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Chapter 8 Arithmetic Laws and Principal Numbers
CHAPTER 8
ARITHMETIC LAWS AND PRINCIPAL NUMBERS
8.1 We now investigate the arithmetic laws of C.0.T.s. Some laws for addition were established in chapter 4 and we first of all strengthen these. 8.1.1 THEOREM. If A , By C are quords, then B < C-A + B < A
+ c.
PROOF.Immediate from theorem 4.2.3.(v) and corollary 3.2.9. However, the other obvious analogue of a classical additive law
BIC*B+AIC+A
(*I
does not hold in general even for co-ordinals. For let A = W, B=O and C = V, then (*) implies W and V are comparable which contradicts corollary 5.2.7. However, (*) does hold if A , B, C are less than what we call a principal number for addition. Classically, a principal number for addition (otherwise called a y-number (BACHMANN, 1955, p. 67) or a prime component (SIERPINSKI, 1958, p. 279)) may be defined as an ordinal number y such that any of the following (equivalent) conditions holds: a
or P = y ,
P
y =0 or y = ma for some ordinal u .
(1") (2') (3") (4")
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ARITHMETIC LAWS AND PRINCIPAL NUMBERS
93
We leave consideration of a recursive analogue of (4') until we have dealt with Cantor Normal Forms, only remarking that we shall show that every co-ordinal of the form W Ais a principal number for addition but that these are not the only principal numbers for addition.
8.1.2 THEOREM. If A is a quord 2 1 then the following implications hold between (l),(2), (3) below:
-
z
(1) (2) - I (3) B < A*B + A = A , B+C=A*C=O or C = A , B, C < A*B + C < A .
(1) (2) (3)
PROOF. (1)*(2). Suppose (1) holds and B+ C = A. Then if C+O, B< A and by (l), B+A=A, hence by corollary 3.2.9, C=A. (2)*(1). Suppose (2) holds and B c A , then there is a C 9 0 such that B + C = A and by (2) we have C=A, hence (1) holds. (1)*(3). Suppose (1) holds and B, C
8.1.3 DEFINITION. A co-ordinal (quord) A is said to be a principal (9-principal) number for addition if A > 1 and B
8.1.4 THEOREM. If A € # ( +), then IAl is a limit number. PROOF. Clearly no finite co-ordinal is a principal number for addition. Suppose IA1 = 1+n where 1 is a limit number and n >0. Then by theorem 5.1.6 B=A-niswell-defined, JBJ=AandBO,B+A=A and 1+n +1= 1 which is impossible.
94
[Ch. 8
CONSTRUCTIVE ORDER TYPES
8.1.5
THEOREM. If P is a principal number for addition and A , By C < P
then A
PROOF. By theorem 8.1.2, A , B, C < P imply A + C < P and B + C t P . Therefore by theorem 4.2.8, A + C and B + C are comparable. Now classically, aa+ysp+
y,
hence by theorem 5.3.4 A + C < B+ C. COROLLARY. If A, B < P and P is a principal number for addition, then B I A + B. Conversely, if A + B < P , then A , B t P . PROOF. Trivially A < P if A + B < P . Since P is a principal number A + B < P implies A + B + P = P . But A < P so A + P = P and therefore A + ( B + P ) = A + P . Hence B + P = P and B I P . Finally we cannot have B =P since then A +B 43'. 8.1.6
8.2 We now give a series of results for quords. All the results in this
section were in a sense obtained in the context of ordinal algebras 1956) by Tarski. Many of the proofs are highly derivative from (TARSKI, Tarski's proofs. We do not know which, if any, of these results can be extended to arbitrary C.0.T.s though a start on this problem was made in 9 2.4 using Morley's lemma. 8.2.1
THEOREM. If B g 0 is a quord and there exist C, D such that then B = A * W .
B+C=A.W=D+B,
PROOF. By theorem 6.2.5. D + B = A - W implies A . W I * B. B+ C = A . W implies B< A . W, hence by theorem 4.2.6, B= A . W. 8.2.2
THEOREM. If C is a quord, then A + B + C = C and ( A + B ) - W = ( B + A ) . W A+C=B+C=C.
PROOF. By lemma 6.3.1, using the first condition, ( A + B).W
+D =C
for some quord D .
iff
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ARITHMETIC LAWS AND PRINCIPAL NUMBERS
95
By theorem 6.2.3.(iii), (B
+ A ) - W = B + ( A + B).W = ( A + B ) . W by the second condition.
Hence by lemma 6.3.1, B * W + E = ( A +B).W
forsomequordE.
Therefore B
+ C = B + ( A + B ) * W+ D , = B + (B.W + E ) + D , =B*W+E+D, by theorem 6.2.2.(iv) = ( A + B)*W D = C .
+
Thus B + C = C and therefore A + C = A + B + C = C . Now suppose A + C = B+ C = C,then A + B+ C = C and B+ A + C = C . Hence ( B + A ) . W + D = C and ( A + B ) . W+E=Cfor some quords D,E, by lemma 6.3.1. By the directed refinement theorem 2.3.2 it follows that for some F either (B + A)+W F = ( A B ) * W , or ( A + B)-W F = (B + A ) . W . (5) Wesuppose the latter holds. Now (A + B ) . W = A + ( B + A ) . W by theorem 6.2.3.(iii), hence (B + A ) . W I * ( A + B ) . W and hence by ( 5 ) and
+ +
+
theorem 4.2.6, (B + A ) . W = ( A + B ) . W . This completes the proof.
8.2.3 THEOREM. If A, C (or B, C) are quords, then A + C.n = B + C.n*A + C = B + C . PROOF. By the directed refinement theorem 2.3.2, there exists an E such that either A + E = B or B + E = A and E + C - n= C - n . Suppose, without loss of generality, that A + E = B . By lemma 6.3.1 we have, for some F, E . W + F = C * n = C + C . ( n - 1) by theorem 6.2.2.(ii). Now by the directed reiinement theorem 2.3.2, there exists G such that E * W + G = C and G + C - ( n - l ) = F
(i)
96
[a. 8
CONSTRUCTIVE ORDER TYPES
or C+G=E.W
and G + F = C . ( n - l ) .
(ii)
If (i) holds then
E+C=E+E*W+G =E*W+G =
c.
by theorem 6.2.2.(iv)
Now B+ C = ( A +E ) + C = A + ( E + C ) = A + C and the theorem is established for this case. If (ii) holds, then by theorem 6.2.5 there exists H such that H + E . W = G. But then C - ( n - 1)= G + F = H + E . W+F=Hf C - n = H + C * ( n- 1)+ C (last step by theorem 6.2.2.(ii)). By lemma 2.4.8 (proof) we therefore have C . ( n - l ) = C * ( n - l ) + C . But then by corollary 3.2.9 we have C=O which reduces the theorem to a triviality. 8.2.4
THEOREM. If A, C, D (or B, C) are quords and A + C - n + D = B + Cqn, then A + C + E = B + C for some (quord) E .
Before we proceed to the proof we observe that if one side (and hence the other) of the k s t equality is a recursive quord or co-ordinal then the theorem may be stated thus: A
+ Can 5 B + C.n=+A + C IB + C .
PROOF(by induction on n). If n = 1, then the assertion is trivially true with B= D. Now suppose the implication holds for n, then A
+ C * ( n+ 1) + D
=B
+ C.(n + 1)
implies
A + C - n + ( C + D ) = B + Can + C . Therefore, by the directed refinement theorem 2.3.2, there is an F such that either A + C - n + F = B + C - n and C + D = F C , or A + C - n = B + C - n + F and F + C + D = C . In the former case, by the induction hypothesis
+
A+C+E=B+C
for some E. In the latter case, by theorem 3.2.7, D=O and hence
A
+ C.(n + 1) = B + C * ( n+ 1)
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ARITHMETIC LAWS AND PRINCIPAL NUMBERS
97
and by theorem 8.2.3, A+ C=B+ C.
This completes the proof.
8.3 By analogy we now introduce principal numbers for multiplication (cf. BACHMANN, 1955,p. 66). 8.3.1 DEFINITION. A co-ordinal (quord) A is said to be a principal (.%principal) number for multiplication if A > 2 and 0
We write X ( . )[3(.)]for the collection of all principal (9-principal) numbers for multiplication. Later on we shall show that A E3(-)implies A is divisible by or divides Wwnfor every n. As in the classical case O < B
A
is a stronger condition than BC=A*B=A
or C = A .
But also, for co-ordinals, the former condition is stronger than 0 < B, C < A* BC < A .
For V satisfies this last condition but is not a principal number for multiplication since 2< V but by lemma 7.4.1,2V = V implies 2w = V and by theorem 7.4.2,2w = W + V. 8.3.2 THEOREM. If A is a co-ordinal in X ( -),then IAJis a limit number. PROOF. Left to the reader (cf. theorem 8.1.4).
+
8.3.3 THEOREM. (i) If B 0, then A I A B when B is a recursive quord, a co-ordinal or B 2 1. (ii) If B > 1, then A < A B whenever ABO for an arbitrary quord B. (iii) If A divides B and IAI =lBl, then A = B . PROOF. We prove only (ii) and (iii) leaving (i) to the reader.
98
mNsTRum ORDJiR TYPES
[Ch. 8
(ii) B > l = > ( E !C ) (B=l+C&C+O).HenceAB=A(l+C)=A+AC, where A C l O i f A 9 0 ; thus A c A B . (iii) By (i) A divides B implies A S B hence by theorem 5.3.4, A =B. 8.3.4 THEOREM. If A and B are isomorphic well-orderings then there is a unique isomorphism f such that every isomorphism between A and B is an extension off. (SIERPINSKI, 1958, p. 264, corollary 3).
We now prove a left cancellation law for co-ordinals using this classical theorem 8.3.4. Later on we shall use the same technique to obtain a cancellation law for exponentiation for co-ordinals. 8.3.5
THEOREM. If A 9 0 and A , B, C are co-ordinals, then AB = AC* B = C .
PROOF. Let AEA, BEB and C E C and suppose p : AB N AC.
Then AB-AC and since AB and AC are well-orderings it follows from the preceding theorem that p is an extension of the unique minimal isomorphism, p,, say, between AB and AC. Now, classically, u 0 &a/? = ay-p = y ,
*
hence there is an isomorphism q, (not necessarily partial recursive) such that
q,:B-C. Now the map r , : j ( a , b)=+,
q,(b)),
defined only on C A B is an isomorphism between AB and AC and therefore, by theorem 8.3.4, p is an extension of r,. Since A+O, there is an element, say a,, in C'A. Let p , be the map p with domain and range restricted to (j(ao,n):nENZ,
then po is partial recursive. Further, if poj(ao,x) is defined, then its value is j(u,, y) for some y.
Ch. 81
ARITHMETIC LAWS AND PRINCIPAL NUMBERS
99
Now let qo be the map 40 :x + IPOj (ao, x)
,
then clearly qo is partial recursive and agrees with qc on C'B (again by theorem 8.3.4). qo is one-one, since 40
(XI = 40 ( Y ) * k o j (ao, x) = b o j (ao, Y ) * P o j (ao, x) = j (UO? c ) Lk P o j (a05 Y > = j (a03 c>
(since ppo E {j(uo,n ) : n e M }by construction) * j ( a 0 , x) = j ( u o , y)
(since p is one-one)
*x=y.
Thus qo is partial recursive and one-one and also agrees with qe on C'B, i.e. q0:B=C from which the theorem follows. We have a stronger version of the above theorem which derives from SIERPINSIU (1948), however, we shall leave the proof of this theorem until appendix A.
8.3.6 LEMMA. If M is a principal number for multiplication, and B, C 0, then
+
BC< M o B < M & C < M .
PROOF. Suppose BC
=> BCM = M
,
and therefore by theorem 8.3.5, CM=M.
Using theorem 8.3.3.(ii) it follows that C
Conversely, C
=M
and B < M a B M = M .
100
CONSTRUCTIVE ORDER TYPES
[Ch. 8
Hence (BC) M
= B ( C M ) = BM = M
and by theorem 8.3.3.(ii) BC
8.4. 8.4.1 THEOREM. If A, C are quords then (i) If A =I= 0, then B < C* AB < A C ,
(ii) BI; C-ABS AC. PROOF.Let AEA and CEC, then there exists BEB such that B
+@
C=B$D. Now the reader will readily verify that AC = A(B $ D) = AB $ AD. But AD
+0,since A, D=k8, hence AB
-= AC
and the theorem follows by taking C.0.T.s. (ii) follows at once from (i).
8.4.2 THEOREM. There exist co-ordinals A, ByC+O such that A < B but AC $ BC.
PROOF. Let A = 1, B= V and C = W, then AC=W
and B C = V W .
By theorem 8.3.3.(i), VIVW.
Hence if
w 5 vw, V and W are comparable by theorem 4.2.8 which contradicts corollary 5.2.7.
Ch. 81
ARITHMETIC LAWS AND PRINCIPAL NUMBERS
101
8.4.3 THEOREM. If there is a principal number for multiplication, My such that B, C < M (or equivalently B C < M ) then A < B*AC
BC
PROOF. If B or C=O there is nothing to prove. Similarly if A=O. Otherwise, by lemma 8.3.6, AC
and B C < M .
Hence, by theorem 4.2.8, A C and BC are comparable. Now classically, for arbitrary ordinals a, /I, y a < P*ar
S
Pr,
hence ACI BC by corollary 5.3.5. 8.4.4 THEOREM. If A, B, C are co-ordinals, then AC < BC* A < B .
PROOF. If C=O, then the assertion is trivial. If C+O, then by theorem 8.3.3.(i), A I A C and B S B C . Hence by the transitivity of Iand theorem 4.2.8, A and B are comparable. By the classical theorem UY
P,
we have IAl .c IBI
and hence by corollary 5.3.5 A
8.4.5 THEOREM. There exist co-ordinals A, B, C such that O
but A $ B .
PROOF.(As in the classical case.) Let A =2, B= 1, C = W. 8.4.6 THEOREM. If By C are comparable quords, then A B
PROOF. Immediate from theorem 8.4.1 .(i).
102
[Ch. 8
CONSTRUCTIVE ORDER TYPES
THEOREM. If A, ByC are co-ordinals and AB< AC then B < C ; similarly with “ s ” replacing “ < ” at both occurrences. PROOF. AB
8.4.7
So A C = A D + A E by theorem 6.2.1. By theorem 4.2.1 we have ( 3 F ) ( A . B + F = A . C ) and again using corollary 5.3.6 ( E ! G ) ( E ! H ) ( G + H = A C& IGl=a/3). We must have then G = A B = A D , and by theorem 8.3.5, B = D . D < C by (*) so the result follows. The other part of the theorem follows from the above and theorem 8.3.5.
It is well-known that, classically, for arbitrary order types
b, z,
0 - n = z - n & n > O*o = z (and similarly with ‘‘5” replacing “ =” at both occurrences, see (1948) for proofs). For quords the analogues are also true SIERPINSKI, provided we use the strong interpretation of I ; thus we have: 8.4.8
THEOREM. If A or B is a quord, then A - n + C = B - n & n > O* (30) (A
+ D = B).
PROOF^^ (by induction on n). If n=1, then the assertion is trivial. Now assume the theorem holds for n and that A-(n + 1) + C = B . ( n Then
A * n+ ( A
+ 1).
+ C ) = Ban + B
and by the directed refinement theorem 2.3.2, there is an E such that either A-n + E = B * n ( & A + C = E + B), or A * n = Ben + E & E + A + C = B . In the former case the assertion follows by the induction hypothesis. In the latter, by the induction hypothesis we have, for some F, A=B+F,
Ch. 81
ARITHMETIC LAWS AM) PRINCIPAL NUMBERS
103
whence E+B+F+C=B
and by theorem 3.2.1, E+B=B,
F=C=O.
We conclude A =B.
8.5 We now introduce principal numbers for exponentiation and investigate the arithmetic laws for exponentiation.
8.5.1 DEFINITION. A co-ordinal (quord) A is said to be a principal (9-principal) number for exponentiation if A > 2 and
1 < B < A - B~ = A . We write &' (exp) [1(exp)] for the collection of all principal (%principal) numbers for exponentiation. Later on we shall obtain an explicit description of all principal numbers for exponentiation (chapter 11). THEOREM. If AE&'(exp), then IAl is a limit number. PROOF. Left to the reader.
8.5.2
The condition in definition 8.5.1 is stronger than the condition 2 5 B, C < A*BC < A
This will be shown later by proving (lemma 11.2.2) that
2A = A *
W divides A ,
whereas V satisfies (6) but not (7). 8.5.3
THEOREM. If A > 1, B, C are co-ordinals, then A~=A~*B=c.
PROOF.^^ Let AEA, BEB and CEC and suppose p : A~
A=.
104
CONSTRUCTIVE ORDER TYPES
[Ch. 8
Then A'-AC and since A' and A' are well-orderings it follows from theorem 8.3.4 that p is an extension of the unique minimal isomorphism p e , say, between A' and A'. Now, classically, u>l&aS=aY+B=y,
hence there is an isomorphism qc (not necessarily partial recursive) such that qc:B C . Now the map N
defined only on 83 e (A, B) is an isomorphism between A' and A'. Hence by theorem 8.3.4, p is an extension of rc. Since A > 1, there is a non-minimum element, say a,, in C'A. Let p , be the map p with domain and range restricted to
then p , is partial recursive. Further if
is defined then its value is
for some y. Now let q, be the map
then clearly q, is partial recursive and agrees with qc on theorem 8.3.4). qo is one-one, since
C'B (again by
Ch. 81
ARITHMETIC LAWS AND PRINCIF'AL NUMBERS
for some u, v , m,n, xl, ..., x,, y,, image of p , is of the form
...,y,,,. But by the definition of p,,
105
any
2j(oo. b )
and hence n =m =0,u = v = a, and
and
Therefore, from qo(x) = qo (Y), we have
from which it follows, sincep, and e are one-one, that x=y. Thus we have shown that q, is a recursive isomorphism between B and C and the proof is complete. 8.5.4
THEOREM. There exist co-ordinals A, B, C, all > 1, such that A C = B C but A + B .
PROOF(as in the classical case). Let A = 2 , B = 3 , C = W, then by theorem 7.4.2, 2" = 3" = W. 8.5.5
THEOREM. (i) A > 1 & B
(for arbitrary quords A,
B, 0. (ii) If C is a co-ordinal, then A > 18z B < C+AB
d>,$,b, hence any bracket symbol in (A,
BT D) is of the form
106
CONSTRUCTIVE ORDER TYPES
[Ch. 8
where the d&'D, b j ~ C ' Band the a,, aj.eC'A ( i = O , . . . , m ;j=O,..., n). Hence from the definition of exponentiation we easily obtain
A0 5 AO'D= AC. (ii) By theorem 7.2.1.(ii), assuming A E A ,etc. as above, A' (and hence AB)is a quasi-well-ordering. Hence it suffices, by lemma 4.2.5, to prove that A~ A'.
+
A a+min(A). Since D+0 there Now A > 1, hence there exists ~ E C with also exists dsC'D. Therefore
(")
e a
CAC
- CAB,
from which we have the required result.
LEMMA.(i) If C is a co-ordinal and A, C > 1 then A < A'. (ii) If C r 1, then A
8.5.6
Now, by the definition of exponentiation we have at once 1 IA D since A 9 0.
Hence
A' = A ( l + E) for some E, by theorem 6.2.1. = A + AE
The required result now follows at once. 8.5.7
THEOFCEM. There exist co-ordinals A , B, C such that A
but A C $ B c .
PROOF.Let A = 2 , B= Vand C= W,then by theorem 7.4.2, A'=C=W and by lemma 8.5.6.(i), v < VW=BC. Now,if IB ~ ,
Ch. 81
107
ARITHMETIC LAWS AND PRINCIPAL NUMBERS
then by theorem 4.2.8, V and W are comparable which contradicts corollary 5.2.7. Thus we see that the analogue of one of the classical laws for exponentiation breaks down in a very similar way to one of the multiplicative laws (theorem 8.4.2). However, the similarity also extends to the cases where the analogues do go over.
LEMMA.If E is a principal number for exponentiation, then A , B 1. PROOF. The assertion is trivial if A , B I 1. Otherwise, if E is a principal number for exponentiation, then A < E*AE= E and similarly for B ; moreover, we must also have that A and B are co-ordinals since E is a co-ordinal. It follows that A(BE) =E.
8.5.8
Since B < E it follows by theorem 2.1.2 that there is a (co-ordinal) C such that B+C=E. Therefore E = A ( B ~ )= A ( B + C ) = A B . A C , But A'> I , since C + O ; hence A'= 1 + D forsome D + O and it follows that i.e.
+
E = A B ( l + D) = AB + ABD where ABD 0 ,
A~ < E . Conversely, suppose A,B>1
and A B < E ,
then by lemma 8.5.6.(i), A
Since E is a principal number for exponentiation,
E=
= ( A ~=) ~
By theorem 8.5.3 it follows that BE=E
108
CONSTRUCIWB ORDER TYPES
[Ch. 8
and hence by theorem 8.3.3.(ii) we have B
PROOF. By the transitivity of 5 and lemma 8.5.8, A C < E and B C < E .
Hence by theorem 4.2.8, AC and BC are comparable co-ordinals. Now, classically, for ordinals a, fl, y, hence
8.5.10 THEOREM. If A, B, C are co-ordinals, then
c B~ =+ A .C B . PROOF. If C=O, there is nothing to prove. Otherwise, by lemma 8.5.6(ii), A I A C and B I Bc
and therefore by theorem 4.2.8 and the transitivity of I, A and B are comparable. Hence by the classical theorem for ordinals ay < pY*u
< 8,
we have IAI < IBI
and hence A < B .
8.5.11 THEOREM. There exist co-ordinals A, B, C such that
1 < AC IBC but A $ B.
PROOF(as in the classical case). Let A=3, B = 2 and C=W, then by theorem 7.4.2, A ~ = B ~W =.
Ch.81
ARITHMETIC LAWS AND PRINCIPAL NUMBERS
8.5.12 THEOREM. (i) If B, C are comparable and A > 1, then A*
=.
3
B
PROOF. By theorem 8.5.5., since C I B means C = B or C < B.
109
ARITHMETIC LAWS AND PRINCIPAL NUMBERS
8.1 We now investigate the arithmetic laws of C.0.T.s. Some laws for addition were established in chapter 4 and we first of all strengthen these. 8.1.1 THEOREM. If A , By C are quords, then B < C-A + B < A
+ c.
PROOF.Immediate from theorem 4.2.3.(v) and corollary 3.2.9. However, the other obvious analogue of a classical additive law
BIC*B+AIC+A
(*I
does not hold in general even for co-ordinals. For let A = W, B=O and C = V, then (*) implies W and V are comparable which contradicts corollary 5.2.7. However, (*) does hold if A , B, C are less than what we call a principal number for addition. Classically, a principal number for addition (otherwise called a y-number (BACHMANN, 1955, p. 67) or a prime component (SIERPINSKI, 1958, p. 279)) may be defined as an ordinal number y such that any of the following (equivalent) conditions holds: a
or P = y ,
P
y =0 or y = ma for some ordinal u .
(1") (2') (3") (4")
Ch. 81
ARITHMETIC LAWS AND PRINCIPAL NUMBERS
93
We leave consideration of a recursive analogue of (4') until we have dealt with Cantor Normal Forms, only remarking that we shall show that every co-ordinal of the form W Ais a principal number for addition but that these are not the only principal numbers for addition.
8.1.2 THEOREM. If A is a quord 2 1 then the following implications hold between (l),(2), (3) below:
-
z
(1) (2) - I (3) B < A*B + A = A , B+C=A*C=O or C = A , B, C < A*B + C < A .
(1) (2) (3)
PROOF. (1)*(2). Suppose (1) holds and B+ C = A. Then if C+O, B< A and by (l), B+A=A, hence by corollary 3.2.9, C=A. (2)*(1). Suppose (2) holds and B c A , then there is a C 9 0 such that B + C = A and by (2) we have C=A, hence (1) holds. (1)*(3). Suppose (1) holds and B, C
8.1.3 DEFINITION. A co-ordinal (quord) A is said to be a principal (9-principal) number for addition if A > 1 and B
8.1.4 THEOREM. If A € # ( +), then IAl is a limit number. PROOF. Clearly no finite co-ordinal is a principal number for addition. Suppose IA1 = 1+n where 1 is a limit number and n >0. Then by theorem 5.1.6 B=A-niswell-defined, JBJ=AandB
94
[Ch. 8
CONSTRUCTIVE ORDER TYPES
8.1.5
THEOREM. If P is a principal number for addition and A , By C < P
then A
PROOF. By theorem 8.1.2, A , B, C < P imply A + C < P and B + C t P . Therefore by theorem 4.2.8, A + C and B + C are comparable. Now classically, a
y,
hence by theorem 5.3.4 A + C < B+ C. COROLLARY. If A, B < P and P is a principal number for addition, then B I A + B. Conversely, if A + B < P , then A , B t P . PROOF. Trivially A < P if A + B < P . Since P is a principal number A + B < P implies A + B + P = P . But A < P so A + P = P and therefore A + ( B + P ) = A + P . Hence B + P = P and B I P . Finally we cannot have B =P since then A +B 43'. 8.1.6
8.2 We now give a series of results for quords. All the results in this
section were in a sense obtained in the context of ordinal algebras 1956) by Tarski. Many of the proofs are highly derivative from (TARSKI, Tarski's proofs. We do not know which, if any, of these results can be extended to arbitrary C.0.T.s though a start on this problem was made in 9 2.4 using Morley's lemma. 8.2.1
THEOREM. If B g 0 is a quord and there exist C, D such that then B = A * W .
B+C=A.W=D+B,
PROOF. By theorem 6.2.5. D + B = A - W implies A . W I * B. B+ C = A . W implies B< A . W, hence by theorem 4.2.6, B= A . W. 8.2.2
THEOREM. If C is a quord, then A + B + C = C and ( A + B ) - W = ( B + A ) . W A+C=B+C=C.
PROOF. By lemma 6.3.1, using the first condition, ( A + B).W
+D =C
for some quord D .
iff
Ch. 81
ARITHMETIC LAWS AND PRINCIPAL NUMBERS
95
By theorem 6.2.3.(iii), (B
+ A ) - W = B + ( A + B).W = ( A + B ) . W by the second condition.
Hence by lemma 6.3.1, B * W + E = ( A +B).W
forsomequordE.
Therefore B
+ C = B + ( A + B ) * W+ D , = B + (B.W + E ) + D , =B*W+E+D, by theorem 6.2.2.(iv) = ( A + B)*W D = C .
+
Thus B + C = C and therefore A + C = A + B + C = C . Now suppose A + C = B+ C = C,then A + B+ C = C and B+ A + C = C . Hence ( B + A ) . W + D = C and ( A + B ) . W+E=Cfor some quords D,E, by lemma 6.3.1. By the directed refinement theorem 2.3.2 it follows that for some F either (B + A)+W F = ( A B ) * W , or ( A + B)-W F = (B + A ) . W . (5) Wesuppose the latter holds. Now (A + B ) . W = A + ( B + A ) . W by theorem 6.2.3.(iii), hence (B + A ) . W I * ( A + B ) . W and hence by ( 5 ) and
+ +
+
theorem 4.2.6, (B + A ) . W = ( A + B ) . W . This completes the proof.
8.2.3 THEOREM. If A, C (or B, C) are quords, then A + C.n = B + C.n*A + C = B + C . PROOF. By the directed refinement theorem 2.3.2, there exists an E such that either A + E = B or B + E = A and E + C - n= C - n . Suppose, without loss of generality, that A + E = B . By lemma 6.3.1 we have, for some F, E . W + F = C * n = C + C . ( n - 1) by theorem 6.2.2.(ii). Now by the directed reiinement theorem 2.3.2, there exists G such that E * W + G = C and G + C - ( n - l ) = F
(i)
96
[a. 8
CONSTRUCTIVE ORDER TYPES
or C+G=E.W
and G + F = C . ( n - l ) .
(ii)
If (i) holds then
E+C=E+E*W+G =E*W+G =
c.
by theorem 6.2.2.(iv)
Now B+ C = ( A +E ) + C = A + ( E + C ) = A + C and the theorem is established for this case. If (ii) holds, then by theorem 6.2.5 there exists H such that H + E . W = G. But then C - ( n - 1)= G + F = H + E . W+F=Hf C - n = H + C * ( n- 1)+ C (last step by theorem 6.2.2.(ii)). By lemma 2.4.8 (proof) we therefore have C . ( n - l ) = C * ( n - l ) + C . But then by corollary 3.2.9 we have C=O which reduces the theorem to a triviality. 8.2.4
THEOREM. If A, C, D (or B, C) are quords and A + C - n + D = B + Cqn, then A + C + E = B + C for some (quord) E .
Before we proceed to the proof we observe that if one side (and hence the other) of the k s t equality is a recursive quord or co-ordinal then the theorem may be stated thus: A
+ Can 5 B + C.n=+A + C IB + C .
PROOF(by induction on n). If n = 1, then the assertion is trivially true with B= D. Now suppose the implication holds for n, then A
+ C * ( n+ 1) + D
=B
+ C.(n + 1)
implies
A + C - n + ( C + D ) = B + Can + C . Therefore, by the directed refinement theorem 2.3.2, there is an F such that either A + C - n + F = B + C - n and C + D = F C , or A + C - n = B + C - n + F and F + C + D = C . In the former case, by the induction hypothesis
+
A+C+E=B+C
for some E. In the latter case, by theorem 3.2.7, D=O and hence
A
+ C.(n + 1) = B + C * ( n+ 1)
Ch. 81
ARITHMETIC LAWS AND PRINCIPAL NUMBERS
97
and by theorem 8.2.3, A+ C=B+ C.
This completes the proof.
8.3 By analogy we now introduce principal numbers for multiplication (cf. BACHMANN, 1955,p. 66). 8.3.1 DEFINITION. A co-ordinal (quord) A is said to be a principal (.%principal) number for multiplication if A > 2 and 0
We write X ( . )[3(.)]for the collection of all principal (9-principal) numbers for multiplication. Later on we shall show that A E3(-)implies A is divisible by or divides Wwnfor every n. As in the classical case O < B
A
is a stronger condition than BC=A*B=A
or C = A .
But also, for co-ordinals, the former condition is stronger than 0 < B, C < A* BC < A .
For V satisfies this last condition but is not a principal number for multiplication since 2< V but by lemma 7.4.1,2V = V implies 2w = V and by theorem 7.4.2,2w = W + V. 8.3.2 THEOREM. If A is a co-ordinal in X ( -),then IAJis a limit number. PROOF. Left to the reader (cf. theorem 8.1.4).
+
8.3.3 THEOREM. (i) If B 0, then A I A B when B is a recursive quord, a co-ordinal or B 2 1. (ii) If B > 1, then A < A B whenever ABO for an arbitrary quord B. (iii) If A divides B and IAI =lBl, then A = B . PROOF. We prove only (ii) and (iii) leaving (i) to the reader.
98
mNsTRum ORDJiR TYPES
[Ch. 8
(ii) B > l = > ( E !C ) (B=l+C&C+O).HenceAB=A(l+C)=A+AC, where A C l O i f A 9 0 ; thus A c A B . (iii) By (i) A divides B implies A S B hence by theorem 5.3.4, A =B. 8.3.4 THEOREM. If A and B are isomorphic well-orderings then there is a unique isomorphism f such that every isomorphism between A and B is an extension off. (SIERPINSKI, 1958, p. 264, corollary 3).
We now prove a left cancellation law for co-ordinals using this classical theorem 8.3.4. Later on we shall use the same technique to obtain a cancellation law for exponentiation for co-ordinals. 8.3.5
THEOREM. If A 9 0 and A , B, C are co-ordinals, then AB = AC* B = C .
PROOF. Let AEA, BEB and C E C and suppose p : AB N AC.
Then AB-AC and since AB and AC are well-orderings it follows from the preceding theorem that p is an extension of the unique minimal isomorphism, p,, say, between AB and AC. Now, classically, u 0 &a/? = ay-p = y ,
*
hence there is an isomorphism q, (not necessarily partial recursive) such that
q,:B-C. Now the map r , : j ( a , b)=+,
q,(b)),
defined only on C A B is an isomorphism between AB and AC and therefore, by theorem 8.3.4, p is an extension of r,. Since A+O, there is an element, say a,, in C'A. Let p , be the map p with domain and range restricted to (j(ao,n):nENZ,
then po is partial recursive. Further, if poj(ao,x) is defined, then its value is j(u,, y) for some y.
Ch. 81
ARITHMETIC LAWS AND PRINCIPAL NUMBERS
99
Now let qo be the map 40 :x + IPOj (ao, x)
,
then clearly qo is partial recursive and agrees with qc on C'B (again by theorem 8.3.4). qo is one-one, since 40
(XI = 40 ( Y ) * k o j (ao, x) = b o j (ao, Y ) * P o j (ao, x) = j (UO? c ) Lk P o j (a05 Y > = j (a03 c>
(since ppo E {j(uo,n ) : n e M }by construction) * j ( a 0 , x) = j ( u o , y)
(since p is one-one)
*x=y.
Thus qo is partial recursive and one-one and also agrees with qe on C'B, i.e. q0:B=C from which the theorem follows. We have a stronger version of the above theorem which derives from SIERPINSIU (1948), however, we shall leave the proof of this theorem until appendix A.
8.3.6 LEMMA. If M is a principal number for multiplication, and B, C 0, then
+
BC< M o B < M & C < M .
PROOF. Suppose BC
=> BCM = M
,
and therefore by theorem 8.3.5, CM=M.
Using theorem 8.3.3.(ii) it follows that C
Conversely, C
=M
and B < M a B M = M .
100
CONSTRUCTIVE ORDER TYPES
[Ch. 8
Hence (BC) M
= B ( C M ) = BM = M
and by theorem 8.3.3.(ii) BC
8.4. 8.4.1 THEOREM. If A, C are quords then (i) If A =I= 0, then B < C* AB < A C ,
(ii) BI; C-ABS AC. PROOF.Let AEA and CEC, then there exists BEB such that B
+@
C=B$D. Now the reader will readily verify that AC = A(B $ D) = AB $ AD. But AD
+0,since A, D=k8, hence AB
-= AC
and the theorem follows by taking C.0.T.s. (ii) follows at once from (i).
8.4.2 THEOREM. There exist co-ordinals A, ByC+O such that A < B but AC $ BC.
PROOF. Let A = 1, B= V and C = W, then AC=W
and B C = V W .
By theorem 8.3.3.(i), VIVW.
Hence if
w 5 vw, V and W are comparable by theorem 4.2.8 which contradicts corollary 5.2.7.
Ch. 81
ARITHMETIC LAWS AND PRINCIPAL NUMBERS
101
8.4.3 THEOREM. If there is a principal number for multiplication, My such that B, C < M (or equivalently B C < M ) then A < B*AC
BC
PROOF. If B or C=O there is nothing to prove. Similarly if A=O. Otherwise, by lemma 8.3.6, AC
and B C < M .
Hence, by theorem 4.2.8, A C and BC are comparable. Now classically, for arbitrary ordinals a, /I, y a < P*ar
S
Pr,
hence ACI BC by corollary 5.3.5. 8.4.4 THEOREM. If A, B, C are co-ordinals, then AC < BC* A < B .
PROOF. If C=O, then the assertion is trivial. If C+O, then by theorem 8.3.3.(i), A I A C and B S B C . Hence by the transitivity of Iand theorem 4.2.8, A and B are comparable. By the classical theorem UY
P,
we have IAl .c IBI
and hence by corollary 5.3.5 A
8.4.5 THEOREM. There exist co-ordinals A, B, C such that O
but A $ B .
PROOF.(As in the classical case.) Let A =2, B= 1, C = W. 8.4.6 THEOREM. If By C are comparable quords, then A B
PROOF. Immediate from theorem 8.4.1 .(i).
102
[Ch. 8
CONSTRUCTIVE ORDER TYPES
THEOREM. If A, ByC are co-ordinals and AB< AC then B < C ; similarly with “ s ” replacing “ < ” at both occurrences. PROOF. AB
8.4.7
So A C = A D + A E by theorem 6.2.1. By theorem 4.2.1 we have ( 3 F ) ( A . B + F = A . C ) and again using corollary 5.3.6 ( E ! G ) ( E ! H ) ( G + H = A C& IGl=a/3). We must have then G = A B = A D , and by theorem 8.3.5, B = D . D < C by (*) so the result follows. The other part of the theorem follows from the above and theorem 8.3.5.
It is well-known that, classically, for arbitrary order types
b, z,
0 - n = z - n & n > O*o = z (and similarly with ‘‘5” replacing “ =” at both occurrences, see (1948) for proofs). For quords the analogues are also true SIERPINSKI, provided we use the strong interpretation of I ; thus we have: 8.4.8
THEOREM. If A or B is a quord, then A - n + C = B - n & n > O* (30) (A
+ D = B).
PROOF^^ (by induction on n). If n=1, then the assertion is trivial. Now assume the theorem holds for n and that A-(n + 1) + C = B . ( n Then
A * n+ ( A
+ 1).
+ C ) = Ban + B
and by the directed refinement theorem 2.3.2, there is an E such that either A-n + E = B * n ( & A + C = E + B), or A * n = Ben + E & E + A + C = B . In the former case the assertion follows by the induction hypothesis. In the latter, by the induction hypothesis we have, for some F, A=B+F,
Ch. 81
ARITHMETIC LAWS AM) PRINCIPAL NUMBERS
103
whence E+B+F+C=B
and by theorem 3.2.1, E+B=B,
F=C=O.
We conclude A =B.
8.5 We now introduce principal numbers for exponentiation and investigate the arithmetic laws for exponentiation.
8.5.1 DEFINITION. A co-ordinal (quord) A is said to be a principal (9-principal) number for exponentiation if A > 2 and
1 < B < A - B~ = A . We write &' (exp) [1(exp)] for the collection of all principal (%principal) numbers for exponentiation. Later on we shall obtain an explicit description of all principal numbers for exponentiation (chapter 11). THEOREM. If AE&'(exp), then IAl is a limit number. PROOF. Left to the reader.
8.5.2
The condition in definition 8.5.1 is stronger than the condition 2 5 B, C < A*BC < A
This will be shown later by proving (lemma 11.2.2) that
2A = A *
W divides A ,
whereas V satisfies (6) but not (7). 8.5.3
THEOREM. If A > 1, B, C are co-ordinals, then A~=A~*B=c.
PROOF.^^ Let AEA, BEB and CEC and suppose p : A~
A=.
104
CONSTRUCTIVE ORDER TYPES
[Ch. 8
Then A'-AC and since A' and A' are well-orderings it follows from theorem 8.3.4 that p is an extension of the unique minimal isomorphism p e , say, between A' and A'. Now, classically, u>l&aS=aY+B=y,
hence there is an isomorphism qc (not necessarily partial recursive) such that qc:B C . Now the map N
defined only on 83 e (A, B) is an isomorphism between A' and A'. Hence by theorem 8.3.4, p is an extension of rc. Since A > 1, there is a non-minimum element, say a,, in C'A. Let p , be the map p with domain and range restricted to
then p , is partial recursive. Further if
is defined then its value is
for some y. Now let q, be the map
then clearly q, is partial recursive and agrees with qc on theorem 8.3.4). qo is one-one, since
C'B (again by
Ch. 81
ARITHMETIC LAWS AND PRINCIF'AL NUMBERS
for some u, v , m,n, xl, ..., x,, y,, image of p , is of the form
...,y,,,. But by the definition of p,,
105
any
2j(oo. b )
and hence n =m =0,u = v = a, and
and
Therefore, from qo(x) = qo (Y), we have
from which it follows, sincep, and e are one-one, that x=y. Thus we have shown that q, is a recursive isomorphism between B and C and the proof is complete. 8.5.4
THEOREM. There exist co-ordinals A, B, C, all > 1, such that A C = B C but A + B .
PROOF(as in the classical case). Let A = 2 , B = 3 , C = W, then by theorem 7.4.2, 2" = 3" = W. 8.5.5
THEOREM. (i) A > 1 & B
(for arbitrary quords A,
B, 0. (ii) If C is a co-ordinal, then A > 18z B < C+AB
d>,$,b, hence any bracket symbol in (A,
BT D) is of the form
106
CONSTRUCTIVE ORDER TYPES
[Ch. 8
where the d&'D, b j ~ C ' Band the a,, aj.eC'A ( i = O , . . . , m ;j=O,..., n). Hence from the definition of exponentiation we easily obtain
A0 5 AO'D= AC. (ii) By theorem 7.2.1.(ii), assuming A E A ,etc. as above, A' (and hence AB)is a quasi-well-ordering. Hence it suffices, by lemma 4.2.5, to prove that A~ A'.
+
A a+min(A). Since D+0 there Now A > 1, hence there exists ~ E C with also exists dsC'D. Therefore
(")
e a
CAC
- CAB,
from which we have the required result.
LEMMA.(i) If C is a co-ordinal and A, C > 1 then A < A'. (ii) If C r 1, then A
8.5.6
Now, by the definition of exponentiation we have at once 1 IA D since A 9 0.
Hence
A' = A ( l + E) for some E, by theorem 6.2.1. = A + AE
The required result now follows at once. 8.5.7
THEOFCEM. There exist co-ordinals A , B, C such that A
but A C $ B c .
PROOF.Let A = 2 , B= Vand C= W,then by theorem 7.4.2, A'=C=W and by lemma 8.5.6.(i), v < VW=BC. Now,if IB ~ ,
Ch. 81
107
ARITHMETIC LAWS AND PRINCIPAL NUMBERS
then by theorem 4.2.8, V and W are comparable which contradicts corollary 5.2.7. Thus we see that the analogue of one of the classical laws for exponentiation breaks down in a very similar way to one of the multiplicative laws (theorem 8.4.2). However, the similarity also extends to the cases where the analogues do go over.
LEMMA.If E is a principal number for exponentiation, then A , B
8.5.8
Since B < E it follows by theorem 2.1.2 that there is a (co-ordinal) C such that B+C=E. Therefore E = A ( B ~ )= A ( B + C ) = A B . A C , But A'> I , since C + O ; hence A'= 1 + D forsome D + O and it follows that i.e.
+
E = A B ( l + D) = AB + ABD where ABD 0 ,
A~ < E . Conversely, suppose A,B>1
and A B < E ,
then by lemma 8.5.6.(i), A
Since E is a principal number for exponentiation,
E=
= ( A ~=) ~
By theorem 8.5.3 it follows that BE=E
108
CONSTRUCIWB ORDER TYPES
[Ch. 8
and hence by theorem 8.3.3.(ii) we have B
PROOF. By the transitivity of 5 and lemma 8.5.8, A C < E and B C < E .
Hence by theorem 4.2.8, AC and BC are comparable co-ordinals. Now, classically, for ordinals a, fl, y, hence
8.5.10 THEOREM. If A, B, C are co-ordinals, then
c B~ =+ A .C B . PROOF. If C=O, there is nothing to prove. Otherwise, by lemma 8.5.6(ii), A I A C and B I Bc
and therefore by theorem 4.2.8 and the transitivity of I, A and B are comparable. Hence by the classical theorem for ordinals ay < pY*u
< 8,
we have IAI < IBI
and hence A < B .
8.5.11 THEOREM. There exist co-ordinals A, B, C such that
1 < AC IBC but A $ B.
PROOF(as in the classical case). Let A=3, B = 2 and C=W, then by theorem 7.4.2, A ~ = B ~W =.
Ch.81
ARITHMETIC LAWS AND PRINCIPAL NUMBERS
8.5.12 THEOREM. (i) If B, C are comparable and A > 1, then A*
=.
3
B
PROOF. By theorem 8.5.5., since C I B means C = B or C < B.
109