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Chapter 9 Cantor Normal Forms
CHAPTER 9
CANTOR NORMAL FORMS
9.1 In this chapter we prove the existence of Cantor Normal Formsg1 for a large class of co-ordinals, but first of all we prove a general decomposition theorem for all co-ordinals less than some principal number for addition. We restrict our attention from now until the end of Part One of this monograph to co-ordinals, returning to the consideration of quords in Part Two. Since we are dealing with co-ordinals we shall repeatedly use the fact (theorem 4.2.2) that A < B e ( 3 C )( A + C = B & C
=+ 0).
9.1.1 LEMMA. A co-ordinal P is a principal number for addition if, and only if, A
9.1.2 THEOREM. If P E S (+), then P. WE&( +) and there is no principal number Q such that P
PROOF. Suppose P E X (+) and A < P -W ,then by theorem 6.2.5, there exist n, D such that
A = P * n + D , where D < P.
Ch. 91
111
CANTOR NORMAL FORMS
Hence A + P . W = A +(P + P - W ) =(A +P)+P-W =(Pen D +P)+P-W = P . ( n + 1) + P a w =P.W.
since 1 + W = W ,
+
sincePsX(+)
and D < P ,
Hence P.WE*( +). Clearly PCP-W since P 9 0. Now suppose Q s X ( +) and P < Q , then by lemma 9.1.1, P*W
hence Q
9.1.3 THEOREM. g2 If O c A < P E X ( +), then there exist principal numbers for addition Ply ..., Pnsuch that P > P,,2 . . * 2Pl
and A = P,,
Pl.
A = Q,
Q1
+ a * * +
Further, if + * * a +
is any other representation of A as a sum of principal numbers Q, (i=l,..., m) such that
Q,
>***
2
Q1,
then m = n & P i = Qi for i
=
1, ..., n.
Conversely, if A is expressible in the form (2) with P,, 2
2 PI where the
Pi(i= 1, ..., n) are principal numbers for addition, then there is a principal number P e.g. P,,. W,such that P > A and P > Pi for i
=
1, ..., n.
PROOF(by transfinite induction on the partial well-ordering I). We assume O < A < P E H ( + ) and take as induction hypothesis: If O c B c A
112
CONSTRUCTIVE ORDER TYPES
[Ch. 9
then B is uniquely expressible in the form (2) where the P, are principal numbers satisfying (1). If A is a principal number for addition then the theorem follows at once from theorem 9.1.2. Now suppose A is not a principal number, then there exist B, C such that B+C=A,C+O,A
and B + C < P .
(3)
By corollary 8.1.6, C
Let P, be the least C satisfying (3). PI exists since { C :c < A }
is well-ordered by I .We show that P, is a principal number for addition. Suppose Pi = D + E , then by corollary 8.1.6, E c P and hence by theorem 4.2.8, PI and E are comparable. But
(El5 lP1l
I
hence E IPi
and by the minimality of P,, E=P,.
Therefore PIis a principal number by theorem 8.1.2. Now let B, be the least B such that B + P, = A . If B, = 0, then A is a principal number which contradicts our assumption. Hence B, + O
and by the induction hypothesis B, has a unique decomposition where
Ch. 91
113
CANTOR NORMAL FORMS
and all the Pi(i = 2, ..., n) are principal numbers. Therefore A = Pn +**.+Pl
and, since p, < p , all the Pi (i = 1, ..., n) are comparable by theorem 4.2.8. Suppose pz < PlB
then, since Pl is a principal number for addition,
Pz +PI =PI; hence A=(Pn+-.+Pz)+P1 =(P.+--+P,)+P,.
Now
*
Pz 0 3 Pn+
9.-
+ P3 < P, + - + P2 = B , , **
whereas Bl was chosen as the least B such that B + P , = A . This contradiction shows that Pz 2 PI and we conclude that A has a decomposition of the form (2). As regards uniqueness, let A
= P,, +..,+PI
and A = Q,
+-.+ Q ,
be two decompositions of A as a sum of non-increasing principal numbers. By theorem 4.2.8, P,,and Q, are comparable. Suppose Pn>Qm,
then, since P,,is a principal number for addition,
Q, Therefore
+ P,,= P,.
+ P,, Pi = Q , + Q , + P,, - ... = Q,.(m + 1) + A .
A = Q,
+ a * * +
+ * a * +
Since A+O, we therefore have Q,.(m
+ 1) < A .
114
[Ch. 9
CONSTRUCTIVE ORDER TYPES
On the other hand, if i Im, then or Qm.2,
Qi+Qm=Qm
according as Qi
< Qm
or
= Qm.
Qi
Consequently, we have Q,.(m
+ 1) + A = A < A + Q;m
5 Q;2m
and by corollary 3.2.9, it follows that A
(5)
which contradicts (4) above. We conclude Q, 4: P,, and similarly P,, 4: Q,,
but since P,, and Q, are comparable it follows that
P,,= Q, and P,-,
+...+ P, = Q,-, +...+ Q1
(by corollary 3.2.9). Proceeding thus we obtain after s( = minimum of m, n ) steps P,,-r = Qm-r r = 0, ..., s and either P,+ - . , + P I= 0 or Q, Q, = 0 , + . e m +
where t = maximum of m - s , n-s. By theorem 2.2.4.(ii) it follows that t =0 and hence that m =n and
Pi= Qi for i
= 1, ..., n ( = m )
For the converse: if A=P,,+**.+P,,
then by the argument which produced ( 5 ) we have A I P;n
and consequently A < P,,. W .
By theorem 9.1.2, P,,. Wis a principal number for addition. This completes the proof.
Ch. 91
115
CANTOR NORMAL FORMS
9.2.
LEMMA. C
9.2.1
The next lemma is a strengthening of theorem 6.2.5. 9.2.2
LEMMA.If A 90,then
c < A B o ( 3 D ) ( D < B & A D 5 c < A ( D + 1)). PROOF.The implication from right to left is clear from corollary 5.2.3, theorem 8.4.1 and the transitivity of < for co-ordinals. Suppose now that C
then there exists FPO such that C+F=AB.
Given AEA and BEB, then by the separation lemma 2.3.1, there exists CEC and F E F such that [C)(Fand]
C $ F=A.B.
Let
f = min(F), d = l(f), D = d) B and D
= COT(D).
We shall show that for this D, ADIC
Now AD E C E AB,
since j ( a , x ) ~ e A D = . xsB d =>j(a7 x )
SABf
=.j(a, X)EC'C.
Further, since AD and C are both initial segments of AB, we must have ADsC. By corollary 5.2.3,
D
116
[Ch. 9
CONSTRUCTIVE ORDER TYPES
and hence by theorem 8.4.1.(ii) it follows that A(D
+ 1) IA B .
From theorem 4.2.8 we now have or C I A ( D + l ) ,
A(D+1)1C
but
c c A * ( DT ( ( 4d ) ) ) and so we have
c IA ( D + 1).
Finally, since
f E C j ( A , d ) - C ' C , C + A ( D + 1) and the proof is complete. 9.2.3
LEMMA.1 1 C < A ~ o ( 3 0 ) ( D < B & A D s C < A D f l ) .
PROOF. The implication from right to left follows at once from corollary 5.2.3, theorem 8.5.5.(ii) and the transitivity of <. Now suppose lIC
C +F
=AB.
Further, there exist A, B such that
A = [AIEA and B = [B]EB and by the separation lemma 2.3.1, there exist C E Cand FEFsuch that
c~
F=A~.
Let f = min(F),
then f
=
( b o ... b:) a,
where bo>
... > b,,,b,eB
... a,
(i=O,...,n), n+-1
(since C21) and93
O+U,EA( i = O , . . . , H).
Also let
D = bo)B and D = COT (D) ,
Ch. 91
CANTOR NORMAL FORMS
117
then we claim ADs C. Now XEC'A" implies
where do
f,
which implies XEC'C. Further, if y€C'ADand < y , x)eAD, then by the same argument Y < A exp B f It follows at once (from corollary 4.2.5) that AD Ic.
Now let E = D T ((bo, bo)} ,
then XEC'CJX=O
or x = e
where bb>-..>b;, biEB and 0 8 a ; e A . Further, by the definition of bo, (bb, bo)
E
B
and hence x€C'AEand
C5AEdD+'. But C 8 AEsince
f E C'AE - C'C and it follows that
AD 5 C < A D + ' . LEMMA. A T W & B< C*AB+ AC= A'. PROOF. Since B < C , (30)( D + O & B + D = C ) .
9.2.4
Hence + A =~ A +~A ~ . A ~ = AB(l + A D ) = AB(l + A E)
+
by theorem 7.3.4, by theorem 6.2.1, by lemma 8.5.6.(i) , where A + E = A D ,
118
[Ch. 9
CONSTRUCTIVE ORDER TYPES
+ W + F + E)
= A'(1
= AB(W
+ F + E)
=A ~ A D =A
9.2.5
+
since W < A* W F = A for some F , by theorem 5.2.6.(i),
~ .
LEMMA. If C < AB, then there exist unique Q, R such that C = AQ
+R ,
where 0 I Q < B & O I R < A .
PROOF. By lemma 9.2.2, C < AB*(3Q)(O I Q < B & AQ I C < A ( Q
Hence AQ I C < AQ
+ 1)).
+A .
Since A , C, Q are co-ordinals, R=C-AQ
+R as required. Now suppose C = AQ + R = AQ, + R , ,
is well-defined, 0 5 R < A and C = A Q
where 0 1 Q,
and O I R < A ,
by theorem 4.2.8 Q and Q, are comparable (and so too are R and R,). By corollary 5.2.3, if Q + Q, then either Q 1 + l I Q or Q + l S Q , .
Suppose the former holds, then AQl I C < A (Q1
+ 1) I AQ < C ,
which is a contradiction. Similarly we cannot have Q
+ 1s Q,, hence
Q = Qi and by corollary 3.2.9, R=R,. 9.2.6
LEMMA.If 1I C< A* then there exist unique D , Q, R such that
C = A D Q + R , where D < B ,
O
and O j R < A D .
Ch.91
119
CANTOR NORMAL FORMS
PROOF. By lemma 9.2.3, 1 IC < AB* ( 3 0 ) ( A DIC < AD+' & D < B ) , hence C < AD+' = A D . A .
Now, if A < C, D + 0. By lemma 9.2.5, there therefore exist Q, R such that C = A D Q + R , where O < Q < A
and O I R < A .
Suppose that we also had C = AEQ, + R , ,
where E < B ,
O
and O I R , < A .
By theorem 4.2.8, D and E are comparable and hence by corollary 5.2.3, if D =kE, then D + l I E or E + l l D .
Suppose the former holds, then by theorem 8.5.5.(i), and using the fact that A E is a co-ordinal, we have A ~ cS < A ~ + I '
C,
which is a contradiction. Likewise we cannot have E + l s D and we conclude D=E.
Now suppose A 2 C, then either trivially A = C or C = A o Q + R , where O < Q < A
and R = O
and Q, R are unique by the preceding lemma. Thus for all A satisfying the hypothesis we obtain D, Q, R uniquely as required. 9.2.7
THEOREM. If 1 IC < AB then C is uniquely expressible in the form C = A B 1 * D+ 1 .-.+ ABr.D,,
(1)
where B > B , > - . . > B , and l l D , < A f o r l l i l r . PROOF. By lemma 9.2.6, 1 Ic < A~ 3 c
=
D,
+ c, ,
where O I D , < A , O I C , < A B ' and B , < B and B,, D,, C, are unique. Since Cl < AB' < A B and AB' I C < A B
120
CONSTRUCTWE ORDER TYPES
[Ch. 9
it follows that C, < C < A B .
Since {E:E< A B } is well-ordered by Iwe obtain, by a finite number of repetitions of the above process, the decomposition (1). Suppose that C is also expressible as C = AF'.H1 +-.*+AFs*Hs, where B> F, > >F, and 1IHi
+
C = AB'*D1 C1 = AB1*Hl+ El ,
where 0
or H,ID,+J
D,+IIH, Suppose the former holds, then
C=A~'.D1+C1
which is a contradiction. Similarly, H, $ D, + 1 and we conclude
D, = H , . By corollary 3.2.9 it follows that C, = El
and hence by induction on the maximum of r, s that Bi = Fi and D i = H i for 1 Ii Ir = s .
This completes the proof. As an immediate consequence we obtain our main theorem of this section. 9.2.8 THEOREM (CANTORNORMAL
FORM).
If O
Ch. 91
CANTOR NORMAL FORMS
121
uniquely expressible in the form
c=~ where A > A , >
~ * +...+ - n , WAp*n,,
(2)
> A , and the ni are finite, non-zero co-ordinals.
COROLLARY. A necessary and sufficient condition that a co-ordinal C+O should have a Cantor Normal Form (2) is that there should exist a co-ordinal A such that C < WA. PROOF. The condition is obviously sufficient. Now suppose C has the given Cantor Normal Form and let A be greater than A , (say, take A =A, + 1); then by n, ...+n, applications of lemma 9.2.4, we obtain C + WA=W Awhence C < WA.
9.2.9
+
CANTOR NORMAL FORMS
9.1 In this chapter we prove the existence of Cantor Normal Formsg1 for a large class of co-ordinals, but first of all we prove a general decomposition theorem for all co-ordinals less than some principal number for addition. We restrict our attention from now until the end of Part One of this monograph to co-ordinals, returning to the consideration of quords in Part Two. Since we are dealing with co-ordinals we shall repeatedly use the fact (theorem 4.2.2) that A < B e ( 3 C )( A + C = B & C
=+ 0).
9.1.1 LEMMA. A co-ordinal P is a principal number for addition if, and only if, A
9.1.2 THEOREM. If P E S (+), then P. WE&( +) and there is no principal number Q such that P
PROOF. Suppose P E X (+) and A < P -W ,then by theorem 6.2.5, there exist n, D such that
A = P * n + D , where D < P.
Ch. 91
111
CANTOR NORMAL FORMS
Hence A + P . W = A +(P + P - W ) =(A +P)+P-W =(Pen D +P)+P-W = P . ( n + 1) + P a w =P.W.
since 1 + W = W ,
+
sincePsX(+)
and D < P ,
Hence P.WE*( +). Clearly PCP-W since P 9 0. Now suppose Q s X ( +) and P < Q , then by lemma 9.1.1, P*W
hence Q
9.1.3 THEOREM. g2 If O c A < P E X ( +), then there exist principal numbers for addition Ply ..., Pnsuch that P > P,,2 . . * 2Pl
and A = P,,
Pl.
A = Q,
Q1
+ a * * +
Further, if + * * a +
is any other representation of A as a sum of principal numbers Q, (i=l,..., m) such that
Q,
>***
2
Q1,
then m = n & P i = Qi for i
=
1, ..., n.
Conversely, if A is expressible in the form (2) with P,, 2
2 PI where the
Pi(i= 1, ..., n) are principal numbers for addition, then there is a principal number P e.g. P,,. W,such that P > A and P > Pi for i
=
1, ..., n.
PROOF(by transfinite induction on the partial well-ordering I). We assume O < A < P E H ( + ) and take as induction hypothesis: If O c B c A
112
CONSTRUCTIVE ORDER TYPES
[Ch. 9
then B is uniquely expressible in the form (2) where the P, are principal numbers satisfying (1). If A is a principal number for addition then the theorem follows at once from theorem 9.1.2. Now suppose A is not a principal number, then there exist B, C such that B+C=A,C+O,A
and B + C < P .
(3)
By corollary 8.1.6, C
Let P, be the least C satisfying (3). PI exists since { C :c < A }
is well-ordered by I .We show that P, is a principal number for addition. Suppose Pi = D + E , then by corollary 8.1.6, E c P and hence by theorem 4.2.8, PI and E are comparable. But
(El5 lP1l
I
hence E IPi
and by the minimality of P,, E=P,.
Therefore PIis a principal number by theorem 8.1.2. Now let B, be the least B such that B + P, = A . If B, = 0, then A is a principal number which contradicts our assumption. Hence B, + O
and by the induction hypothesis B, has a unique decomposition where
Ch. 91
113
CANTOR NORMAL FORMS
and all the Pi(i = 2, ..., n) are principal numbers. Therefore A = Pn +**.+Pl
and, since p, < p , all the Pi (i = 1, ..., n) are comparable by theorem 4.2.8. Suppose pz < PlB
then, since Pl is a principal number for addition,
Pz +PI =PI; hence A=(Pn+-.+Pz)+P1 =(P.+--+P,)+P,.
Now
*
Pz 0 3 Pn+
9.-
+ P3 < P, + - + P2 = B , , **
whereas Bl was chosen as the least B such that B + P , = A . This contradiction shows that Pz 2 PI and we conclude that A has a decomposition of the form (2). As regards uniqueness, let A
= P,, +..,+PI
and A = Q,
+-.+ Q ,
be two decompositions of A as a sum of non-increasing principal numbers. By theorem 4.2.8, P,,and Q, are comparable. Suppose Pn>Qm,
then, since P,,is a principal number for addition,
Q, Therefore
+ P,,= P,.
+ P,, Pi = Q , + Q , + P,, - ... = Q,.(m + 1) + A .
A = Q,
+ a * * +
+ * a * +
Since A+O, we therefore have Q,.(m
+ 1) < A .
114
[Ch. 9
CONSTRUCTIVE ORDER TYPES
On the other hand, if i Im, then or Qm.2,
Qi+Qm=Qm
according as Qi
< Qm
or
= Qm.
Qi
Consequently, we have Q,.(m
+ 1) + A = A < A + Q;m
5 Q;2m
and by corollary 3.2.9, it follows that A
(5)
which contradicts (4) above. We conclude Q, 4: P,, and similarly P,, 4: Q,,
but since P,, and Q, are comparable it follows that
P,,= Q, and P,-,
+...+ P, = Q,-, +...+ Q1
(by corollary 3.2.9). Proceeding thus we obtain after s( = minimum of m, n ) steps P,,-r = Qm-r r = 0, ..., s and either P,+ - . , + P I= 0 or Q, Q, = 0 , + . e m +
where t = maximum of m - s , n-s. By theorem 2.2.4.(ii) it follows that t =0 and hence that m =n and
Pi= Qi for i
= 1, ..., n ( = m )
For the converse: if A=P,,+**.+P,,
then by the argument which produced ( 5 ) we have A I P;n
and consequently A < P,,. W .
By theorem 9.1.2, P,,. Wis a principal number for addition. This completes the proof.
Ch. 91
115
CANTOR NORMAL FORMS
9.2.
LEMMA. C
9.2.1
The next lemma is a strengthening of theorem 6.2.5. 9.2.2
LEMMA.If A 90,then
c < A B o ( 3 D ) ( D < B & A D 5 c < A ( D + 1)). PROOF.The implication from right to left is clear from corollary 5.2.3, theorem 8.4.1 and the transitivity of < for co-ordinals. Suppose now that C
then there exists FPO such that C+F=AB.
Given AEA and BEB, then by the separation lemma 2.3.1, there exists CEC and F E F such that [C)(Fand]
C $ F=A.B.
Let
f = min(F), d = l(f), D = d) B and D
= COT(D).
We shall show that for this D, ADIC
Now AD E C E AB,
since j ( a , x ) ~ e A D = . xsB d =>j(a7 x )
SABf
=.j(a, X)EC'C.
Further, since AD and C are both initial segments of AB, we must have ADsC. By corollary 5.2.3,
D
116
[Ch. 9
CONSTRUCTIVE ORDER TYPES
and hence by theorem 8.4.1.(ii) it follows that A(D
+ 1) IA B .
From theorem 4.2.8 we now have or C I A ( D + l ) ,
A(D+1)1C
but
c c A * ( DT ( ( 4d ) ) ) and so we have
c IA ( D + 1).
Finally, since
f E C j ( A , d ) - C ' C , C + A ( D + 1) and the proof is complete. 9.2.3
LEMMA.1 1 C < A ~ o ( 3 0 ) ( D < B & A D s C < A D f l ) .
PROOF. The implication from right to left follows at once from corollary 5.2.3, theorem 8.5.5.(ii) and the transitivity of <. Now suppose lIC
C +F
=AB.
Further, there exist A, B such that
A = [AIEA and B = [B]EB and by the separation lemma 2.3.1, there exist C E Cand FEFsuch that
c~
F=A~.
Let f = min(F),
then f
=
( b o ... b:) a,
where bo>
... > b,,,b,eB
... a,
(i=O,...,n), n+-1
(since C21) and93
O+U,EA( i = O , . . . , H).
Also let
D = bo)B and D = COT (D) ,
Ch. 91
CANTOR NORMAL FORMS
117
then we claim ADs C. Now XEC'A" implies
where do
f,
which implies XEC'C. Further, if y€C'ADand < y , x)eAD, then by the same argument Y < A exp B f It follows at once (from corollary 4.2.5) that AD Ic.
Now let E = D T ((bo, bo)} ,
then XEC'CJX=O
or x = e
where bb>-..>b;, biEB and 0 8 a ; e A . Further, by the definition of bo, (bb, bo)
E
B
and hence x€C'AEand
C5AEdD+'. But C 8 AEsince
f E C'AE - C'C and it follows that
AD 5 C < A D + ' . LEMMA. A T W & B< C*AB+ AC= A'. PROOF. Since B < C , (30)( D + O & B + D = C ) .
9.2.4
Hence + A =~ A +~A ~ . A ~ = AB(l + A D ) = AB(l + A E)
+
by theorem 7.3.4, by theorem 6.2.1, by lemma 8.5.6.(i) , where A + E = A D ,
118
[Ch. 9
CONSTRUCTIVE ORDER TYPES
+ W + F + E)
= A'(1
= AB(W
+ F + E)
=A ~ A D =A
9.2.5
+
since W < A* W F = A for some F , by theorem 5.2.6.(i),
~ .
LEMMA. If C < AB, then there exist unique Q, R such that C = AQ
+R ,
where 0 I Q < B & O I R < A .
PROOF. By lemma 9.2.2, C < AB*(3Q)(O I Q < B & AQ I C < A ( Q
Hence AQ I C < AQ
+ 1)).
+A .
Since A , C, Q are co-ordinals, R=C-AQ
+R as required. Now suppose C = AQ + R = AQ, + R , ,
is well-defined, 0 5 R < A and C = A Q
where 0 1 Q,
and O I R < A ,
by theorem 4.2.8 Q and Q, are comparable (and so too are R and R,). By corollary 5.2.3, if Q + Q, then either Q 1 + l I Q or Q + l S Q , .
Suppose the former holds, then AQl I C < A (Q1
+ 1) I AQ < C ,
which is a contradiction. Similarly we cannot have Q
+ 1s Q,, hence
Q = Qi and by corollary 3.2.9, R=R,. 9.2.6
LEMMA.If 1I C< A* then there exist unique D , Q, R such that
C = A D Q + R , where D < B ,
O
and O j R < A D .
Ch.91
119
CANTOR NORMAL FORMS
PROOF. By lemma 9.2.3, 1 IC < AB* ( 3 0 ) ( A DIC < AD+' & D < B ) , hence C < AD+' = A D . A .
Now, if A < C, D + 0. By lemma 9.2.5, there therefore exist Q, R such that C = A D Q + R , where O < Q < A
and O I R < A .
Suppose that we also had C = AEQ, + R , ,
where E < B ,
O
and O I R , < A .
By theorem 4.2.8, D and E are comparable and hence by corollary 5.2.3, if D =kE, then D + l I E or E + l l D .
Suppose the former holds, then by theorem 8.5.5.(i), and using the fact that A E is a co-ordinal, we have A ~ cS < A ~ + I '
C,
which is a contradiction. Likewise we cannot have E + l s D and we conclude D=E.
Now suppose A 2 C, then either trivially A = C or C = A o Q + R , where O < Q < A
and R = O
and Q, R are unique by the preceding lemma. Thus for all A satisfying the hypothesis we obtain D, Q, R uniquely as required. 9.2.7
THEOREM. If 1 IC < AB then C is uniquely expressible in the form C = A B 1 * D+ 1 .-.+ ABr.D,,
(1)
where B > B , > - . . > B , and l l D , < A f o r l l i l r . PROOF. By lemma 9.2.6, 1 Ic < A~ 3 c
=
D,
+ c, ,
where O I D , < A , O I C , < A B ' and B , < B and B,, D,, C, are unique. Since Cl < AB' < A B and AB' I C < A B
120
CONSTRUCTWE ORDER TYPES
[Ch. 9
it follows that C, < C < A B .
Since {E:E< A B } is well-ordered by Iwe obtain, by a finite number of repetitions of the above process, the decomposition (1). Suppose that C is also expressible as C = AF'.H1 +-.*+AFs*Hs, where B> F, > >F, and 1IHi
+
C = AB'*D1 C1 = AB1*Hl+ El ,
where 0
or H,ID,+J
D,+IIH, Suppose the former holds, then
C=A~'.D1+C1
which is a contradiction. Similarly, H, $ D, + 1 and we conclude
D, = H , . By corollary 3.2.9 it follows that C, = El
and hence by induction on the maximum of r, s that Bi = Fi and D i = H i for 1 Ii Ir = s .
This completes the proof. As an immediate consequence we obtain our main theorem of this section. 9.2.8 THEOREM (CANTORNORMAL
FORM).
If O
Ch. 91
CANTOR NORMAL FORMS
121
uniquely expressible in the form
c=~ where A > A , >
~ * +...+ - n , WAp*n,,
(2)
> A , and the ni are finite, non-zero co-ordinals.
COROLLARY. A necessary and sufficient condition that a co-ordinal C+O should have a Cantor Normal Form (2) is that there should exist a co-ordinal A such that C < WA. PROOF. The condition is obviously sufficient. Now suppose C has the given Cantor Normal Form and let A be greater than A , (say, take A =A, + 1); then by n, ...+n, applications of lemma 9.2.4, we obtain C + WA=W Awhence C < WA.
9.2.9
+