:: The Ordinal Numbers. Transfinite Induction and Defining by Transfinite Induction
:: by Grzegorz Bancerek
::
:: Received March 20, 1989
:: Copyright (c) 1990-2021 Association of Mizar Users


theorem :: ORDINAL1:1
canceled;

theorem :: ORDINAL1:2
canceled;

theorem :: ORDINAL1:3
canceled;

theorem :: ORDINAL1:4
canceled;

::$CT 4
theorem Th1: :: ORDINAL1:5
for X, Y being set st Y in X holds
not X c= Y
proof end;

definition
let X be set ;
func succ X -> set equals :: ORDINAL1:def 1
X \/ {X};
coherence
X \/ {X} is set ;
end;

:: deftheorem defines succ ORDINAL1:def 1 :
for X being set holds succ X = X \/ {X};

registration
let X be set ;
cluster succ X -> non empty ;
coherence
not succ X is empty ;
end;

theorem Th2: :: ORDINAL1:6
for X being set holds X in succ X
proof end;

theorem :: ORDINAL1:7
for X, Y being set st succ X = succ Y holds
X = Y
proof end;

theorem Th4: :: ORDINAL1:8
for X being set
for x being object holds
( x in succ X iff ( x in X or x = X ) )
proof end;

theorem Th5: :: ORDINAL1:9
for X being set holds X <> succ X
proof end;

definition
let X be set ;
attr X is epsilon-transitive means :Def2: :: ORDINAL1:def 2
 for x being set st x in X holds
x c= X;
attr X is epsilon-connected means :Def3: :: ORDINAL1:def 3
 for x, y being set st x in X & y in X & not x in y & not x = y holds
y in x;
end;

:: deftheorem Def2 defines epsilon-transitive ORDINAL1:def 2 :
for X being set holds
( X is epsilon-transitive iff for x being set st x in X holds
x c= X );

:: deftheorem Def3 defines epsilon-connected ORDINAL1:def 3 :
for X being set holds
( X is epsilon-connected iff for x, y being set st x in X & y in X & not x in y & not x = y holds
y in x );

Lm1: ( {} is epsilon-transitive & {} is epsilon-connected )
;

::
:: 4. Definition of ordinal numbers - Ordinal
::
registration
cluster epsilon-transitive epsilon-connected for number ;
existence
ex b1 being set st
( b1 is epsilon-transitive & b1 is epsilon-connected ) by Lm1;
end;

definition
let IT be object ;
attr IT is ordinal means :: ORDINAL1:def 4
IT is epsilon-transitive epsilon-connected set ;
end;

:: deftheorem defines ordinal ORDINAL1:def 4 :
for IT being object holds
( IT is ordinal iff IT is epsilon-transitive epsilon-connected set );

registration
cluster ordinal -> epsilon-transitive epsilon-connected for number ;
coherence
for b1 being set st b1 is ordinal holds
( b1 is epsilon-transitive & b1 is epsilon-connected ) ;
cluster epsilon-transitive epsilon-connected -> ordinal for number ;
coherence
for b1 being set st b1 is epsilon-transitive & b1 is epsilon-connected holds
b1 is ordinal ;
end;

notation
synonym Number for object ;
synonym number for set ;
end;

registration
cluster ordinal for number ;
existence
ex b1 being number st b1 is ordinal by Lm1;
end;

registration
cluster ordinal for Number ;
existence
ex b1 being Number st b1 is ordinal by Lm1;
end;

definition
mode Ordinal is ordinal number ;
end;

theorem Th6: :: ORDINAL1:10
for A, B being set
for C being epsilon-transitive set st A in B & B in C holds
A in C
proof end;

theorem Th7: :: ORDINAL1:11
for x being epsilon-transitive set
for A being Ordinal st x c< A holds
x in A
proof end;

theorem :: ORDINAL1:12
for A being epsilon-transitive set
for B, C being Ordinal st A c= B & B in C holds
A in C
proof end;

theorem Th9: :: ORDINAL1:13
for a being object
for A being Ordinal st a in A holds
a is Ordinal
proof end;

theorem Th10: :: ORDINAL1:14
for A, B being Ordinal holds
( A in B or A = B or B in A )
proof end;

definition
let A, B be Ordinal;
:: original: c=
redefine pred A c= B means :: ORDINAL1:def 5
 for C being Ordinal st C in A holds
C in B;
compatibility
( A c= B iff for C being Ordinal st C in A holds
C in B )
proof end;
connectedness
for A, B being Ordinal st ex C being Ordinal st
( C in A & not C in B ) holds
for C being Ordinal st C in B holds
C in A
proof end;
end;

:: deftheorem defines c= ORDINAL1:def 5 :
for A, B being Ordinal holds
( A c= B iff for C being Ordinal st C in A holds
C in B );

theorem :: ORDINAL1:15
for A, B being Ordinal holds A,B are_c=-comparable
proof end;

theorem Th12: :: ORDINAL1:16
for A, B being Ordinal holds
( A c= B or B in A )
proof end;

registration
cluster empty -> ordinal for number ;
coherence
for b1 being number st b1 is empty holds
b1 is ordinal by Lm1;
end;

theorem Th13: :: ORDINAL1:17
for x being set st x is Ordinal holds
succ x is Ordinal
proof end;

theorem Th14: :: ORDINAL1:18
for x being set st x is ordinal holds
( union x is epsilon-transitive & union x is epsilon-connected )
proof end;

registration
cluster non empty epsilon-transitive epsilon-connected ordinal for number ;
existence
not for b1 being Ordinal holds b1 is empty
proof end;
end;

registration
let A be Ordinal;
cluster succ A -> non empty ordinal ;
coherence
( not succ A is empty & succ A is ordinal ) by Th13;
cluster union A -> ordinal ;
coherence
union A is ordinal by Th14;
end;

theorem Th15: :: ORDINAL1:19
for X being set st ( for x being set st x in X holds
( x is Ordinal & x c= X ) ) holds
( X is epsilon-transitive & X is epsilon-connected )
proof end;

theorem Th16: :: ORDINAL1:20
for X being set
for A being Ordinal st X c= A & X <> {} holds
ex C being Ordinal st
( C in X & ( for B being Ordinal st B in X holds
C c= B ) )
proof end;

theorem Th17: :: ORDINAL1:21
for A, B being Ordinal holds
( A in B iff succ A c= B )
proof end;

theorem Th18: :: ORDINAL1:22
for A, C being Ordinal holds
( A in succ C iff A c= C )
proof end;

::
:: 6. Transfinite induction and principle of minimum of ordinals
::
scheme :: ORDINAL1:sch 1
OrdinalMin{ P1[ Ordinal] } :
ex A being Ordinal st
( P1[A] & ( for B being Ordinal st P1[B] holds
A c= B ) )
provided
A1: ex A being Ordinal st P1[A]
proof end;

:: WP: Transfinite induction
scheme :: ORDINAL1:sch 2
TransfiniteInd{ P1[ Ordinal] } :
for A being Ordinal holds P1[A]
provided
A1: for A being Ordinal st ( for C being Ordinal st C in A holds
P1[C] ) holds
P1[A]
proof end;

::
:: 7. Properties of sets of ordinals
::
theorem Th19: :: ORDINAL1:23
for X being set st ( for a being object st a in X holds
a is Ordinal ) holds
( union X is epsilon-transitive & union X is epsilon-connected )
proof end;

theorem Th20: :: ORDINAL1:24
for X being set st ( for a being object st a in X holds
a is Ordinal ) holds
ex A being Ordinal st X c= A
proof end;

theorem Th21: :: ORDINAL1:25
for X being set holds
not for x being set holds
( x in X iff x is Ordinal )
proof end;

theorem Th22: :: ORDINAL1:26
for X being set holds
not for A being Ordinal holds A in X
proof end;

theorem :: ORDINAL1:27
for X being set ex A being Ordinal st
( not A in X & ( for B being Ordinal st not B in X holds
A c= B ) )
proof end;

::
:: 8. Limit ordinals
::
definition
let A be set ;
attr A is limit_ordinal means :: ORDINAL1:def 6
A = union A;
end;

:: deftheorem defines limit_ordinal ORDINAL1:def 6 :
for A being set holds
( A is limit_ordinal iff A = union A );

theorem Th24: :: ORDINAL1:28
for A being Ordinal holds
( A is limit_ordinal iff for C being Ordinal st C in A holds
succ C in A )
proof end;

theorem :: ORDINAL1:29
for A being Ordinal holds
( not A is limit_ordinal iff ex B being Ordinal st A = succ B )
proof end;

::
:: 9. Transfinite sequences
::
definition
let IT be set ;
attr IT is Sequence-like means :Def7: :: ORDINAL1:def 7
( proj1 IT is epsilon-transitive & proj1 IT is epsilon-connected );
end;

:: deftheorem Def7 defines Sequence-like ORDINAL1:def 7 :
for IT being set holds
( IT is Sequence-like iff ( proj1 IT is epsilon-transitive & proj1 IT is epsilon-connected ) );

registration
cluster empty -> Sequence-like for number ;
coherence
for b1 being set st b1 is empty holds
b1 is Sequence-like ;
end;

definition
mode Sequence is Sequence-like Function;
end;

registration
let Z be set ;
cluster Relation-like Z -valued Function-like Sequence-like for number ;
existence
ex b1 being Sequence st b1 is Z -valued
proof end;
end;

definition
let Z be set ;
mode Sequence of Z is Z -valued Sequence;
end;

theorem :: ORDINAL1:30
for Z being set holds {} is Sequence of Z
proof end;

theorem :: ORDINAL1:31
for F being Function st dom F is Ordinal holds
F is Sequence of rng F by Def7, RELAT_1:def 19;

registration
let L be Sequence;
cluster proj1 L -> ordinal ;
coherence
dom L is ordinal by Def7;
end;

theorem :: ORDINAL1:32
for X, Y being set st X c= Y holds
for L being Sequence of X holds L is Sequence of Y
proof end;

registration
let L be Sequence;
let A be Ordinal;
cluster L | A -> rng L -valued Sequence-like ;
coherence
( L | A is rng L -valued & L | A is Sequence-like )
proof end;
end;

theorem :: ORDINAL1:33
for X being set
for L being Sequence of X
for A being Ordinal holds L | A is Sequence of X ;

definition
let IT be set ;
attr IT is c=-linear means :Def8: :: ORDINAL1:def 8
 for x, y being set st x in IT & y in IT holds
x,y are_c=-comparable ;
end;

:: deftheorem Def8 defines c=-linear ORDINAL1:def 8 :
for IT being set holds
( IT is c=-linear iff for x, y being set st x in IT & y in IT holds
x,y are_c=-comparable );

theorem :: ORDINAL1:34
for X being set st ( for a being object st a in X holds
a is Sequence ) & X is c=-linear holds
union X is Sequence
proof end;

::
:: 10. Schemes of definability by transfinite induction
::
scheme :: ORDINAL1:sch 3
TSUniq{ F1() -> Ordinal, F2( Sequence) -> set , F3() -> Sequence, F4() -> Sequence } :
F3() = F4()
provided
A1: ( dom F3() = F1() & ( for B being Ordinal
for L being Sequence st B in F1() & L = F3() | B holds
F3() . B = F2(L) ) ) and
A2: ( dom F4() = F1() & ( for B being Ordinal
for L being Sequence st B in F1() & L = F4() | B holds
F4() . B = F2(L) ) )
proof end;

scheme :: ORDINAL1:sch 4
TSExist{ F1() -> Ordinal, F2( Sequence) -> set } :
ex L being Sequence st
( dom L = F1() & ( for B being Ordinal
for L1 being Sequence st B in F1() & L1 = L | B holds
L . B = F2(L1) ) )
proof end;

scheme :: ORDINAL1:sch 5
FuncTS{ F1() -> Sequence, F2( Ordinal) -> set , F3( Sequence) -> set } :
for B being Ordinal st B in dom F1() holds
F1() . B = F3((F1() | B))
provided
A1: for A being Ordinal
for a being object holds
( a = F2(A) iff ex L being Sequence st
( a = F3(L) & dom L = A & ( for B being Ordinal st B in A holds
L . B = F3((L | B)) ) ) ) and
A2: for A being Ordinal st A in dom F1() holds
F1() . A = F2(A)
proof end;

theorem :: ORDINAL1:35
for A, B being Ordinal holds
( A c< B or A = B or B c< A )
proof end;

:: moved from ORDINAL2, 2006.06.22, A.T.
definition
let X be set ;
func On X -> set means :Def9: :: ORDINAL1:def 9
 for x being object holds
( x in it iff ( x in X & x is Ordinal ) );
existence
ex b1 being set st
for x being object holds
( x in b1 iff ( x in X & x is Ordinal ) )
proof end;
uniqueness
for b1, b2 being set st ( for x being object holds
( x in b1 iff ( x in X & x is Ordinal ) ) ) & ( for x being object holds
( x in b2 iff ( x in X & x is Ordinal ) ) ) holds
b1 = b2
proof end;
func Lim X -> set means :: ORDINAL1:def 10
 for x being object holds
( x in it iff ( x in X & ex A being Ordinal st
( x = A & A is limit_ordinal ) ) );
existence
ex b1 being set st
for x being object holds
( x in b1 iff ( x in X & ex A being Ordinal st
( x = A & A is limit_ordinal ) ) )
proof end;
uniqueness
for b1, b2 being set st ( for x being object holds
( x in b1 iff ( x in X & ex A being Ordinal st
( x = A & A is limit_ordinal ) ) ) ) & ( for x being object holds
( x in b2 iff ( x in X & ex A being Ordinal st
( x = A & A is limit_ordinal ) ) ) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def9 defines On ORDINAL1:def 9 :
for X, b2 being set holds
( b2 = On X iff for x being object holds
( x in b2 iff ( x in X & x is Ordinal ) ) );

:: deftheorem defines Lim ORDINAL1:def 10 :
for X, b2 being set holds
( b2 = Lim X iff for x being object holds
( x in b2 iff ( x in X & ex A being Ordinal st
( x = A & A is limit_ordinal ) ) ) );

:: WP: Generalized Axiom of Infinity
theorem Th32: :: ORDINAL1:36
for D being Ordinal ex A being Ordinal st
( D in A & A is limit_ordinal )
proof end;

definition
func omega -> set means :Def11: :: ORDINAL1:def 11
( {} in it & it is limit_ordinal & it is ordinal & ( for A being Ordinal st {} in A & A is limit_ordinal holds
it c= A ) );
existence
ex b1 being set st
( {} in b1 & b1 is limit_ordinal & b1 is ordinal & ( for A being Ordinal st {} in A & A is limit_ordinal holds
b1 c= A ) )
proof end;
uniqueness
for b1, b2 being set st {} in b1 & b1 is limit_ordinal & b1 is ordinal & ( for A being Ordinal st {} in A & A is limit_ordinal holds
b1 c= A ) & {} in b2 & b2 is limit_ordinal & b2 is ordinal & ( for A being Ordinal st {} in A & A is limit_ordinal holds
b2 c= A ) holds
b1 = b2 ;
end;

:: deftheorem Def11 defines omega ORDINAL1:def 11 :
for b1 being set holds
( b1 = omega iff ( {} in b1 & b1 is limit_ordinal & b1 is ordinal & ( for A being Ordinal st {} in A & A is limit_ordinal holds
b1 c= A ) ) );

registration
cluster omega -> non empty ordinal ;
coherence
( not omega is empty & omega is ordinal ) by Def11;
end;

definition
let A be object ;
attr A is natural means :Def12: :: ORDINAL1:def 12
A in omega ;
end;

:: deftheorem Def12 defines natural ORDINAL1:def 12 :
for A being object holds
( A is natural iff A in omega );

registration
cluster natural for Number ;
existence
ex b1 being Number st b1 is natural
proof end;
cluster natural for number ;
existence
ex b1 being set st b1 is natural
proof end;
end;

definition
mode Nat is natural number ;
end;

registration
Nat ex b1 being number st
for b2 being Nat holds b2 in b1
proof end;
end;

:: from ARYTM_3, 2006.05.26
registration
let A be Ordinal;
cluster -> ordinal for Element of A;
coherence
for b1 being Element of A holds b1 is ordinal
proof end;
end;

:: missing, 2006.06.25, A.T.
registration
cluster natural -> ordinal for number ;
coherence
for b1 being number st b1 is natural holds
b1 is ordinal ;
end;

:: from ZF_REFLE, 2007,03.13, A.T.
scheme :: ORDINAL1:sch 6
ALFA{ F1() -> non empty set , P1[ object , object ] } :
ex F being Function st
( dom F = F1() & ( for d being Element of F1() ex A being Ordinal st
( A = F . d & P1[d,A] & ( for B being Ordinal st P1[d,B] holds
A c= B ) ) ) )
provided
A1: for d being Element of F1() ex A being Ordinal st P1[d,A]
proof end;

:: from CARD_4, 2007.08.06, A.T.
theorem :: ORDINAL1:37
for X being set holds (succ X) \ {X} = X
proof end;

:: from ARYTM_3, 2007.09.16, A.T.
registration
cluster empty -> natural for number ;
coherence
for b1 being number st b1 is empty holds
b1 is natural by Def11;
cluster -> natural for Element of omega ;
coherence
for b1 being Element of omega holds b1 is natural ;
end;

registration
cluster non empty natural for number ;
existence
ex b1 being number st
( not b1 is empty & b1 is natural )
proof end;
end;

:: from ARYTM_3, 2007.10.23, A.T.
registration
let a be natural Ordinal;
cluster succ a -> natural ;
coherence
succ a is natural
proof end;
end;

:: from DILWORTH, 2011.07.31,A.T.
registration
cluster empty -> c=-linear for number ;
coherence
for b1 being set st b1 is empty holds
b1 is c=-linear ;
end;

registration
let X be c=-linear set ;
cluster -> c=-linear for Element of bool X;
coherence
for b1 being Subset of X holds b1 is c=-linear by Def8;
end;

theorem :: ORDINAL1:38
canceled;

::$CT
definition
::$CT
func 0 -> number equals :: ORDINAL1:def 13
 {} ;
coherence
{} is number ;
end;

:: deftheorem defines 0 ORDINAL1:def 13 :
0 = {} ;

registration
cluster 0 -> natural ;
coherence
0 is natural ;
end;

definition
let x be Number ;
attr x is zero means :Def14: :: ORDINAL1:def 14
x = 0 ;
end;

:: deftheorem Def14 defines zero ORDINAL1:def 14 :
for x being Number holds
( x is zero iff x = 0 );

registration
cluster 0 -> zero ;
coherence
0 is zero ;
cluster zero for Number ;
existence
ex b1 being Number st b1 is zero
proof end;
cluster zero for number ;
existence
ex b1 being set st b1 is zero
proof end;
end;

registration
cluster zero -> natural for Number ;
coherence
for b1 being Number st b1 is zero holds
b1 is natural ;
end;

registration
cluster non zero for Number ;
existence
not for b1 being Number holds b1 is zero
proof end;
end;

registration
cluster zero -> trivial for number ;
coherence
for b1 being set st b1 is zero holds
b1 is trivial ;
cluster non trivial -> non zero for number ;
coherence
for b1 being set st not b1 is trivial holds
not b1 is zero ;
end;

definition
let R be Relation;
attr R is non-zero means :Def15: :: ORDINAL1:def 15
 not 0 in rng R;
end;

:: deftheorem Def15 defines non-zero ORDINAL1:def 15 :
for R being Relation holds
( R is non-zero iff not 0 in rng R );

definition
let X be set ;
attr X is with_zero means :Def16: :: ORDINAL1:def 16
 0 in X;
end;

:: deftheorem Def16 defines with_zero ORDINAL1:def 16 :
for X being set holds
( X is with_zero iff 0 in X );

notation
let X be set ;
antonym without_zero X for with_zero ;
end;

registration
cluster empty -> without_zero for number ;
coherence
for b1 being set st b1 is empty holds
b1 is without_zero ;
end;

registration
cluster empty Relation-like -> non-zero for number ;
coherence
for b1 being Relation st b1 is empty holds
b1 is non-zero ;
end;

registration
cluster non empty without_zero for number ;
existence
ex b1 being set st
( b1 is without_zero & not b1 is empty )
proof end;
end;

registration
let X be non empty without_zero set ;
cluster -> non zero for Element of X;
coherence
for b1 being Element of X holds not b1 is zero by Def16;
end;

registration
cluster Relation-like non-zero for number ;
existence
ex b1 being Relation st b1 is non-zero
proof end;
cluster Relation-like non non-zero for number ;
existence
not for b1 being Relation holds b1 is non-zero
proof end;
end;

registration
let R be non-zero Relation;
cluster proj2 R -> without_zero ;
coherence
not rng R is with_zero by Def15;
end;

registration
let R be non non-zero Relation;
cluster proj2 R -> with_zero ;
coherence
rng R is with_zero by Def15;
end;

registration
let R be non-zero Relation;
let S be Relation;
cluster S * R -> non-zero ;
coherence
S * R is non-zero
proof end;
end;

:: to be removed
registration
cluster without_zero -> with_non-empty_elements for number ;
coherence
for b1 being set st b1 is without_zero holds
b1 is with_non-empty_elements
proof end;
cluster with_zero -> non with_non-empty_elements for number ;
coherence
for b1 being set st b1 is with_zero holds
not b1 is with_non-empty_elements
proof end;
cluster with_non-empty_elements -> without_zero for number ;
coherence
for b1 being set st b1 is with_non-empty_elements holds
b1 is without_zero ;
cluster non with_non-empty_elements -> with_zero for number ;
coherence
for b1 being set st not b1 is with_non-empty_elements holds
b1 is with_zero ;
end;

definition
let o be object ;
func Segm o -> set equals :: ORDINAL1:def 17
o;
coherence
o is set by TARSKI:1;
end;

:: deftheorem defines Segm ORDINAL1:def 17 :
for o being object holds Segm o = o;

registration
let n be Ordinal;
cluster Segm n -> ordinal ;
coherence
Segm n is ordinal ;
end;

registration
let n be zero Ordinal;
cluster Segm n -> empty ;
coherence
Segm n is empty by Def14;
end;