Z c= Z1) ; theorem :: CARD_FIL:14 (for Y1 st Y1 in F holds Y1 meets Z) implies ( Z in Extend_Filter(F,Z) & Extend_Filter(F,Z) is Filter of X & F c= Extend_Filter(F,Z) ); reserve S,S1 for Subset of bool X; definition let X; func Filters(X) -> non empty Subset-Family of bool X equals :: CARD_FIL:def 9 {S where S is Subset of bool X : S is Filter of X}; end; theorem :: CARD_FIL:15 for S being set holds S in Filters(X) iff S is Filter of X; reserve FS for non empty Subset of Filters(X); theorem :: CARD_FIL:16 FS is c=-linear implies union FS is Filter of X; theorem :: CARD_FIL:17 for F ex Uf st F c= Uf & Uf is being_ultrafilter; reserve X for infinite set; reserve Y,Y1,Y2,Z for Subset of X; reserve F,Uf for Filter of X; definition let X; func Frechet_Filter(X) -> Filter of X equals :: CARD_FIL:def 10 { Y : Card (X \ Y) <` Card X}; end; definition let X; func Frechet_Ideal(X) -> Ideal of X equals :: CARD_FIL:def 11 dual Frechet_Filter(X); end; theorem :: CARD_FIL:18 Y in Frechet_Filter(X) iff Card (X \ Y) <` Card X; theorem :: CARD_FIL:19 Y in Frechet_Ideal(X) iff Card Y <` Card X; theorem :: CARD_FIL:20 Frechet_Filter(X) c= F implies F is uniform; theorem :: CARD_FIL:21 Uf is uniform being_ultrafilter implies Frechet_Filter(X) c= Uf; definition let X; cluster non principal being_ultrafilter Filter of X; end; definition let X; cluster uniform being_ultrafilter -> non principal Filter of X; end; theorem :: CARD_FIL:22 for F being being_ultrafilter Filter of X for Y holds Y in F iff not Y in dual F; reserve x for Element of X; theorem :: CARD_FIL:23 F is non principal being_ultrafilter & F is_complete_with Card X implies F is uniform; begin ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :: :: :: Inaccessible and measurable cardinals, Ulam matrix :: :: :: ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: theorem :: CARD_FIL:24 nextcard N <=` exp(2,N); definition pred GCH means :: CARD_FIL:def 12 for N being Aleph holds nextcard N = exp(2,N); end; definition let IT be Aleph; attr IT is inaccessible means :: CARD_FIL:def 13 IT is regular limit; synonym IT is_inaccessible_cardinal; end; definition cluster inaccessible -> regular limit Aleph; end; theorem :: CARD_FIL:25 alef 0 is inaccessible; definition let IT be Aleph; attr IT is strong_limit means :: CARD_FIL:def 14 for N st N <` IT holds exp(2,N) <` IT; synonym IT is_strong_limit_cardinal; end; theorem :: CARD_FIL:26 alef 0 is strong_limit; theorem :: CARD_FIL:27 M is strong_limit implies M is limit; definition cluster strong_limit -> limit Aleph; end; theorem :: CARD_FIL:28 GCH implies (M is limit implies M is strong_limit); definition let IT be Aleph; attr IT is strongly_inaccessible means :: CARD_FIL:def 15 IT is regular strong_limit; synonym IT is_strongly_inaccessible_cardinal; end; definition cluster strongly_inaccessible -> regular strong_limit Aleph; end; theorem :: CARD_FIL:29 alef 0 is strongly_inaccessible; theorem :: CARD_FIL:30 M is strongly_inaccessible implies M is inaccessible; definition cluster strongly_inaccessible -> inaccessible Aleph; end; theorem :: CARD_FIL:31 GCH implies ( M is inaccessible implies M is strongly_inaccessible); definition let M; attr M is measurable means :: CARD_FIL:def 16 ex Uf being Filter of M st Uf is_complete_with M & Uf is non principal being_ultrafilter; synonym M is_measurable_cardinal; end; theorem :: CARD_FIL:32 for A being being_limit_ordinal Ordinal for X being set st X c= A holds sup X = A implies union X = A; theorem :: CARD_FIL:33 M is measurable implies M is regular; definition let M; cluster nextcard M -> non limit; end; definition cluster non limit infinite Cardinal; end; definition cluster non limit -> regular Aleph; end; definition let M be non limit Cardinal; func predecessor M -> Cardinal means :: CARD_FIL:def 17 M = nextcard it; end; definition let M be non limit Aleph; cluster predecessor M -> infinite; end; definition :: infinite matrix let X be set; let N,N1 be Cardinal; mode Inf_Matrix of N,N1,X is Function of [:N,N1:],X; end; reserve X for set; reserve M for non limit Aleph; reserve F for Filter of M; reserve N1,N2,N3 for Element of predecessor M; reserve K1,K2 for Element of M; reserve T for Inf_Matrix of predecessor M , M, bool M; definition :: Ulam matrix on nextcard N; let M,T; pred T is_Ulam_Matrix_of M means :: CARD_FIL:def 18 (for N1,K1,K2 holds K1<>K2 implies T.(N1,K1) /\ T.(N1,K2) is empty ) & (for K1,N1,N2 holds N1<>N2 implies T.(N1,K1) /\ T.(N2,K1) is empty ) & (for N1 holds Card (M \ union {T.(N1,K1): K1 in M}) <=` predecessor M ) & (for K1 holds Card (M \ union {T.(N1,K1): N1 in predecessor M}) <=` predecessor M ); end; :: this is pretty long theorem :: CARD_FIL:34 ex T st T is_Ulam_Matrix_of M; :: also a little longer theorem :: CARD_FIL:35 for M for I being Ideal of M st (I is_complete_with M & Frechet_Ideal(M) c= I) ex S being Subset-Family of M st Card S = M & ( for X1 being set st X1 in S holds not X1 in I ) & for X1,X2 being set st (X1 in S & X2 in S & X1 <> X2) holds X1 misses X2; theorem :: CARD_FIL:36 for X for N being Cardinal st N <=` Card X ex Y being set st Y c= X & Card Y = N; theorem :: CARD_FIL:37 for M holds not ex F st ( F is uniform being_ultrafilter & F is_complete_with M ); reserve M for Aleph; theorem :: CARD_FIL:38 M is measurable implies M is limit; theorem :: CARD_FIL:39 M is measurable implies M is inaccessible; theorem :: CARD_FIL:40 M is measurable implies M is strong_limit; theorem :: CARD_FIL:41 M is measurable implies M is strongly_inaccessible;

Back to top