A
S A U L A. B A S R I Department of Physics Colorado State University Ft. Collins, Colorado
1966
0
-
- 1966
All rights reserved No part of this book may be reproduced in any form by print, microfilm or any other means without written permission froin the publisher
PREFACE
do upon single upon
I”;
on.
up
PREFACE
VI
symbolic Eogic (SL). book upon
SL A.
by
by I ,
C,
by P ,
by A ,
by T,
by D . by
on
by
by
5
TI14.5 11,
T5
4,
114. on p
q.
book
on by
by 119581,
v
x,
[1924],
[1914]. IV
on
V.
[1938], IV,
PREFACE
VII
I
SAULA. BASRI
1965
I. W H A T IS A DEDUCTIVE THEORY?
1. Deductive physical theory A (2)
physical concepts
physical larcs. explained
A
a
jied every concepts
every primitive
postulates;
on
experimental
3
4
WHAT IS A
[I
12
A deductive physical theory. 2. Deductive abstract theory by by
by
by
interpretations,
deductive abstract theory ; ubstracted by An
by
5 8, pp.
9 ([1950]
A
concepts. strings. If s s.t
s
t
by
121
5
DEDUCTIVE ABSTRACT THEORY
A
by
...,a, b
(1)
(1) a,,
...,
(2) a,, . . .,
A b ‘4’
(2)
by s,, . ..,sk
,
x,,
. .. , x k -
...,
sk- * x k -
* x l ~s2-x2~..:
si
ask’,
xi
no
p. si
As d,
6 Concepts: 1, %, = . Axioms: . Postulates: %m, %n 4 % m - n ,
%n+n=n. 1, ‘%n’
Interpretations: ‘m=n’
“n
m,n ‘m’
‘m’ ‘m’
on.
‘n’
‘n’ ‘n’ by ‘=’ %= =.
81111, %=
... 3. Definitions
‘d’
A
9,
gf,
9’
d, 9‘
9 9)
9
9,
9‘
9
9’.
‘d’
9, p. ‘d’
s
9.
9’
9
xdy+xsy, xsy+xdy, x, y
9
d for
s.
9‘ by
d
(2)
7
DEFINITIONS
by
‘x’,‘~’
9
9, xsy
xsy.
9 by
a?’,
11, 3 f o r 111, 4 f o r 1111, ... . = 2’,
= 3’,
‘d’ ‘d’
‘f(d)’
s
9.
9 ‘d’ by ‘f(d)’;
f(d) f o r s. by
d: (m+n)
‘+’, ‘(’,
man.
(5)
‘)’,
(5),
by
A
‘d’ ‘d’ a?’
for n, for x
x 3) +
+
(C)
x 2* x 1* x
+ 5).
+ + 5).
+ + + 5). + + + 5).
+ 5)’ :
8
WHAT IS A DEDUCTIVE THEORY?
on on
[I
by
394
by
on
+
((4 x 2.1) 5).
‘(A)’ 4. Theorems
As
A theorem
a,,
b proved.
(2.2, l)]:
provedproduction
..., a , 4 b .
(1)
a,, ...,a, without a proof, P,, ...,pk, [1950] p. 163) { a ; , ..., E ...,an), P,:a;, ...,ah, 4 b , ; { a : , ...,uL2}E ...,a,, b l } , P 2 : a ; , ..., aL2 4 b,; ...; {a:k),..., umk} (k) E ..., a,, b , , ..., b k pk: a?’, ..., 4 b .
proof
(1)
proved by Pl, ..., Pk.
by
importance
d
Theorem. %m, %n+(m+n)=m*n Proof. Pl:iJlm, %n+%m*n. P2 :( A ) + T?Z * n = m . n . (3.5): (B)-(m + n ) = m .n.
I
451
9 p.
5. Symbolic logic and deductive theories 2
a
SL,
SL
SL.
SL.
SL SL book by
11. OBJECTIVE UNIVERSE
1. Introduction
upon
pp.
XX,XXI).
do
go.
macroscopic
:
subjective no outside
10
I1
11
11
INTRODUCTION
by :
book
by by by no
book
14, ‘4’
by T ‘F+G’, G by Con.
by Ant ‘Ant+Con’.
by
F
A,
C , D , I , P,
by
A I, P, T.
(T)
A, C, D,
[11 2,3
12 2. Observers
c1.
2.
Z 1. 2 organs, bias.
class living humans adequately functioning sense communicate do honestly without ;/i observers. A not
do ‘X
Al.
by
as. A
X
do ‘ ~ H E Z ’ , ‘H
3. Sensations
II
31
13
SENSATIONS
by no by
3.
c2.
Z2.
‘ S is a simultaneous set of sensations of
‘(S,
observer H’. ‘ ( S , H ) E 3’
9
(Il2)t
(S, H ) by
9
3.
A2.
3
by
P1.
%29.
D1.
S 9 H for
class of sensations of observer H 02.
SoHfor
by
X3X3H.
‘X3X3H’
X
12
P12
X
8,
H(113).
a ‘SE3,’
‘S3H’. ‘S3H’, H X
(P2).
‘(112)’ denotes ‘Interpretation 12’ in Appendix A.
%”
14
OBJECTIVE UNIVERSE
P2.
[I1
3,4
(~X)XBH.+HE.@'. S
A
(D35),
9 P3.
S ~ G S.ZH-iG= A H. S
H,
G
H.
4. Subjective entities As
c3.
8.
13.
do by
diferent by looks jeels. '(R, S, H ) d' 'The tMio simultaneous sets of sensations R and S are attributed by the observer H to t h P same source; H R S '(R , S, H ) E &'
y',
( R , S, H ) by A3
P1
6.6.
A3. P1.
%32
D1.
R.6HS for
(R,S,H)EB.
All by ' ( R , S, H ) ' H,
P2.
P2 by
R8&3--+RZTHA S T H .
13,
11
41
15
SUBJECTIVE ENTITIES
‘R#HS’
by D 1
TfHS,
H (Tl).
RfHS, 13,
R S
R%”H SSH. ‘ R B H ’ by D 3.1 12. R$GS SjGR, SYl1T by by
S
T
H
H, S
T1. T1.
R$,S
v S#,R.
A
.S&,Tv
T$HS:
-+
G =H .
Proof.
As
R&,S-+R9G A S S G . RTG A SBG-+SBG.
The
on
on
on R$GS-+SZFG.
SfHT-tSSH. by
(3),(4),T(1.7,2.33):R$GS~Sf,fT-+S2G ASBH -+G=H.
S S G A S Z H - + G =H , A S$,T+G=
(5), (7),T1.2: R$,S
H
(5)
16
[I1
OBJECTIVE UNIVERSE
4
on. T1.2 P(2, 3.3)
T(2.14, 1.2)
R#,S AT$,S+G=H, SY,R AS$,T+G=H, S&,R A T&HS+ G = H . (8)-( lo), T1.7 :(6’) A ( 8 ) A ( 9 ) A (ll), T(2.37,l.l): RY,S A S&HT. v .R&,S A T Y H S . v . S f , R A S&HT. v .SY,R A T&,S:+G=H. T(2.42,1.2): T.
(9)
(10) (11) (12)
T1:
‘T’ R&,S v S f , R .
A
yH
.S&HTV T$HS:+G= H . H
YH (D32)
yH
(D25),
H , TH(O3.2). (D27),
(D30).
equivalence subclasses, no A by
fH, P3.
S
STH+S$HS.
f Hby (D31).
P4.
RjHS
RyHS
A
R$,T+S&HT,
RJHT, SYHT. R, T
H
R, S S, T
11 41
17
SUBJECTIVE ENTITIES
‘S$,R’
‘RyHS’. by (T2).
$H
XH
T2. Proof.
yH
P4,
TH.
equivalence
P3, P4, T7.1.
H
S
by : 02.
YH(S) for
X3XY1A
X
X3XfJ
S(113).
yH(S) fH(S)
H A2).
H,
by
A D21,02, X$,S’. ‘REA’
XH(S), A XH(S). P12, ‘ A= $H(S)’ X, A ‘RfHS’ ‘ A = $H(S)’ ‘R,
A’,
‘R H
A R
S.
A T3.22,
SZH,
X
P5.
fI2(S),
SETH + X )
x Ey H( S ) . D10):
P5
T3. Proof.
S 9 ” H +( 3 !
x)x 5 2
H
(S).
P5,T(3.22,1.2).
X
STH,
X - $H(S).
subjective entity associated with S , by :
03.
3 ( H ) for ( 7 X ) X = Y f 1 ( S ) , ‘( 7 X ) X - y H ( S ) ’
the
X
fH(S)’
18
OBJECTIVE UNIVERSE
[lI
0(17, D l l ) .
4
0 3
by : S9"Hr\A=S"(H)*S9Z"Hr\A~~$,(S). T3, T (4.3,1.2):S9"H +. A = (7 X ) X 3 $H (S)*A
T4. Pro0
+ . A = S"(H)*A
03 (l),T2.32: T.
T4 'A=S"(H)', 'A H class of subjective entities H is
S(H) for
04.
(s) (1)
H, S'.
S
Y).YBHAX=P(H).
~ 3 ( 3
X
9(H)
$H
=f H ( S ) .
Y.
H
04,
D5.
9 for x 3 ( 3 U ) X ~ 2 ( U ) , 2
X
U,
X E2(U). subjective,
H,
A
(T5).
H
T5. Proof.
A E 2 (G)A A E 2 (H)+G = H . 0 4 , T4,D2, D21, T3.(15,4):Ant + ( 3 X , Y ) X ~ G YA z H A(VU): U E A ~ + U $ ~ X . A U .E A - U ~ H Y T1.14 X , Y).X%"G. A .(V U ) . U y G X * U y H Y P37 P4+(3 x , Y ) . x $ ~ Ax .x$,x4xyHy T2.7 +(3 X , Y ) . X Y G XA X f , Y T1 +(3X, Y)G=H T3.13 +Con. by
XII
by j H .
H
G
II
19
OBJECTIVITY
S”(H).
by
by H.
5. Objectivity by do by
x. 14. ‘ ( A , G , B, H ) E X’
‘The two diflerent observers G, H establish
n correspondence between the subjecfive entity A perceived by G and the sub-
jective entity B perceived by H,
A
B
H G
A B.
A
B ‘X
As
by
a
‘Y
P1.
by
:
%z,Y.
:
D1.
for
A,%,B
( A , G, ByH ) E%.
‘A,%,B’
P2.
A,%,B+A A,%,B,
E 9 ( G )A
B E 2 ( H ) A G# H,
B
A
H,
H. A, B
H
T1. Proof.
A G X H B + A# B. P2 :Ant-+A E 2 ( G ) A B E 2 ( H ) A
#H.
(Tl).
H
[I1
OBJECTIVE UNIVERSE
20
T5.3 :A E 9 A B E 9 ( H ) A A = B+A E 22 A AE2 2(H) T4.5 -+G=H. (2), T2.32: A E 2 A B E 2 ( H ) A G # H + A # B. (11, (3): T.
5
(2) (3)
:
A,X,BA AHS,C-+F=H. P 2 : Ant+ A E $ ( F ) A A E 2 ( H ) T4.5 +Con.
T2. Proof.
‘A,.X,B’
A,
B, H P3.
AGXHB+BHXGA.
F H A
A
B F F# H.
B C
F P4.
A,X,B
A
B,.Y,C-+A,S,C
v A = C.
A=
F= H , by T2 objective by
P3.
H F= H, A diflerent
;
by
A E B for
D2. ‘A
B’
A = B, A B ‘f ’ (T3)
T3. Proof.
A = B v (3 U , V)A,X,B.
U, V
(T4),
~fA . 02:AZA++A=Av(3 U ,V)A,X,A T1,TI.II + + A = A . (l),T ( 5 . 1 , l . l ) : T.
T4.2.
11
51
21
OBJECTIVITY
A&BAA~C+B~C. 0 2 : A n t o : A = B v (3 U , V)A,Z,B.
Proof.
A
.A=
c V (3 U ' , T/')A,,x,,C
T(~.~~,~.~),P~,T~.~~+B=CV(~U,V)C,.X,B. V (3
(3 U , V , W).A,.X,B
A
u, V , W).A&f,B A A , s , v C . A,%&. +B=CV(~ V , W)B,X,C.
(1)
(2)
T. 5
9.
Proof. by :
A 03.
Z ( A ) for
X3.XE?z?AX~AA.
As
X
A
X(A). '4
E 9+(3
! X ) X E %(A).
Proof. objective entity (object) A,
04.
for
d A
A'
'A
(AE~?),
A
AE~+AEA. Proof.
T6,
Ant +(V X ) .
x = A-x
EZ ( A ) -(VY).YEX-YE~AY&A
+.AE~AA~A+AEX +.A
E2
+A
E
= AA A € 9 - + A +AEA
A
x=A
'AEA'. A d .
22
[I1 5
UNIVERSE
T3. (19,15)+ : AE 2 A (3 X ) X =A.-+AE A T1.10 +.AE2+AEA T2. (32,18)-+Con.
A, B
A, B
[(I U , V)A,.X,B], T8. Proof.
(T8)
(3 u, V)A,X,B.+A=B. T2.15,02: A n t + A z B . (l), T2.12: Ant A X E A + A s B A X & A K T5 -+X=B. K T2.32: Ant-+.X 5! A + X = B. K P 3 , (3): Ant-+ .X $! B-t X = A . (3), (4) :Ant +(V X ) .X A o X f B. D21,03 :Ant A X = X ( A ) - t ( V Y ) .Y E X o Y E 2 A Y 5 A ( 5 ) , T1.14 ++YES/\Y 5 - B -+ X = X (B). 0 3 T2.32, P 3 , P3, P5: Ant +(VX).X = . X ( A ) t t X E X ( B ) T4.2, 0 4 -+Con.
T8
T9. Proof.
A=B A A , B E
2 A A Z B + ( ~U , v)A,X,B.
T6, T4.3,04: A E 2?+(VXX).X=A++X~X((A). B E 2?+(VX).X =B-X = X ( B ) . Ant 4(V X ) . X = 3'"(A)++X X ( B ) (1)-(3): D21, T1.14 +(V Y).YEX(A)4+YE%(B) P4 -+ . A E X ( A )+A E X ( B ) D3,2 3 . A E 2 A A = A +Con T(1.10,2.32) -+Con. objective universe
D5.
O ( H ) for
O(H) Y
H
U Y
Z
observer H ; ~ 3 ( Y).X 3
=
H :
PA
U , Z) Y,,.x"z.
X, X(X= p), Z
X
U
H T8,
U
2= f = X ) .
II
51
23 (macroscopic) objective universe
U(D6). 06.
8 for
X~(~U)XEO(U). by 0 5
06
A by G[.~ELO(G)], by @(HI,...,H,) (D18) @(IT,),..., O(H,). b ( H , , ..., H,)
07.
for
I1
on
A H. 07, H , ,..., H,
by
(A€&),
...nC"(H,).
07,
upon by
on
111. P A R T I C L E S
1. Parts
no (116)
(D19),
(DI15.4), @(T6.24),
c ‘AEB’,
~(D19).
A, B
B = ( A , C,, ..., C,], A
A, D19
B. A
‘AzB’ ‘ Ac B’ (D20)
B’, B, B’,
‘A
B constituent
A proper subset
c,
no ‘ C consists
{A,B}’
A
:
B’
‘C=AuB’. C,
A, B E
C,
A
A, B A
c
B,
B B on
24
B
C, C,
A A
not
111
11
25
PARTS
-
good ' A c B' A B. (D19-2t; DI15.5):
01. 02. 03. As
A c , B ,for A G B A A , B E O ( H ) . A c H B for A c B A A, B EO(H). A-,B f o r A-BAA,BELO(H). T6.(3,4), D(20, 21),
-
T1. T2. T3. T4. T5. T6.
Ac,B-Ac,Bv A-,B. AcHB-AGHB A B G ~ A , A = B-A E B A B c H A . WACHA. Ac,B+-Bc,A. AcHBABcfrC+AcHC. fI
T4, 5 , 6 (D30). H, @ ( H ) .
(D26),
cIr
(D28),
cIr
H
H A
B,
A
A
U B,
B, c
A
:
( 3 U ) A c , B . A A , B E O ( H )+ A c H B . 02:Ant-+AcBr\A,B~O(lf) 02 +Con.
T7. Proof.
In 3
particle
B, U
A
B. In
26
[m 2,3
PARTICLES
2. Connection and separation
AxHB for N A GB , A - B E HA A ( 3 x ) xC HA A x C HB. ‘A=,B’ B(
N
X
‘A B A = H B ) ,A A
connected H’, B, B B.
-
02.
A x H B for - A c , B A B s H AA
‘AxHB’ B, A
‘A
A
B B, B B.
A A,
X ) X C H A A X cHB.
separate
A
H’, A,
X
-Dl,2;T6.3:AxHAvAxHA-t-A-A. AxHAA
Proof.
AxHA.
A x HB++BxHA. Proof. .~~cHB*BxI~A. 02.
Proof.
A, B
B c H A , ( 4 )A x H B , ( 5 ) A x H B . Proof. H,
A - ,,B v A c f I Bv B c H Av A x H Bv A x H B . ‘ A s f I B ’ ,‘Bs.A’, ‘ ( 3 X ) X c H AX ~c H B ’
F, G,
3. Particles by by H(D1).
no P(H) f o r
X3XEo(H)A
by do
X
H
y)Yc~x.
P, Q, R , S , T, U .
111 31
27
PARTICLES
by
..,H,
(018)
B ( H , ) , ...,P(H,).
...n Y ( H , ) .
Y ( H , , ..., H,) f o r
02.
02.1,
no (Tl). T1. Proof.
PEy(H)+ 0 1 :Ant + -(I T3.17 T2.15 T(3.17,2.38) T2.15, 0 2 . 1
D1 T1, (3) (4) T2.4
X)XXHP. Y )Y c H P +(v Y)" YCHP +(v Y)." Y c H P v Y c , { x +"(I Y ) . Y c H P / Y\ c H X -+Con.
B
A B
(2)
P, Q (T2). T2. Proof.
P,QEP(H)+PE~QVP~C~Q. Dl:Ant+P,QE U ( H ) T2.4 + P s H Q v P C HQ v QcHPv P x H Q v P ~ C H Q . D l , T l : A n t + - P c , Q r \ - Q ~ , P A- P x H Q . (2),T2.11:T.
not
( z100 'H',
H, macroscopic
microscopic
(1) (2)
IV. EVENTS
1. Introduction
do changes point-events,
sudden
cles
If
2. Appearance and disappearance events
inC 5
of 28
C6
1v 21
29
cs.
d.
15. ‘ ( a , P , H ) E ~ ’ ‘the observer H perceives the event a of the sudden appearance of particle P ’ . H, by
.d
by
%&.
PI.
:
D1.
ad,P do
for
(a,P,H)E.d.
‘a
P’
H,
a d P for ( 3 a Q, R H, a
P(adHP), a
Q ,R ‘a.d,P’
.Y(H):
dr,
P2.
(3
R,
P,Q P,Q
R
P,Q
three one R. C-T
p. p. by
p.
a at most one
X,
30
[IV
EVENTS
P3.
u.d,P
A
2
ala/,Q+P=Q
P
at most one
adoZ,P A b d H P + a = b.
P4.
P by G,H
P4,
P3,
is do
same
All
9.
C6.
16. ‘ ( a ,P, H ) E 3’ ‘the observer Hperceives the event a of the sudden disappearance of particle P’. H, P
G9.
A6. P5. 02. P6. P7. P8.
93p. P g f 1 a for. < U , P , H ) E ~ x)P 9 H x . +P E 9( H ) . P a H ar \ Q g H a + P = Q . P g G a A P g H b+a = b.
a
P,
Q,
19
a T1. Proof.
-
P
.ad,PA QgHa. P9: a d $ A QBHa+a # a. (l),T2.20:a=a+T. ( 2 ) ,T5.1:T.
Q(T1).
a#b.
I V 2,3j
31
OF EVENTS
by 03.
H
x3(3 Y ) . x d H Yv Y ~ ’ , x ,
for
x
b(H)
HI, ..., H,
Y.
& ( H I ,..., H,) for x3(3 Y).Y E ~ ( H ,..., , H,,) A ( X d , , Y A .. . A XdI,,,Y. V .Yg,,X A ... A Y ~ H ~ X ) ,
04.
b(H,, ...,H,)
x . ., H,,,
Y a
P
a
P
T2. Proof.
Y
x
Y H,, ...,H,,. G, a H,
to
H, H (T2).
a d G P A a E &(G, H ) + a d H P A P E y(G,H ) . 04:Ant + + a d I , P r \ ( 3 X ) X E 9 ( G , H ) A (a.d,X A a d 9 , X . v . X B G a A X g H a ) T I , T2.11++(3 X ) X p ( G , H ) A a.d,P A a.d,X A a.d,X P3, +Con. by
T3. Proof.
P%,a A a E b ( G , H)+Pg?,a 0 4 , T1,P7, T2.
AP
E ~ ( GH ,) .
3. Coincidence of events coincidence before.
c7.
K.
Z7. ‘ ( a , b, H ) E % ’
‘the observer Hperceives the events a,b to occur at the same place and the same time’, ‘H a coincides b’.
32
3 17
no
on V
A7.
gq,
P1.
%,%.
D1.
a=,b
for
(a,b,H)Eq.
by H,
‘a=-,b’ by :
P2.
a=-,b+a
E b(H).
xf,( T1). G,H(D2.4),
a,b
P
H
a,b(a=,b),
a
b
H
a,b(a=-,b),
G,H (P3). P3.
a--,b
P4.
a E b(H)-+a--,a.
A
a , D E &(G, H)+a=-,b.
b(H).
x-,
(D31) :
P5.
a=,b
T1.
XH
Proof.
A
a v H c + bx,c.
is P4, P5, T7.1.
&(H).
IV
3,4]
33
TIME ORDER
a x Hb +a , b E €(El). TI: a v H b - + b + p P2 +b E & ( H ) . P2, (1) :T.
T2. Proof.
4. Time order
by
particles
inside
on [1958] p.
by light
[1958] $21) 2.
‘a precedes 6’ Z , , ...,Z,(n> t ) Z, Z, b. b.
Z,,
D1.
a
(Dt)
Z,, ...,
Z , , . . .,Z,
U
1 A (3 xl, y l , Z,, . ..,x, Y,, Z,).
a x H x l A X , ” Q I ~AZ ,Z , ~ ~r\y,=r,x, J I ~ A x,dHZ, A Z,,9Hy,, A y,xHb.
‘a<,b’ DV3.2
‘a precedes b,
Proof.
a<,,b+a, b E b ( H ) . D1 :Ant+(] x)c?XHx A T3.2 +Con.
A b
T2. Proof.
a d H PA P B H b + a i , b . 0 2 . 3 :Ant+a, b E & ( H )
...
‘a
H’. by H(T1).
T1.
A
y)y=rHb D1, P,
bejore b’
34
[IV
+aXHa
D1
A a d H P APBHb A
b--,b
-+Con.
<,
5
02.
for
a
UXHb
v a<,{b.
ax&
02, xH
<, a<,b
A
b<&. v . a i , b
proof.
A
b<,c:+a
b
-*(3 x).aX:,x
T3.1 D1
A
...
...
-+a<,c.
D1,
a<,b
A
b<,c
++(IU ,V , X, ) J ) . U X H U A VXHb A b X H X A J ’ X H C U ,V , X, y ) . a x H u A V X ~ , X A
A
...
y x H cA ,..
D1 +a<,c. aXHb V
D2:U<,b
a
a
A b
A b
A b
(5),
T. b<,c,
T3
b<,c,
a<&. xH,
<, P1.
D1,
xfi
(PI). .uxHb A a
(D26).
4
IV
41
<, T5. Proof.
(T5).
a
(T4),
+f,
(T5), b(H);
<. on
A
03.
‘afHb’
35
TIME ORDER
a T H b ,for
ax,b v a<,b
‘a,b are time-like f o r H’.
v b+f,li.
(T3),
V.
SIGNALS
1. Change, coincidence and dissociation 2 P I ,. . . , P ~ O H Q ...,~Q,n AptttgHxm y , d H Q 1A ... A Y , ~ ~ ~ Q , A HX ., .V. ~ H ~ m ~ ~ l y l ~ l l . . . ~ H y n ,
( ~ X l i . . . , X m , Y l r . . . r Y n ) P l ~ H x IA . . .
A
‘P,, ...,P m O HQ , , ...,Q,’
x,,..., x, P , , ..., P,,
y , , ...,y ,
‘P,, ..., PrnOHQ’ ‘P dissociates
Q,, ..., Q,, ‘P,, . .., P, go into Q , , ...,Q,, H’. ‘POHQ’ ‘Pchanges Q, H’; ‘ P 1 ,.. .,Pm coincide Q, H ’ ; ‘ P O H Q l .., ., Q,’ . .., Q,, H’.
OH. 2. PI, .. .)PmOH
...>Qn+P,,
.. . > P m ,
.. .)
E~(H).
Prwf.
i,, ...,i, 1, ..., j , , ...,j , 1, ... , n , P , , . . ., P , O H Q 1 , . . ., Qn-Pi,, . . ., P i m O HQ j l ,.. ., Qj,. Proof. D1,
Proof.
PI, . . . , P p O ~ Q .i ., . , Q q ~ R 1..., , R r@IfS1,...,Ss A .{ P,, . . ., P,} n{ R , , ... ,R,} # 0 v ..., * P i , .. ., Pp’ R 1 , . . ‘ 9 RrOH Q i , . . ., Q q , S , , .. S,. D1,
P,, .. .,PmOH .. .,
A
. ..,S,} # 0 :
i E { 1, .. ., m ] ~j E { 1, . . ., n}+Pi@, Qj.
36
” 11
37
[email protected] A R O H Q-P (3), T.
= Q.
PI, ...,P,,
Pl@HP;?, P2@& ,...,Pn-l@HPnr (T8).
a=Hcr\cdol,Pi A P , @ H P ~ A... r \ P n - l @ ~ P n A P,,gHd A d=r,b-+a<,b. D1, DIV4.1.
T8.
Proof.
P@HQ, R, Q=
Y,P Q
II
R c Y,
Qc Y
Y
:
R
,
P O HQ, R A (V Y).Pc,Y-+Q c Y A R c H Y:-tQ = R.
P1.
2. World lines
P
P P
([email protected]), Q
Q
P
continuation P ; (2) (POHQ), Q
X, P Y Q by
P
P
P
continuation Q’ D1
‘P
0 2
on D1. ‘PYQ’
P y Q for ( 3 X ) P , Q c HX. A P@, Q : v :( 3 X ) P c H X .A P o HQ A (V Y).Pc,Y -+Q c f l Y . N
‘ P continues
Q,
H’
‘Q
continuation
P,
H’.
v 21
39
WORLD LINES
T1. Proof.
(PyQ)-+p,Q E p(H). D1, T1.l.
T2. Proof.
( P y Q ) + (QyP). D1: Ant+P@, Q T1.6 -+-Q@IIP D1 + Con.
T3. Proof.
P y Q AP~R.+Q=R. D1,T2.42, TI .3: Ant+ :P@H Q A P v @ .R~. V . P * @ * H AP@,R. Q V ,P@H Q A P.@*NR. V Q, R A(V Y).Pc,Y-+Qc,YARc,Y 2-1.7, P1.l +Con.
N
P 1 y P 2 ,...,P,,- ly
T1.8, P,, ...,P,
PA T4). T4.
Proof.
aXHCACcdHP1 A P l y P 2 A . . . A P n - l T P n r\P,B,d~d=,b+a<~b. D1, D1. (2-4):Ant+axHc A c.0lHP, /\P~@HP~~...~P,,-~@HP,, A P,gHd A dXHb +Con. P
P 02.
W((P), for
~ 3 ( 3Y):x=:,Y
A
P.
. y d H P vP9,y
v (3 n ) . n > O ~ ( 3 Z,, ..., Z , ) ( P y Z , A Z ~ A ... ~ Z ~ A - lyz,,.v .z,y;;*z, - 1 A .. . A I ';tP) A . y d ~ zv, ,z,,g,y.
z,
W(P),
z
world line 02,
... A p,-
P.
T5. Proof.
PlYP2 A 02.
T6. Proof.
a EW(P)H+a E €(If) A P E B ( H ) .
As
lyPn.+W(P1),, -W(p,),.
0 2 , TIV3.2,PIV2.(2,6), TI.
a,b on
P
40
293
SIGNALS
a b
a,
b,
a
b,
aF,b
T7. Proof.
a , b EW(P)H+Uc?Hb.
T8. Proof.
a--,b 02,
Proof.
P= v P y Q v Q Y P . + W ( P ) , =w(Q)H. 02.
bEW(P)H+a € ~ ( P ) H .
A
3. Signal relation
So
by by
no
C8.
Y. E, #j’
1 8 . ‘(a,b, H ) E Y’
a
p’.
on
a, b ‘a is the start of a signal that ends at
‘ ( a ,b, H ) E Y ’
b, for H ’ . by
0
by book.
C8 by
eight
C8. .Y
A8.
by
69, by
%,9.
C1
v 31
41
SIGNAL RELATION
:
D1.
U.YHb f o r Y,,
P2.
(a,b,H)E.v. B(H),
a-Y,b+a, b e b ( H ) , by
a,b a.Y,b+
P3.
-UyHb.
9,. a.Y,b+
P4.
N
b.Y,a,
9,:
P5
a .y,b A bcyHC+ a Y
P5.
H C .
Y, T1. Proof.
P4
P5:
-a.YHa.
P4, T7.2.
9,
8,.
by Y,, by ,Y,(P6).
is
P6.
a x H bA b.Y,c. v .a.y,b
A
b=r,c:-+aY,c. by <,i
02.
by 9,. A
uaHb U 0 A X I , . .,X,): a
‘aB’,b’
‘a
before h,
H’. .%!,
T2. Proof.
UgHb-tU, b b ( H ) . 0 2 , TIV4.1,P2.
T3. Prooj.
aap,a. 0 2 , TIV4.4, T1.
T4. Proof.
aaHb+ bBHa. 0 2 , TIV4.5,P4.
-
-
S,:
42
Proof.
fl.!%HbA b g H C - + d H c . 02,
Proof.
a X H bA b g H C . v . u a H bA b=r,,c:-+ad,c. 02,
4. First signals
tfirst signal’
by
([1958] p.
:
UPHb f o r
UYHb
(v A
+ ( X a H b + X < H a . A .bgHX+a
‘u the start of a j r s t signal that ends at b, f o r H’. ‘aFHb’ 1. ‘aF,b’ a.Y,b; a,x xg’,b x<,a, bgf1xa
‘aPHb’
aFHb +a , b E b(H). Proof.
:: N
a-F,a.
Proof.
Fig. 1.
signals
v 41
43
F1RST SIGNALS
-
acFHb+ b F H a .
Proof.
Fll 2FH
ax-,b D1,
Proof.
T5.
A
b-FHc.v .a.FHb A b=Hc:+a.FHc.
a p H bA C F H d-+ :a v H cA b r H d + b x H d . A . u F H c A bxHd-a=CHc. A a F H b + c x H a A ag',b -tC%Hb. :b
Proof.
As
a F H b A C 9 ~ Ad
A d
--f
PI.
b,
a F H bA c F H d + : a i H cb ~F H d - +-d<,b. A . b i w d A a Y H c + -c
aFHb d,
cFHd,
a a.
c,
d
44 T6.
Proof.
SIGNALS
a F H b A c F H d + : a < H C h bCYHd-+b<,d. A .biHd~aYHc-+a<,c. P1V4.1:aFHb~ c F - , d ~ a < , c ~ b 9 - , d -+a.F-,b A C 9 ~ Ad - C I V H C A b9,d A a<& T5,PI + - b = ~ [ ~ -dd ~< , b ~ b F , d DIV4.3 -+ b
P2.
a 3 - , b A a . ~ , c A c ~ , d ~bF&-+b<-,d.
after P3.
(P3
a F H bA bFt,C AaF,c-+a<,c. A ( ~ / X ) . ~ ~ , , ~ ~ , , C - + - . X . ~ ~v Ib) SVHXX~. ~ ~
t
Fig. 2. trip a
signal
a,
P3 no
a
.Y
6.
[1958] 5 22). a c,
b, a
c,
a
b, a
45
FIRST SIGNALS
b, a
VI.
CLOCKS A N D TIME INTERVALS
1. Introduction
A clock output particle; a , b by
a, b
on by
no
a
b
on by
on
g,,),
by by 49
50
1
CLOCKS AND TIME INTERVALS
do good
(2)
by by
by by no by upon
by by no
up
51
CLOCKS
VI
no
2).
by up 1963
by
on
A
by 3
6.
2. Clocks
by by
by by
A
...,u, P a,, . ..,an
P, A(Pc,A) P(a,, . . . , u , E W ( P ) ~ )
< H ( a , < H . . .
by P
on {a,, ...,a,) X
A.
no
P
52
[VI
CLOCKS AND TIME INTERVALS
A
a, on
a,
2,3
A,
X by
P
( a l , ..., a , 929P , A)H for a 1, .. ., anE w ( P ) H A a1 < H . ..
D1.
*
x,
(andHXn
XngHa~)
€ ~ ( P ) AH x)x#P A X C H AA ( U d ~VxX ~ H U ) A a , < , u < H a , - + u E { a , , ..., a,,}. A (vU).U
‘(a,, ...,a,,
P , A)H’
...,a, is a sequence of events at P associated
P(PcHA)
u on P ,
with A’. A xl, ..., xk,, P x~(x,<~u<~x~).
k, A,
x1
u
P u,
on
(Yl,...,Yn+i
u
P , A ) H ,Y ~ < H V < H Y ~ , on P
Y,,
u X, Y A,
on
w.
( X = Y v X;;, Y v Y;;,X).
X, 02.
W(U<~~W),
y l , ...,y,+
n
(AVGP), for P c , A A ( V U , ~ ) . U E W ( P ) , Ak E 9 . . - , x k +,)(x1, -..)x k + l s p P , A)H A X 1 < H u < H +(I A ( v V , W).V, ...,4’n+l
A
‘(AVL‘P),’
~aplA)HAyl~HU
(v U l , . . ., urn,211, .. .)On, x,Y ) . ( U , , . ..)
A (U1,
..., U,
x2:
W E ~ ( P ) H A U < H W ~ ( ~ ~ ) ~ E ~ A ( ( ~ ~ ~ ,
Y , A),-+X=
V
x,A)H
xv;;*Y V Y T X .
‘ A is a clock with output particle P ’ . 9
3. Relative instability clocks A , B are compared, starting with event a
53
31 of A
b of
of A
of
n
A', Y
A,
...,
Z ( X ,Y @ . H Z ) , ul,
...,u,+
a,u,,
A
u,
u,,
...,u,,,+
B, b u,~,+( U , , , < ~ ~ V , < ~ U , , , + A,
h,vl, ...,u,+ a b
u,
b
a
u,
urn,
by
A.
B.
A
( A , a , m V B , b, n)H f o r m ,n €9 A ( 3 X , Y , Z > ( A % / X ) , A (B%?fY)HA X , Z A (3 u 1, . . urn + 1 , ut > . . . > u,+ 1) (0, u 1, .. .)urn + 1 .%(/Z , A)H A (b, u1 ...>un+ 1 .4"r(/ z, B)H A a BHb< H u 1 A um
DI.
.$
9
fill
m B,
k k
A
..., nk
...,&(A=ni/m) lo6. by k - 1
k A :
km
+ ( k - 1)j + 1 = k (m + j ) -j + 1 .
Fig. 3.
54
[VI
CLOCKS AND TIME INTERVALS
...,A,
sample
3
:
k
1
J'
S
X P(x)
F,
p.
<X <x
+
by
=
X.
probability density ,function
X
1,
: fm
=1
(- co,co).
f(x) p. 170):
(f(X)>=
-x
true (population) mean (5),
f ( x >p > .( dx
(5)
9
expectation
(true) moments
P- = (fi>>
~m
=
( ( A - I*>">
9
(x
>
r 7 = JP-2,
p
mean
r7
standard deviation. i,
...,f,
If p
p. 208),
6.
P For z,%oP(n) =1', and Eq. ( 5 )
=
(- +x'>,
Eq. (3) ' P { X = x , } = P (n)', Eq. (4) < f ( X ) >=Z',"=O f ( x n ) P (n).
(8)
VI
31
55
RELATIVE INSTABILITY
pp. 382, 387,
pp. 219,
< p
xf
xz
p
k -1
pp. ‘ X < C< Y ’ ,
X,Y
C
‘P{X< C< Y } (X,Y )
1-p, ( X , Y ) 1- p ;
C
C
(X,Y ) C
( X , Y).
PDF p
k
c
central limit theorem p
p. c
PDF, k+ m
P { p - g p o / J k
+ g p a / J k } = 1- p ,
(1
p
gp
p.
P{lfi - pI < g,a}
(- +x2) dx = 1 - p .
= - QP
p.
p4 k+
P {c’ - g p [(p4 - c4)/k]* < S’ < cz + g p [(p4 - c4)/k]+} k, f
=
1- p .
S2
k
pp. 366, p.
P{f< p
PDF.
pay
56
CLOCKS AND
3
INTERVALS
p = (fi)
= (r
'The A , B have the frequency ratio M j N and relative instability o [ ( A , B %?eJM / N , o)~]' B as u A m, u B n ...,A D 1, k
fk N/M, 1- p .
S2
g,[(S,
- S4)/(k-
02,
E,
k
q,r,s
2 s, j2 q
j
m ;
ni
..., ui(,,,+ j ) - j +
,'A Y
A,B;
...&A= ( A ,u ( t n + j ) + I
~ z , ~ z ) H. . ,. ) ( A ,u(k- 1 ) ( n l + j : +
0 1, M/N,
p
fk (k-
(9
02,
l'k,nk)ii;
S k g,[(S4 1-p
c.
- S4)/
41
57
STANDARD CLOCKS
4. Standard clocks
by
by by on by
by
by
by
1 upon by
C J ~ .
3,
C J ~
10-l2.
D1.
( A Y%?P), f o r ( A V L P ) , A r ) A 2 A ( ~ x Y, ) . x # Y A X , Y E { X ,,..., X r } C J ) ( XY, c ) H A CJ I ‘A
‘(AY’gP),’
02.
Y%(A),{ f o r
03.
a,,
..., a,,.Y%,A
standard clork (SC) reliability index, ‘ # ( X , , ..., X,)’
P’.
x)(ALYvX)H. for
(~X)(A.Y%X)HAU ...,~u, , E W ( X ) H .
58
CLOCKS
4,5
TIME
A , , ...,A ,
&.Y%?(A,,..., A,)H for y g ( A l ) H A ... A ~ y g ( A n ) H A # ( A , , ..., A , ) A ( V X , Y ) . X # Y A X , Y E{ A l , ...,A , } -+(30)(x, Y%‘~>I/I,o)~Ao<~~.
04.
‘A1, ...,A ,
‘€Y%?(A,,...,A,)w’
equivalent standard clocks’.
1
:10-’4:10-40.
A on p.
c?,u%?(A, B)HA 8 u % ? ( B ,C ) H - + b . y % ( Ac)fi. ,
PI.
5. Time metric
A
%?
metric space z(a, b)
%?
:
T(a, b)>0, z ( a , b)=O++a=b, r ( a , b)=T(b,a), T ( a , < z(a, b ) T (b,c).
+
t
metric in the space %?. on 8,
by
D1.
S C , A , by
A a, b by A
a, b A,
on a
b.
s(A;a,b;n), for n d A a , b . S P q , A A (3 X , ~ 1 . ..* , un+ 2) (U I ...)U n + 2 %’/ X , A),, 9
59
51 A :u1
hUn+1
v -u1 d < H u 2
A
<
un+ 1 < H a
2.
'The time interval between a,b by clock A is n'.
't(A;a,b;n),'
T1 T2 T1.
a, bYqHA+(3 n)'C(A;a , b ; n ) H .
Proof.
04.(3,l):Ant +(3 X)(A%/ X ) , 02.2+(3 n)n €Y A ( 3 U 1 , ..., U n + 2 ) ( U 1 , ..., U,+2 %a X , A ) H A : 1
T2.
(3 n) 'C ( A;a , b ;l l ) ~ . 01,02.(1.2),TV2.9,TIV(4.5,4.3): 7 ( A ;a, b ;m)H h 'C ( A ;a, b ;n ) H +(j X,u1,* * . , u m + 2 ,~ 1 ...,vn+2)* , x,A ) H A (u1, ...)un+ 2 ( u ] ,. ..) urn+ 2
Proof.
A
(u 1 < H a
*H...
A 01 < H a < H V 2
*
x,A ) H
1
2 2'
v .u, < H ~ < H ~ ~ < , . . . < H ~ ~ + ~ < H ~ < H U , + Z A 01 < H b
*
1
2)
TV2.7,TIV4.3+(3X,u1, ...,u , + ~ , v ] ..., , u,+~). (u1, . . . , u m + 2
~ ~ X , A ) H A ( U ...,un+2 l ,
A(u1
X,A)H
vv1
+ 1
A
2)
...)on+ 2 )
+m=n.
(1)
(l), D9 :T.
T3.
a, b 9 F H A + ( 3! n )
Proof.
T1,2;D10.
02.
z,(a, b ) H
zA(a,b)H
for
(7
a , b ; n)H.
n ) 7 ( A;a, b ;n ) ~ .
b by clock A .
the
T4.
a , b%%HA+.'C,(U,
Proof.
T3,T4.3,02.
b)~=n++'C(A;a,b; n)~.
60
CLOCKS AND TlME INTERVALS
5
[VI
sA(a,b)If a , bY%?HA +t~(a, b), 2 0.
T5. Proof.
Tl:Ant+(3n)z(A;u,b;n), T4, D1 +(I n ) sA( a , b)If= n A II 2 0 T5.6 +Con. a
T6 T6. Proof.
b
zA(a,b),
a , brY’VI,Ar\a=,b-,TA(a, b),,=O. T1,4:Ant+(I n ) z A ( U , b),=n AUXHb -+(II I > T , ( ~ , b)H= n A n 1 = 1 T5.6 +Con.
+
z,(a, b)H= 0,
T7
a, b
A.
n
no
A
T7.
b a
6.
a, ~.Y%?~*AA T A ( u , b)l,=O
+ ( ~ X , ~ , U ) ( ~ , U , ~ P ~ ~ / X , b
Proof.
T4,
a,b T7 ‘=’
A
by
‘.-II / ’. :
T8.
a, b.4PFHA- + z A ( ab)H , = z,(b,
Proof.
D 1 :T ( A;a , b ; n),++z(A:b, a ;I I ) ~ .
(1) =
T1.14, T4: Ant+(V r I ) . ? ~ ( ab)tI , = II++TA(b, T5.4 +Con.
z,(a,b), T9.
a , b, c Y V H A A (a
Proof.
T l , D l : A n t + ( 3 X , u , , ..., U , , , + ~ , U ~..., , u,+~) (U1, . . ., u,,+ 2 %r/x,A)t, A (01 .. ., U,+,,sr/x, A)H
-fz,4(a, C)H=TA(a,
b ) H +T.4(b,
c<,,b<,,u) ‘)If‘
I
VI
5,6]
COMPARISON
61
TIME INTERVALS
6. Comparison of time intervals
zA(a,b)H
As
by SC's (~,(a,b)~)
:
t E
k
6
0 r,
1,
r
X , , ..., X,,
62
CLOCKS AND TIME INTERVALS
[VI
6
b)H= n,, . .., zxk(a,b)H= nk,
ini’
lz= k 1
t,
E,
16,
in probability
ti
p.
t
n,
pp.
~ ) ( V ES).F, , 6 E LZ A& A < 1+(3 r ) r €9A (V k). k d A k 2 r +x1, ...)x,,n,, ...)n k ) . B Y V ( X , , ... X& A zx,( a , b)Ii= n I A . . . A z x k( a , b)H= nk-+ P{ I? -i t 1 > E } I 6. (z,(a, b),) z,(a, b)H. (T,(u, b),)
for
(7
)
02.
V{t,4(a, b)H}
for
b)H-
variance
V{z,(a,b),}
P1. -+
b)H))z).
t,(a,b),,.
n)z,(a, b ) H = f l (3! t ) (T, (a, b ) ~=)t A
1
(Q,~)H 5 tit.
K oT by ~T=~(fi)=~(n~/m)=~(ni)/m=((ni>2/~)[~(ni)/(~i>~] 2 ic ~(n,) ic2
do bound on
by
P1
by p.
Ti.
ZA ( a , b)H = f? A (t,
Proof.
P1,
( a , b), ) = t
--f
P { In - t I < ic (tip)’ } 2 1- p . ic(f/p)+,
In-il
n.
T1
‘p-’’
by
p
61
COMPARISON
63
TIME INTERVALS
p - * = 10,
p=
‘g,,’.
9,%3.
T2.
tA(a, b)H= I I
A
z E(c, d), = n2 A 8 y g (A , B)H
A (T,(a,b)H)=ti A ( T ~ ( c , d ) ~ ) = t 2 +P{ K . 1 - n,)-(t, - t 2 ) l < X[(tl + t2)/Pl’>2 1 -P.
Proof. P1,
+t2)
by
1
K<<
p (l(111 T3.
TA(U,
+ 772) -
+ t2)l < K [ ( t l + t2)/p]+}2 1 - p .
b ) ~ = n ,A T s ( C , d ) ~ = n zA b . y g ( A , B)H
A (TA ( a , 6 ) H ) = t 1 A ( T g ( C ,
Proof.
+n2).
- t2)l
’P{ I(n1T1,2.
d)H) = t 2 A UT << 1 + n2>/Plt) = 1 -P.
T3, ~ ( ab) ,
by T5.6, h=Hd, axHd
z(c, d ) .
cxHd,
a=,b
axHc
bxHc. T3,
( t l =t 2 )
03.
1- p ,
[a, b=c, d ] , , ~f o r a x H bA c x H d .V . a=cIIc A b x H d . V . UXHd A bXHC. v ( 3 X , Y , m , n ) G “ Y W ( X , Y ) HA T ~ ( u b, ) H = m A z X ( c ,d),= A I m - n I < K [( m + n ) / p ] + .
‘ [ a ,b = c, d],,,Ff’
‘(a,b)
p-equal
d)‘.
n
64
6
CLOCKS
‘[a,b < c, d],, I,’
‘(a,b) 03 a =6 ,
d)’. 04.
Prvof.
a , b E &,+[a, 03,
Proof.
[ a , b=c, d ] , , , , t t [ b , n = c , d ] , , , ~ [ a , b=d,c],.,iC-’[~,d=a, b l p . ~ . 03.
A
[ a , b = b, alp,,,.
not
do
[a,b=c,d],,, no
T6. Pmof. m,n T7. Pro0f.
[ a , b
A
[Cd<e,fIp,H+[a,
[ a , b = e , f ] , .,
b<e,fI,,,
mn. X , Y,Iw,~)&YV(X, Y),r\t,(~,b),=m~z,(~,d),=n + [ ~ , b = c , d ] , , , v [ a , b
VII.
LENGTH MEASUREMENT A N D SPACE GEODESICS
1. Introduction
by
by
do
:
(R,S),
(P,Q) __
(Q,P)
~
space
on a 65
1,2
66
(R,S),
(R,S) ( P ,Q ) ( P ,Q )
(P,Q); (R, ( R ,S ) (T,U>.
on do
(2)
(R,S)’
‘(P,Q) P,R
P
Q,S
Q
PV4.3).
Q
P
R,
P a
S
R; Q
S.
P P (P,Q)
2. Length measuring instruments
R (R,S).
Q
21
67
LENGTH MEASURING INSTRUMENTS
A on on
As on
I1
I1
1
on by
A
on
D1.
~ / ( P , Q , ~ , N )for H (3X)(XYgp),AQEPp, A (3 X, 8, A Y C 8, A X n J = 8 A(VU).UEXU~-+UEW(P)H:
A(V~).VE~~(P)~-+I(U)E(O,~, ...,N ) 3
A I(U)
‘@(P,
y
3
XA
A
Q H
N)H’ x, y
0
on P,
v
P.
on
N,
v,
x y
value +
P, Q
on
v. u.
P,Q
(S)’denotes
11,
x,y
number
Q end particles, set S, i.e.
number
scale elements of S.
68
LENGTH MEASUREMENT AND SPACE GEODESICS
‘a no
02.
b
P, P
a(P)-,b
a
on P b[~(P)-~~b]’.
2
a
ad’,Pr\a<,br\bEW(P),
U,x,Y ) U E % f ( P ) H r\a<,u<,,b A U X H U A Pgn,V V U d ~VxUtd1fY. A
[VII
U,
AP@,x, Y A x #
D3 ‘(P, Q ) congruent with ( R , S ) during (a,b), and flze ratio of scale values during this intervaf is mln’ P R X u), Q S Y
Fig. 4.
u),
X,
a on X ,
ii
b on X
M?
a, b
on Y. Y.
w n
m 03.
(a,b).
m ( a , b)NR, n]H for N ) H A * ( R , 1, N ) , A X , Y).P, R@.,X A Q , [email protected] A ( 3 u , u, w).u (X)-,b A u( Y)- H w A a E * T ( X ) , AVBHaA U < H a < ~ b AbgHW A (V x).x E Y Y ( X ) , A a ,x< l,b +€(x) = 111 A l(x) = n .
garz [P,
<
(P,Q> :
(R,S),
DV13.2, ( P ,Q ) ( R ,S )
VII
21
69
LENGTH MEASURING INSTRUMENTS F,
q,r
r
k
m (P,
n,.
. ..,nk
k
(R, by
q),
m,
gpSJ(kS 2 5 g,[(S4'@
04.
1- p
c2
3. (1, 2, 6, 7, 14,
E
by
S4)/(k-
[ P ,Q(,, N ) R , S]f,'.
Y [P , Q (0,N ) R , S],, for c E 3 A 0 I o i)cen,l[P,Q,m(u,U),R,S,
A(VWl).WlE{O, . . . , N ) . - t ( 3 U , U ,
i]H
A ( 3 x , J ' , J ) g o j ? [ R , S~, ( ~ , Y ) N P , Q , J ] H : A (VE).EE<%' A E > O - + ( ~ q, r ) q , r d A (b' k ) . k d A k > r n1, . . . $ k , u k , n k ) : . m E ( O , ...) N } A : g a / z [f, In u ~ ) ~S ,R , ~~~?~/~[R,S,M(U~,U,)NP,Q,IZ~]I~. A _.. A
s, n k ] H
.FcJd[P> m ( m k , O k ) N R , V@fi?[R>m(uk, u k ) N P ,
s,
A
A
u1
Q , nk]H. ..< HU k
X ) : ( X Y % P ) , v (XY%'R)H. A ~ X ( ~ I ~ U ~ ) H ATX(uk-l?uk)H>q.: > ~ A . . .
A
( V p ) . pd ~ A O
IP{ fi - g,S/,/k-l<
+
m
A
IP{S2- gp[(s4 - S4)/(k
(LI), D5.
-
<E
-~)I<E.
DV14.1, by
9 X ( P , Q, I), for N ) W ( P , Q, N ) H A N 2 NL A r).r A 2 < rLI r A X , , Y 1 ,. . .,X,, Y,) # ( X , , Y,, ...,X,, A < P , Q ) . { < X ~ ,Y ~ ) > . . . ? ( xYr>, ) A (v u, y ) .( u , # y> ~ ( uv>,(X, , Y>E{(XI,Y~),*..,(X Yr>) ~,
v,x,
v) (x,
+(3O)@[[u,
V ( C , N ) X ,Y],AO
(P,Q) !V
o I oL.
range
70
LENGTH MEASUREMENT AND SPACE GEODESICS
[VII
3. Distance P,Q by
P,
P
a
7. 01.
PI [ P ( a , b), Q ; .]H
for
u, v,x7Y ) Z X ( u ,~ / , I ) H A U , P @ ‘ H X A ~ , Q @ ’ I ~ Y
(x)-
(3 U , U, W ) U b A U ( Y)- W, A U
A
A Ug$l
‘pl[P(a,b),Q;n],’ is n’.
(x), <
A
a €w(X)~f (X) = / I .
‘the distance between P and Q during (a,b) by 02.3 4. n by D1
2,3
VII
3,4]
the
T2,
P,Q
(a,6 ) on P,
by 02.
p l [ P ( a , b), QIH
T3.
1?1)/1i[P(a,b), Q w ]H +*/[1[P(a3 b), Q ] ~ = n * l l l [ P ( a , h), T2, T4.3,02.
Proof.
71
RIGIDITY
(7n)/l1[P(a961, Q ; ~
IH. ~I]H.
02.(1,5),
0
NL,
~ ~ 2 0 ,
T4. Proof.
(3 n ) ~ r [ P ( a , b ) , Q ; n ] H + P I [ p ( a , b ) , Q 1 , ~ 2 0 D1,02.1.
by measurable by
P,Q
L,
4. Rigidity
'the distance between two particles P, Q is a constant r b y an P Q by
E,
72
LENGTH MEASUREMENT A N D SPACE GEODESICS
6
0 1, n,, ...,nk
i,
6,
Iii-qI<~,
p. 252),
q in probability
ri
4
k
i,
Q by [,
P
[VII
q
r
P 1( p ,Q ;T ) H ( P AI Q ) U A (YE, 6 E 9A E > O A 0 5 6 < 1 +(I A ( v k ) . k E 3 A k > i + ( v t f l , 01, n l , ... uk?u k ? nk). 7
1~
H
v
l
~
H
U
2
~
H
U
2
~
H
~
~
~
~
H
U
~
~
H
u
k
[ P (u 1, v I ) , Q ]H = n 1 v P L [Q (. u I ), = n 1 . A .. . A.l*I[P(uk,%),Q ] ~ = ~ k v ~ l [ Q ( ~ k , u k ) , ! 4 ) 4 E 9A P{ 12 - 41 > E } 5 6 A r = ( 7 S) P { Ifi - S I > E } 5 6. A
./J
7
r
T1. Proof.
T2. Proof.
p[( P, Q ) H .
the
P, Q ; r ) H ,
(3' r ) / ~ l ( PQ, ; r)H. D1, p r ( P , Q ; r ) , Q p ( P , Q; r), A p1( P , Q ;s ) -+ ~ r=
Q ; r)H+(j!
r by Q. E . D.
r)pr(P,Q; r ) H -
D10.
02. P I ( P , Q ) H for ( ~ ~ ) P LQ(; P~ ),H . p r ( P ,Q)H is the (constant) distance between P and Q measured by
T3. Proof.
(3 ~ ) ~ I ( P , Q ; ~ ) H + . ~ ~ ( P , Q ; S ) H ~ ) C I I ( P , Q ) H = S .
T4. Proof.
(3 ~ ) P I ( P Q ,; ~ ) H - ~ P I ( P , Q ) H ~ O . D1, T3.4.
T5.
/J I( P , Q ;~ ) H + + P
Proof.
T2, T4.3, D2.
P ;r ) ~A. . ~ ( p ,; T ) H . +P I(P, Q ) H = T3, TV15.8.
(Q, P)H.
by :
P1.
P ~ ( P , Q ; ~ )FH "A~ ~ ( ~ , I ) H ~ ~ I ( P , Q ; ~ ) H . by by 1.
VII
4,5]
73
CONGRUENCE
P2.
€9
P)P~(P Q ,; P ) H A P ~ ( RS,; P>H A ( P ~ I Q ) H . + ( R ~ ~ S ) H .
no
SC’s,
P3.
P,4 , r , S ) P f ( p ,Q ; P)HALLf(R,S ; q ) H h I * I ( P , A
P [ ( R ,S ;S ) H . &P
f3‘ (
Q)H/P
f ( R >S)H
(‘,
Q ; ~ ) H
Q)H/PI(R,
s>H.
rigidly connected pair of particles. 03.
LI T6. Proof.
P
PBVHQ-Q9VHP. 03.
(P4). P4.
PWVfiQ A ( P ~ ~ Q ) H P + )(P~~ ( PQ ,; P ) H .
04.
.%V(Pl, ...) P,), for Pl.?h?%?lfP2A P2-%%,P,
‘9%?(Pl,,. .,PJli’ T7. Proof.
‘ P I , .. .,P,
A
... A P,- I ~ % ? H P , , . sequence of rigidly connectedparticles’.
c%g(Pl, ...)p n ) H c ) ~ % ? ( P...) n ~PI),. 0 4 , T6.
5. Congruence
02.3 by
DI.
P, Q&R, S for P , Q , R , S E 9,i A :P=Q A R = S . v . P # Q A R # S A :P=RAQ=S.V.P=SAQ=R. P)Pt(P,Q ; P)H A P I ( R , S ;P)H A ay$(f,r)~.
‘P,QHHR, S’
‘(P, is congruent to ( R ,S)’,
P=
74
LENGTH MEASUKEMENT AND SPACE GEODESICS
R = S, P# R#S P=S Q= R ,
[VII
:P =R R,S,
P,
5
=S,
by
T1. Proof. T2. Proof. T3. Proof.
T4. Proof. T5. Proof,
T6. Proof: (T2.40) H , v ... v
H,
i
Fir\
H,+Con
k. T5.1: P , Q, R , S , T, U e;PHA P=Q A R = S +P, Q, T, U €.PI,A P= Q A T= U D l -+Con. 6 T1.11.
A
R=S
A
T= U
'H=SA R#S',
T5.1:P,Q,R,S, T , U E Y ~ A P # Q A R # S AT # U A P = R AQ=S A R = T A S = U -+ P, Q, T, U E .PHA P # Q A T# U A P = T AQ = U D1 -+Con.
Con
VII
51
75
CONGRUENCE
4
4
T5.1: P,Q,R, S , T, U E 8,A P # Q A R # S A T# U A P = R A Q =S A( 1, P)Pf( R , ;p ) H A p 1( T, ;p ) H A ‘9y -+P,Q, T , U E . Y ~ ~ A P # QTA# U A 1, Q ; P ) H A P I ( T , u ;P)H A I)H Dl+Con. (€9
1+6+4+4(
= 15)
P4.( 1,2,4),D4.(1,3):P# Q A R # S A T # U A 1, p ) k ( P , Q ;P)H s;P)H A 11, u ;4 ) H 89y(m?n)H A n, 4)Pnt(R, s;4 ) H A -+ P#QAR#SAT#U 11, A 1, p ) P f ( P ,Q ;P)H A p I ( R , ;P)H A ; 4 ) H A (3r)udT?U ;r)H A m q ) p , n ( R , s;4 ) H A pin( T, P4.3-+ P # Q A T# U A P , Q, T, U E .Yrr A(~~,I,P)I*~(P,Q;P), U ; P ) H A & ~ ~ ( ~ , ~ ) H Dl+Con. €9
T3, T5,
T6
gH
04.4
by 02.
V h ( P , , ..., P,,), f o r # ( P I , ...,P,,) A PZ HH PZ p3 A pz P3g.Y p3 p4 A . .. A PI,- Z > pn 9
9
“e/;.(P,,..., PJH’ ‘ P i,...,P,,’ ‘ # ( P I , . . .,P,)’
9
chain.’ no
- 1 :HPn
- 1 9 ‘n’
76
[VII
6
6. Space geodesics by on q ( p l ,..., pn)H for ...,f,)~A I 1 A (V k €9A k > 1+(V X i , .. .,X k ) .%‘d ( X i , .. .,X,)H A X, A X,=P,, A X2&,P,, P,+k 2 M .
...,Pn),,’
...,P, lie on a spare geodesic (SG).’
..,P,
. ,X ,
Y(P1, . . . , P n ) , P 3 ( P f .l ., . , p 11,.
P z + X I , X 2 H f r P , , P f l -I .
>
.. ., p n ) H A (PI A f p 2 ) H ...
~ ( 3 p ) ~ ~ ( P l , P Z ; P ) I I A ~ I ( P 2 , P 3 ; ~ ) f , AA/ - ~ ~ ( p r r - l > p f l ; P ) H ~
on by
SG,
...>P ~ ) H A ( P ~ ~ P ~ ) , I - , P ~ ~ % ‘ , , P , .
..., P J H A k, ..., n } A i # k A I -jl > 1 A i - j = k - / A (pidpj)H+pi, PjHfiPk, PI. y(P1, . . ., P f l ) H A i , j , k , 1 E A
..., n } ~-~=~-IA(P~~P~)~’P~,P~==,~P~,P~A(P,~P~)~. on
LINEAR
VII
77
MEASURING INSTRUMENTS
A V1113.
7. Linear length measuring instruments 03.2
04.2
linear
A . ., X,,
on 1
D1.
for
xm)H
1
m -
5
x2)H.
29(1)ll
X = Y++jLr(X,Y ) , = 0. / \ @ X i , ..., Xn,??~).%(Xl, ..., ..., n } A(vx,
YE9,A
--f/il(xl?
- l)pl(xl> X 2 ) R .
Xm)H=(m
linear length instrument 02.
A I ( P ? Q ; ~ )fH or P ~ ( P , Q ; ~ ) H A \ ~ ~ ( ~ ) H .
03.
A [ ( P ,Q ) H
for
(9
Y).PI(P,
=
r~ P I , ...,P,,
)Ll(Pi,Pj),
on
li-jl
B ( P , , ..., P , , ) ~ ~ 2 2 ~ ( I ) ~ ~ i , j , ..., k +n l} ~ { l , A i<jA(piAlpj)H+A1(Pi, P J ) H = ( ~ - ~ ) A I ( P ~ > O HP.~ +
n,
78
LENGTH MEASUREMENT A N D SPACE GEODESICS
8. Space metric
by VI5). Proof.
T4.4,07. (2,3).
r2.
P , Q E b, D7.1.
Prouf.
A dip23(I)H+.
Ar(P, Q),
= 0-P
T3. Prmf.
(3 r)AI(P, Q ; r)H.+Ai(P,
T4.
9 ( P , Q, R)H A A (P*AIR)H R ) , = I ~ I ( P ,Q > H + ~ ~ I ( Q , R)H. ~7.2.
P~OOf.
Q)H=AI(Q,
= Q.
P)H.
T4.5,07. (2,3).
i,,,
R),LAI(P~Q > H + ~ I ( Q , R ) H ,
TV1113.2.
[VII
7,8
G E O D E S I C GEOMETRY
VIII.
1. Introduction I by by by by
by by
on up
2. Linear order p.
p.
pp.
Q
D1.
P R.
P,
R,
between P
R
R
Q,
[ P , RIH for +(P, Q, R ) A ( X P t q , r ) ~ I ( P , Q ; p )A, It(Q,R;q)H A A I ( ~ , R ; ~ A P+ ) H 4 =r. Q7
79
no on
80
[vrrr 2
GEODESIC GEOMETRY
on
on
SG
D Proof. Proof.
[J', Q, R I H + [ R , Q,PIEi. Dl, [ p ,Q , R]H+ -[Q, p , R ] HA [P, R,Q ] H . { P , Q , R I HA P, RIH +P# A 1)Ai(p9 R)H=A,(Q, R)Hf I L I ( PQ, ) H
=J-I(Q,R ) H - ) L 1 ( p , +P # A 1)IL,(P, = 0 +P#QAP=Q. (l),T2.(8,22):[P,Q,R]H-+-[Q,P, :[ P , R ] H + [R, P ] H + [Q, R,P ] H [ P , R , Q]ii.
Q)H
(
-
+ N
(3)
T. [P,Q,R]~I~[Q,R,~]H~(P~~~"S),+[P,Q,S],. on (Pd9S) P S P R, no P by ' R ' Proof.
Proof.
81
R R,
S,
P
R
Q
R,
P
P [P,
S ] HA [ p , R , S ] H + Q = R
V
[p,
R ] , , V [ p ,R , Q ] H .
by 'S'
'P'
by : [P,Q,S],A[P,R,S]H-+Q=R
V
[ Q , R , S ] H v [R,Q,S]H.
Proof.
R, S
on
P,
[ P , Q , R ] H A [ P , Q , S ] H ~ R = S V [ P , R , S ] H V[ P , S , R ] H .
[p, Proof.
S ] H - + R = S V [Q, R , s ] H A [P , R , s]H . v . [ P , S]11 A [ p , S, R]H.
R I HA [ P ,
V
S, R ] H .
:Ant-, R = S v .[ p , Q , R ]
+Con. 02.
for
...,Pn]H
[Pl~P~~P~1HA[P2~P3~P4]HA~~~A[[Pn-~~Pn-l~Pn~H~
T7.
[ P I ,. . .,Pn]H A i , j , k { 1, . . . , n } A ( i < j < Ic V k < j < i)
Proof.
02,
A (pid2Pk)H+[pi,
pj, P k l H .
..., P n l H - + g @ ( P 1.. .,P")". 3
Proof. 3. Linear order and space geodesics
T1. Proof.
3 ( p 1 ,.. .,p,,), A
P , , ...,P,
(p1-421p,)H
A
( P ,A'S 1 p 3 ) H
on
A [ ( P , P ) H v .[p>R I ,...)Rm, QIH A R~ ; A J I ( R ~ ~ & ; P ~ .) .H. AA J I ( R ~ , Q ; P ~ + ~ ) H A -I-p2 -I-... + Pm + 1 p :
-+
[PI, .. .,P,]H.
...,P,,lH.
82
3
GEODESIC GEOMETRY
Fig. 5.
5.
P1
P
Q,
by
P1
An
P1 T2
T2 p<s+ q5, s l rl
+ r p , r l Iq1 + q2,r 2 < q 3 + q4. rI,r2,
5,
s
T2
PI. T2
TI :
+
p 5 q1 .. . + q 5 .
VIII 31
LINEAR ORDER AND SPACE GEODESICS
83
AA[(P~,~~)H=...=~~I(P,,-~,P~)H x 2 ) H = . . . = Al(Xk- xk)H
=
9
+(3l)(n-
1 9
1)~.I(P,,P2)HI(k-1)~~l(PI,P2)H
(1)
+n
(l), DVI 16.1 :T. on on
(T4).
on
(T5).
on
PI
T5.
--*
*.> p n ) ~A g(pn- 1, pn, .. pn + k ‘7
A ~ P ~ ) H
) A~
3(Pl?. . ., P,,+,),.
Tl,P2.1, T2.3:Ant+[P1, ..., P,,+k]H. DVI16.1:Ant+V&(P,, ..., p , + k ) H . (1),(2), T 3 : T.
Proof.
(2)
SG [ P , R 1 9 .. .> R,, Q ] H A [P , S 1 .. ., S,,, Q ] H A r l , . . ., ri,sl, . .., s j ) r l ... r i = s l .. . + s j A A ~ ( P , Rr ~l ;) HI ~. I ( R , , R ~ A; ... ~ ~A A ) ~l ( R i - l , R i ; r i ) AH ism
P2.
7
+ +
+
...
A A ~ ( p , ~ l ; S , ) ~ A ~ ~ [ ( ~ l , S 2 ; s 2A) ~ ~A ~ i ( ~ j - l , ~ j ; S j ) H A ~ ~ n .
-+
R i =S j . S,, . . .,S,, P
R , , . . ., R, P P
T6. Proof.
Q, Sj,
Ri
Ri=Sj.
[P, R , Q ] H A [ P , S, Q ] R P2, DVI15.1.
A
P , R=IHP,S + R = S .
T6,
P2,
T6 by :
84
g ( P j R I .. ., Rn, Q ) H A g ( P , S I , S n , Q ) H A ( ~ I ) ( P ~ L ~ ~ R , ) , A ( P ~ ~ ~ R , ) , +AR... ,= AR,=S,. S, Tl:Ant+[P,R,, ..., Rn,Q]Hh[P,S1,..., S,,Q]H. Ant+( 3 p , q)A,(P, R , ;P ) A~... A A,(R,, Q ; P ) ~ A~~(P,S~;~)HA...A~~(S~,Q;~)~, PI A ,~)~~(P,R~;~)HA.,.A~~(R~,Q;P)H A ~ ~, S(~ P ;~)HA...A~~(Sn,Q;P)~, +Con. 9
Proof.
3,4
GEODESIC GEOMETRY
...?
P,Q
P,Q,
4. Collinearity
on SG, s ( P , Q , R ) H f o r P , Q , R E P ~ -A# ( P , Q , R ) . v [If', R]H v P , RIa v [ P , R , ' L f ( P ,Q, R)/,' ' P , R are collinear' P, Q, R P R ; (3)P Q R; R
P € P H + L f ( P ,P , P)/{ Proof. Proof. 9(P,
R)
by : Z"(, R)H-Lf(Q, R , P)Hc*Lf(R,P, - y ( R , Q , P)H++Lf(Q, P , R)H-Lf(P, R , Q ) H . Proof.
P
(2) Q Q.
VIII
4,5]
85
THE SIDE RELATION
T4 T4. Proof.
[ P , R ] H + ~ ( P , R)HA # ( P , T2.15, Dl,D2.1.
T5.
9 ( P , R)HA f ( P , Q , R). + [pj Q, R ] HV [Q,R,p l H D1, T2.2.
Proof.
Cp,
R).
R , QIH.
(T8).
5. The side relation
D1. pTHQ, R for P # R A 2 ( p , R)H A ‘PEHQ, R’ ‘Q,R are on the same side of P’.
P, R]H.
TH. T1. Proof.
PXHQ,R~Q=R~[~,Q,~]H~[~,R,QIH. Dl7D4.1.
T5.
86
5,6
GEODESIC GEOMETRY
P, Q e g f IA P#Q+P%,Q,
Q.
Proof. PXHQ, R++PXt^,R, Q. Proof. P X H R ,S + P S H Q , S . Q, A Q = R -+ P s r ,Q, S . A Q#R+ -[R, A . [ P , Q , R ] , v [ P , R,Q],. [f', Q , R ] HV R , Q ] H . A [Q, P, S ] , : + [ R , P, S]H. (4) (4),T2.32:[P,Q,R],v[P,R,Q],. A -[R,P,S],: [Q, S]a. (5) A Q # R + [Q, P, . P%,Q, R
Proof.
A
+
N
N
T. ~ ( P , Q , R ) , AP # Q - + P = R v [ Q , P , R],v P2YHQ,R. Proof. [ P , Q, R],+PX,Q,
R A R X H P ,Q.
Proof. [ P , Q,
N
Q%z",P,R .
Proof. [P, Q, R ] HA [P, Q , S]t^,+P%,R, S A Q%,R, S . Proof. [ P ,Q ,
A
[P, R , S],+PX,Q,
R A S%,Q, R.
Proof. 6.
In Sec.VI1 5,
Proof.
-
P , QHHP, R+ . [ P , Q, RIH v [ P , R , Qlr,. 02.1, [P, Q, R]H v [ P , R , Q ] H -+(2 1, P, q)J-i(P,Q ; P), A AI(P,R ; q ) H - P , Q&P, R
A
(P< 4 v 4
VIII 61
87 R
Proof.
D1.
on
P T H Q ,R A P, QHHP, R +Q = R. T5.1:PEHQ, R+Q=RV [P,Q,R]HV[P,R,Q]H. : P, Q # H P , R-t .[P, Q, R ] HV [P, R , Q I H . T.
Proof.
(1)
(2)
A i d ( P , Q, R)H for p ( p , Q , R)H A P # R A P , QHHQ,R .
' d i d ( P ,Q, R)H' R. Proof.
P,
midway
P, Q, R
A i d ( P , Q, R)H++Aid(R, Q, P)H. D1, d i d ( P , Q, R)H-'[P, Q ~ R I H P= Q v Q = R . A P , Q&Q, R +P=QAQ=R +P=R. D1 :A n t + p ( P , Q. R ) H A # ( P , Q , R ) + [ p , Q , R ] HV [ Q , p , R ] Hv [ p ,R , Q ] H . +Con.
6 to 10
Proof.
P X H Q ,R A P'!ZHQ',R' A P, Q#HP', Q' A P, R'&P', - + . Q= RHQ' = R'. Ant A Q = R+P, Q E H P ' ,Q' A P , QHHP', R' -+PI, Q ' E H P ' ,R' A P'XHQ', R' +Qr =
Ant A
=R'+Q
= R.
R'
88
T7. Proof.
T8. Proof.
GEODESIC GEOMETRY
[VIII
6
[ p ,Q, R ] HA [p’, Q’, R’]HA p , QHHP’, Q’ A Q , RHHQ’,R’+P, RHHP‘,R’. 0 2 . 1 , PVII4.3:Ant +(I I ) A I ( p , R)H=Al(P, Q ) H + A l ( Q , R>H AA~(P’,R’)H=AI(P’~ Q’)H+AI(Q’,R’)H ~Ai(p9 Q)H=AI(P’,Q’)HAA[(Q,R)H=AI(Q’,R’), +(~I)Af(P,R)H=A~(P’,R‘)H DVII5.1+Con. [ P , Q, R ] HA P’SHQ’, R‘ A P, QEHP’, Q‘ A P, REHP’, R’ +[PI, Q’, R’]EIA Q, R H H Q ’ ,R‘. T5.1: P‘SHQ‘, R‘ -+Q’ = R’ V [P’, Q’, R’]HV [P’, R‘, Q’]H. Ant A Q’ = R’ +P’, Q’HHP, Q A P’, Q’gE1P,R TVI15.6 +P, Q g H P , R T1, T2.39 + [P, Q, RIH (2) (2),T2.9:Ant-t-Qt=R’. (3) 02.1,PVII4.3:Ant+(3 I)AI(P,Q)HA[(P’,R’)H. (5) (4),(5): Ant+ [P’, R‘, Q’],. (6) (3), (6) : Ant+ [P’, Q’, R’]H (7) +[P, Q , R ] HA [P’, Q’, R’]HA P , QHHP’, Q’ A P , R#,P’, R’. As in T7 +Q, REHQ‘, R’. (7),(8): T. N
N
[ P , Q , R ] HA P , QHHP’, Q’ A Q, RHHQ’, R’ A P , R H H P ’ ,R’+[P’, Q’, R ] H . Proof. 0 2 . 1 , PVI14.3 :Ant + ( 3 I)AI(P’,R’)H = Ai(P’, Q’)€I + AI(Q’, R’)H 02.1 +Con. If Q is midway between P and R, and is midway between P’ and R‘, then if any corresponding distances are equal, all the other corresponding distances are equal (T10). T10.
d i A ( P , Q, R), A A i d ( P ’ , Q‘, R‘)H + : P, QEHP’, Q’+Q, RHHQ‘,R’ A P, RHHP’, R’. A .Q, RHHQ’r R’ Q m H P’, Q’ A P , R H H P ’ ,R’. A .P, R H H P ’ ,R’ +P, QHHP‘, Q‘ A Q, R H H Q ’ ,R’. +Pj
61
89
Proof. P,Q,
S
P,Q Proof.
9 ( P , Q , R)H A P f Q A P , RKHP, S A Q , R s H Q S, + R = S . Ant A ( P = R v Q = R ) + R = S . Ant A # ( P , Q , R ) + [ P , Q, RIH V P , RIHV [ P , R , Q ] H . Ant+P, Q E H P ,Q A Q, R E H Q ,S A P , R+HP, S . Ant A [ P , Q , R ] , { + [ P , Q , S]H -+PXHR, S A P , R#HP, S +R=S. Ant A P , RIH-+R= S . (3), A~~A[P,R,Q]H+[P,S,Q]H +R=S. T. [QI
(1)
(2) (3)
(4) (5)
IX.
S P A C E GEOMETRY
1. Non-collinearity non-collinear
P, Q, R D1.
Jlr(P,
for P ,
A
-9(P,
(TI).
T2.
P),,++.N(R, P , Q), P)H++Jlr(Q, P ,
Jlr(f'3
Proof.
(T 3). T3. Proof.
N
. y ( p , Q,
~Jlr(f',
(T4). T4. Proof.
E ~ ~ , + ~ Q( , P , Ant + :P, A9 ( P ,
P,
v .P,
v N ( P , Q ,R)" T3
-+Con.
A
(T5). T5. Proof.
9 ( P , Q), A-N(P~ S, Ant A R = S + 2 ' ( P , T3, T2.8: T .
f S.
AN(P,
90
A
-2'(P,
IX
11
91
NON-COLLINEARITY
(T6). T6. Proof.
P , Q , R EY p H A # ( P , Q , R ) A P d V H Q + N ( P , Q , R)H. DV114.4, TVIII2.8: Ant+ # ( P , Q , P ) A [ P , Q, RIH A [ Q , P , R ] , A [ P , R , QIH DV1114.1 -g(ppQ , ,R)H T4 +Con. N
N
N
+
the TV I I I3.2(T7).
T7. Proof,
N ( P , Q , R ) HA Q ;P ) H A A I ( Q , R ;q ) H /\A,(P,R;r),+r
SG's
Fig. 6 . T8
(T8,
7.
T9
6).
92
[IX
SPACE GEOMETRY
P
P
Fig. 8. 7‘10
9.
Proof,
Proof.
Proof,
10. P1
T11
1
1x
93
COPLANARITY
P1 S,T, U . 10 do 2. Coplanarity
A , B, C,
D
ABC.
on by A , B, p.
A , B, C
D A , B, C, E
A , B, D.
A
A , B,
on
S,
B C. geodesic surface AB
E
A.
D
on E
E
AC,
on S,, 11, pp.
on S, S,
D on S,
D
E
E. by
pp.
by
by
pp.
94
[IX2,3
SPACE GEOMETRY
on
3. Perpendicularity PQ
perpendicular
P, X X
PR Q P,
P ( P;,,Q, R, Y
P,
R)
X, Y P , R, X
P on
t
R
Y
X
1I .
pp. 103,
X P
foot of the perpendicular.
R
Q
IH Q , R for J’”(p, R>uA .dg(Q,f‘, RIM A(vx, Y , I , ~ , ~ ) . ~ ( P , Q , X ) H A ~Y( )PH, ARP, # X , y
P
Dl.
AAI(x?p;p)HAAI(x3y ; q ) H + p < 4 . PR,
PQ
S
P
Q,
PS
P R(T1). TI. Proof.
P i HQ, R A A P#S-+P 1 R. Antr\Q=S+P!-J,R. (1 1 T 1 . 9 : J I / ’ ( f , Q , R ) , , r \ ~ ( fQ, , S ) H ~ f # S - t . / ~ ( P , K . S ) r r . T1.l :Ant A S # Q - 9 ( P , S), A # ( P , S)fi TVIII(4.5,2.8) -+.#g(R, P, S),. TVIIT4.6: 2 ( P , S), A 2 ( P , Q , X), A P # -t9(P, s, x)H. (4) (2)-(4) D1:AntA Q # S + P 1H S ,R. (5) 1, T.
IX
31
95
PERPENDICULARITY
no
A priori
(Pl). P1.
P
L
R + P L HR,
PI, TI T2. Proof.
P 1H Q , R A 2 ( P , R , S), P1, T1.
A
P#S+P
S
(P2). P2.
P .L&, R A P ’ ~ ~ H Q , S A ~ ( P , P ’ , R , S ) H + P = P ’ .
T3. Proof.
P -L H Q , R P1,2.
A
P‘
,S, R A 2 ( P , P‘,
S),+ P = P’. 2
P2
(T4,
P2 12).
Fig. 12.
T4
96
SPACE GEOMETRY
3,4
[IX
on P,Q,R
P,
PQ
~ ( P , Q , R ) , A P ~ , S ; Q , R A P I . T ; Q , R ~ ~ ( P ,TS) H , .
Fig. 13. P3
4. Parallel displacement on
:
by 11, pp. 98-101),
pp. by
11,
pp.
S,
P, Q , PQ.
RS P
PQ
14).
PR
P, P, R Q,
V on RS on
U on PQ on S, PV
41
IX
97
PARALLEL DISPLACEMENT
14.
displacement
W.
RU R
R , U, V X
P, V. PY
P
Y
U,
X
RX
Y
P, on
RV
P.
P U
R, U
V
U
R S so
PV P X
V
PU by R V, P R.
RV U V
initial PQ RS PQ’ RS
P, RU
Y.
RS’, R
PR
PQ RS
PQ
P S‘,
PQ Q‘ Q’
S’
D1.
P,QII’HR,s for N ( ~ , Q , R ) H A J ’ ” ( R , S , P ) H A ( 3 U , V , W)PXHQ, U A RX%,S, V A [P, W , V I HA [ R , W , U I H A P , U&R, V A P , R E H U , V A ( V X , Y).[P,X , u ] H A [ R , Y , v ] H A P , X ~ H R Y, + p , R=IHX, Y A (3 2 )[ P , 2,Y ] , A [ R ,2,X ] H .
‘P,Q l E H R ,S’
‘RS
displaced parallel to P Q
PR’.
98
SPACE GEOMETRY
T1.
P,QIFHR,SHR,SI~HP,Q-
Pro0
D1.
[IX 4
‘P.QllHR,S’
T2. Proof.
T3. Proof.
P , QIPHR,S A P S H Q , T-tP, TIEHR, S . DVI115.1, T 1 . 9 : Y ~ ( P , Q , R ) H ~ P X H Q , T R- ~, T( )PH, . TVIII5.(3,4): P X H Q ,U A PX^,Q, T+PIHT, U . (1),(2),Dl:T. P , Q I P H R , S ~ R X HT-P,QIPHR, S, T. TI ,2.
Fig. 15. Antiparallel displacement
(1)
(2)
IX
99
41
Proof.
02. P,
S
A
S
A
PTf,Q, T-+P, TIl,R,
S.
Proof.
P , Q I111
9
T-1 P,
11NR, T.
As
(a) N
Fig. 16.
.P,
S A P,
(b)
T7
H R ,S .
Proof.
U = U’ A
P,Q~~HR,SAP,Q~C,R,S+(~U,V,W,U’,V’,W ...’ ) A : u = u’ A v = v’.V .[P, u’, A [ R , v’, V .[P, u, A [ R , I/, V’],. V = V’, [P, W , A [R, W , A [ U , w’, A [P, w’, ~ V I I I ~ . ~ , T ~ . ~ : ~ ( P , Q , R ) ,UA- P +X ~ (, PQ, R , , U),. U ) , A [ R ,W , U),. w, U ) , A [ P , V ] , , + x ( u , I/, W),.
w,
P~.~:J~~(U,V,W),A[L~,W’,~/]HA[U,W,R]HA[V,~,P]H
- + N ( P , W’, R),, [P, U‘, A [ R , V’, A P , U‘#,R, +(I 2 )[ p ,2, A [ R ,2, U ’ ] H ,
[P,W’, R],.
(2)
V‘
(3)
0 2 : [ P , U , U’], A [ R , V , - [ U , W‘, V ] , A [ P , W’,
(1)-(4): T.
A
P , U#,R,
V
100
[IX
4
IX
101
4,5]
5. Dimensions
P, on p,q
R, R' P
R'. R # R'
R
17.
no
R S P
p ,q
R'
S' on R R' R
r
S'. S
P
p , q, r
S'
R
S
no P , Q, R. (Pl) by U, V, W,
X,Y U, V, W .
18, U , V, W
P,Q, R
X,Y
S,S'.)
102
P1.
[IX
SPACE GEOMETRY
5,6
w,x,y)-'@-(u, I/, w),A -@g(u, v,w, u)H (u,I/, w ,x,Y )A u, X R H Y~A, T/, X#,T/, Y
( 3 u, I/, A
#
A
w,xg=,w,Y .
space is at most 3-dimensional (P2) S
P , Q, R P,Q, R,
X
S
S
(X
18). ,
/ -
--_
X
P,Q, R
.
Fig. 18.
P2.
-'@-(P,Q,R)H
A
a q ( P , Q,R , P)H A S ~ p pA ,S # P , Q,R
+(~'X)X#SAP,X~,P,SAQ,X~,,Q,SAR,XF,,R,S.
&(X, P),, Ai(X, Q),, space coordinates X S SP, SR X by is 4-dimensional.
J.[(X,R)[* P, Q, R, S, by X , space-time
6. Geodesic space coordinates
A by
IX 61
103
GEODESIC SPACE COORDINATES
by
([1956] p.
by by
A
CS
pp. 263-264).
by
1).
CS CS by
up
CS
CS
CS.
CS by
119651
9A, 10A
11).
CS
An OA, O B
O,A,B,C
OC P = A ) , X,,
...,Xi,
( O I A ,B ; O I B , C ;O j C , A). 0,A , B, C, OA 0, . ., X , on
by (SG)
[Y(O, X I , ...,
0 OB by
OC
SG X, XiUi XiVi Y,( = Ui),Y 2 ,..., Yj
A. (04.1). 19). Y,, ...,Y,)H,
SG XiUi
104
[IX
SPACE GEOMETRY
6
I
19. ‘Rectangular’CS
SG 52,. P( = zk),
XiK Z , , ..., zk i,j, k CS, O,A,B,C. OB
by
YjZ,
9(yi, Z , , .. ., Z&.
OA,
CS
SG by by Y ( X i , Y,, ...) by Y ( Y j , Z l ,...,Z,)
9(O,X,,X, ,...)
SG’s on
CS
Y
CS,
The space coordinates of particle P are i,j , k with respect to the geodesic rectangular coordinate system O , A , B ,C. CS positive i,j , k
D1.
P ( i , j , k)- W U U ( 0 ,A , B , C ) , for A 0 1H A ,B A 0 1H B ,C A0 1 A : j =o-+.i = O-+[k=O-*P= 0. h .k > z , , .. .,zk)z,= c A .. ., zk)H A zk = P ] A .i > ...,xi)[ X I =A A 9(0,X I , ...,xi), A .k=O-+Xi=P. A .k>0+(3 ..., vi,zl,...,z,).
a
IX
61
105
02
0 3
X.
S P A C E - T I M E GEOMETRY
1. Introduction on
no upon
;
by upon
on do
As on
upon 106
x 1,21
107
A
P
B,
A
P
B P
A
go
Zag time
B.
upon
by
2. The fundamental space-time relation
A , B, a,
B
by B b C
6,
A P A
c,
A A
(d'(
A.
6'
b by go
on 6'
by B
d A
a
B
A
by by
b'
by
z,(b,d) P,
by by
c
108
2
SPACE-TIME GEOMETRY
=
dxo =
=
line element. CS(xi
xi-&,
a; C
5 Fig. 20.
2
= =
-g - goo PV
- 2goi
dxo -
1, 2, 3, CS, =
J(-
0, 1, 2, 3.
dxo ,
=
J(-
by
.
by
\I( dxo
by do
tA(c,d), TA
d ) = T A (b'>d ) - z A = J(-
c,
-
3
(4)
x 21
109
THE FUNDAMENTAL SPACE-TIME RELATION
B
(2)
A.
- goo
-
dx3 = 0 ,
-
:
J(-
+ (yij
= yi
.
(6)
(6)
+ 2c-'t,(~,d)(yi~dx'dr')~,
7p(b,d)'=
go
(7) Z~(U,C).
B
A
+ by ' -
by
J'(-
=
+ (yij
- yi
(6), (4, =
'
.
bond Eq. =Za(c,d)'
+
(9)
110
SPACE-TIME
3. Neighborhood no
on
R,
on by
r
p. 116):
(r/R).
= 2nR
C+27rr,
on
2rtr by Q,
6,
p. R
=
(1
11
R
R
.
c2 = a 2 + b 2 ,
space-time
p
q -Y upon on
q, 0 1, by J p ,
lq-rl 1 -p(Dl). p. ;
4%,
111
D1.
E q= r for
1-P.
' q 2 r'
'q
equal within the experimental error'.
r
P
Fig. 21. PI
by e,
(Pl). by
P1, P e
u, u
on P,
u
X, x , z
e u on P ; y , z
x, :
on P, e, x,y,z on X ;u < x < e < z < u ; x . F y r\y.Fe y, e), y
112
[x 3
SPACE-TIME GEOMETRY
P1 PV16.1.
P
Fig. 22. 0 2
D1. Events a, b, c are said to be in the neighborhood
e on P
:P
U,
on P , a S b ~ b F e , U V
b,c
z x ( b ,c)’ E T p ( e , c)’
on P , a<x<e
a, e, c
X
V
b
on X
+ z p ( e ,c ) z p ( a ,e ) .
x,y,z x F y A y S e , Y is y,z on Y,
Y y
W, x , z W
113
NEIGHBORHOOD
I
G
23. 0 4
Two events a, b are in the space-time neighborhood of each other on X(D3). 03. aFA’”,b for (3 u , u, w ) : u, u, w x h , b , X A a E {u,
x,
T1.
a.TMM bc*b.YMHa .
Proof.
03.
w } . v . u, u, WJl’-d,a, X A b
event b is in the space neighborhood of events a, c on P, i f s a, b,x c on P(D4, 04.
b Y X d , P ( a , c) for
( 3 x ) a , b, xA’”FHc,P.
Two particles P,Q are in the space neighborhood of each other, on of on (D5).
{ Z I ,u,
w}.
114
[x 3,4
SPACE-TIME GEOMETRY
4. First-signal speed
by
~ ~ ( a , P
23, b
by
a 3 x lo8 P
[1965]
P
DVII(3.2,
a D1.
for ( l r ) ( 3 X ) ~ l [ X ( a , a ) , P ; r l H~ A~ y ( 1 ) “ . p)H
,$(a;P)H
a
P.
b
(Pl)
a,c on P
23),
r
bcYMff~P(a, h (3 ! r ) 2 i [ ( b; P ) H = rTp(a,
P1.
x, y , z ,
02.
21,(b;P)= r ~ ~ ( a ,
r P, x, z on P ,
y
cIP
by
YYX,
2 4 ( y ;P ) = r t p ( x ,z).
for ( l r ) . ( 3 U)(U~”%P)HA=~LZY(I)H A ( V X , ~ z, ) . y . ~ N e H P ( x , z ) , + 2 1 , l ( ~P; ) H = r ~ p ( x , ~ ) H .
P1
02
T1.
~YM&HP c )(A~~, Y ~ ( I ) H
Proof.
P1,02.
+211(
; p ) H = clP5P(a,
c)H.
x 51
115
METRIC COEFFICIENTS
5. Metric coefficients = A , ( 6 ;P ) ,
T4.1
yij
(Pl).
16 - x i .
D1.
xph f o r
PI.
~ [ Q ( ~ Q ) ] - ~ ~ ~ C),>H ~ ( ~ > A , B , 6.!ZLV4HP(a,c ) + ( 3 ! r l r 1 2 r. 1 3 , r 2 t , ..., r 3 3 ) . r l l , ..., r 3 3€99A r 1 2= r 2 1A r , = r 3 t A r 2 3= r 3 2 (p(Xp),
,,
A
A
P)H=rij&&Q.
23, ( l ) , D1,
P1
P1,
riJ,
(rij=rji),
by no
rij r . . = t(r..+ r j . )+ L2 ( rI .J - r 3.1. )= s.. + 0 . . q.. IJ
IJ
IJ
= 4(rij
+ r j i )=
IJ
I
a I.J . = + ( r i j -
. . 2a,jxlxJ = (aijxixi = (Uij
+ Ujifjf') + 'lji)X'Ij
= (aij- uij)xiIj = 0.
P1
IJ
r . . )= JI
116
on y i j do by
P
Fig. 24.
P, b,d
on P,
a, c
P
c,
by
b
go
c ( r b - 2),
c(Td
= A ( b ;P )
- T,) =
( d ;P )
+ Yi(p,a ) X i Q . +
yi
c
d
(PIc)
Xp’Q.
x 51
117
METRIC COEFFICIENTS
y;.
rQ(b,d)
y;
a;,
t p ( a ,c) upon
ai do
ai 3a’s. 10 goo,
upon
P2.
{ P ( x p ) ,Q ( ~ Q ) - % q y A (, B, o , A P,YJ’”,Q A aYN’,c A U , C E W ( P ) ,A 6, de?V-(Q), A a<,c A aF,b A rZ, r3).cIQZQ(b? d),,=(l+r,X~Q)CIPZP(a,c)f~
+ 1 ( d ;P)H - 2 1( b ;
P2 03.
24):
“ ; ( P ,a ; 0 ,A , B , C, for ( 7 r ) . ( 3 X , U , U , W , r l , r 2 , r 3 , ~ 1 , x 2 , x 3 , ~ 1 , qt )32) ., r = r i A ( P ( ~ ) , X ( I ~ ) - - ~ . Y ( O , A , B , C )A~X) , c~ Y N H P ~ ~ . T . H ~ A a, u e W ( P ) , A u , W E W ( X ) ~ a<,v A A aF,u A u ~ , , w A CI,y”(U,
W ) H = [ ~+ r j ( $ - X ’ ) ]
+ q w ;
CIpTp(a, u), (u;P),.
118
[x 5
SPACE-TIME GEOMETRY
on on
x1,xz,x3, u y1,x2,x3,
P X
f
u,
of r
yijpi,
u)--f(P4)1/($-x')=rpartial differelice o f f with respect to x ' , P3 D4 (V X , U ,s'). UYNHU A aFHu A (U[P(X',X2,X3)], U [x(f)',X2,X3)] -auu(o,A , B, c),}, a ; 0 ,A , B , c,l)H> A f ( p , a)€ {?i,(P> 0 ; 0,A , B,C?I)HI A {yij(x,u ; 0 , A , B , C,I)H,%(x, u ; 0 ,A , B , c,I)H) +(3 ! r ) . [ f ( x , u ) - f ( ~ u, ) ] / ( y ' - x ' ) = r.
P3.
( P , a ;0,A , B , c, (7r)r
04.
for
$,I
P3. x',112,x3,
As P3, C f ( x , U ) - f ( P , a ) 1 / ( y 2- x 2 ) = r .
P4.
D5.
f,2(P,
a;
( w )r As
P5.
A , B?c,1 ) H f o r P4.
X
P3,
[f (X, ). - f (P , ( p ,0 ; (7r) r
06.
f,3
on
a
x1,x2,q3,
- x 3 )=r .
A , B?c,1)H f o r P5. on P,
u
P, r
[ f ( P , .)-f(P, a)l/crPTP(a>u ) = r. time partial difference o f A
P6.
(v u). a, u E w(P ) HA ~ A
f(P,a)
Y N H U
f(P,u)
P3.
+ ( j !r).[f(P,U)-f(Pja)]/cW'TP(a,u)H=r.
07.
f,o(f',a;O,A,B,C,I)rr for (lr)r
P6.
by f,(P,a).
x 5,61
119
on,
‘H’,
Fig. 25. P7
on
a
on
6‘
f(P, a)
P(xp), b b‘Fb
on 25),
by
no
6. Equations of a space geodesic
...,
=P1), P,
,(yr- ,),
Pl(xl), . . .,P ,(x ,) =Pr) on
on
W%(Q1,..., = xf
p i 2 0, E i 2 0
< n < r),
P,
120
SPACE-TIME GEOMETRY
on
1
r-1
-" a&
=
,~ ( Q , , Q , + J / O
1 '8'
= E' = c3 =
A(P1,P2)= ... = A X ': = x,+i
P,.-,) = A l , 1
- I:,
y i j do yij,o (pn,a ) = 0 *
n, P n 9 J r P n + ,
(2),
Q,3.YMQ,,+I
r- 1
(1)
C
05.(4-6),
r- 1
aei
[(y$)
+ y : . k , l ~p , ) ( A x i . + ~ j A p ~ ) ( A +x kE ~ A P : ) ]=~ 0( .~ , ) I 1
1 I-
'i'
no
i.
p i =pi
+
ci=0 way
x 671
121
TRAJECTORIES
).-
p,"
a xi.
AZ
by [1953] p.
+ See
on p. 120.
122
9
P
-i
r-1 ‘-1
a
p1
a,
P?
p3
Q3
7-1
QP-1
P,
1
26. 0 2 ,
... = c ~ ~ ( u , - , , u= , )A S ,
C T ~ ( U ~ , = U ~ )
AX;=
/I:
1
-
=
9
= Ax:/As,
= y i j ( p n , an) 3
c ~ p ,(bn, , an)
2
A4
‘
(4)
/?, =
(5)
Y:;,; = ~ i j , k ( P n ,0,).
T5.1,
26, c tg(Un,lln+1)2
(3)
3
= ( A ~ I , O ++ ~ )2~A s , O + 1 [ ~ j k ( Q n , u n ) d t l , j d t ) , k ] * , i
= qn+ 1
- sf,
A
0 dsO, = A X , , A1)f
= CTQ, (on, u,)
+ ciAp,.
.
123
x 71
4 r-'I
/
3, "r
s
k
/ k
4
/
d
3 3
3 '2
)2I "2
C
p2
Pl
p3
p, - 1
p 4
Fig. 27. 0 2 ,
2
P5.1, Y j k (Qn,
un>
= Y j k (pn, = Y!"' + jk
r-
I
En (bt+ 1
On)
+ Yjk,l
9 1 t ' (
On)
1
-
y!"'EI jk,l P n .
A p Y ) + f$li
~ { p ~ p ~ '= A 0. s] /?,"+
Ar):, (6.3),
(6.4):
124
SPACE-TIME GEOMETRY
27 A 9; Y i j (Qnj
A ')"+
1
02,
= AX:,
un) = Y i j = IJ 0
2 an>
+ Y i j , o (Pn, an) c t p , , (an, u n )
+ y!?'IJ,O&
4,1 3
= CT (U? u),+ 1
=
+
c[~(b,a) S(U,U) - t ( b , ~ ) ] ~ + I . ~ ( hu),+ , z 7(a,u),,.
27,
A1):+i =AdX:+,
f
&(4n+1
- 4,)
(7),
qn
T5.1, AS'
= (AX:,
,)'
+ 2(Ar;+
AS.
by As2, 1
=
(p:+ 1)'
+. 2p:+
1
p, ,
0
8. Inertial systems
D1.
inertial system
P,, ..., P,. P,
a,, 6,
a,
a,
~ ~ ( a , , , ~ M a , ~ b,, ),
bn(bnlTN'bn).
by
x 8,91
125
CONCLUDING REMARKS
P1. y c. .j = 6.. rj’
Ei = 0 ,
hij
1
i=,j
0
i#j.
T1. by
Proof: P I , (1) no
9. Concluding remarks
on. no
by
go on go go
do
I N T R O D U C T I O N T O THE A P P E N D I C E S
on
A
129
F O U N D A T I ON
A.
1. Concepts
2,0,u,( , ),,,
Term concepts Class concepts Formula concepts
C,3 . 8, = , E , I,v, I-.
1;
B, 0,8,I .
2. Interpretations
Term Concepts: set class
A von 56-57).
pp. 11. ‘Zt’
‘t
A term
by
m
4
2, n, 2m + n
n
7.
by 12. (3 13. a u b
a . . , an}
‘;’
null set, no term kvhose elements are the term b, and the elements of term p. is ( a l , ...,a,) ‘u’,
‘Y’
14,5.
‘(’ ‘(A)’
left parenthesis ‘A 131
‘)’ A
132
FOUNDATION
[A
2
it’
pp. 16. 17. ( i x ) ( F , a ) a. by 18. ‘Bx’ ‘23s’
‘,’
comma OL. x F, ( ? x ) ( F , a ) description symbol ‘ I ’ p. ‘x variable, by u, u, w, x , y , z. string of bars, : , , .. .’.
x;
,,
‘D,’,‘nil’, ,.. .
‘z)’
Class concepts: 112. ‘Ed’ ‘d class’.
on
n-place relation by class of all sets x that satisfy F ;
113. x 3 F
114. ‘5F’
Formula concepts: ‘F
by ‘ { x l F } ’
A formula
$2, $
by
7.
by 115. ‘a=b’
H , I, J, K , L.
F,
‘a
b
‘a’
‘= ’
116. ‘ a ~ d ’ d’.
identity defined
element
member
‘b’
A
2,31
133
PUNCTUATION
1~{1,2)
A ‘{{a,b}}’
‘{a,b)’; { { a , b } } a, b. ‘b~a’,
‘a~b’ ‘aEc’.
{a,b}, ‘aeb’
‘bEc’
b={a,x,, ...,x,},
‘aEb’
{a,b}
E
117. ‘FIG’ ‘Fstroke
xl,
...,.Y,
F [1913]
by
05.1-6). 118. ‘(Vx)F’
‘V’
x,F
uni-
x’
versal ‘V’ D5.7. ‘ F true’. 8.
119. ‘I-F’
by ‘x=x’,
x
‘kx=x’
x
x=x.
‘x
‘y’
‘x’
by
3. Punctuation A point
dots. A right-point left-point
attached
by
scope senior
q
p,q
p
p up
q. up [1950] p. 32).
‘a€&, ‘ x 3 F ’ , ‘a=b’. D5.1-6,
134 +
FOUNDATION
1,
c,
-.
A,
v,
[A
v,
v (GlG)>>-((a = b) v (0 E -4)
(((F A (-
v .GIG:-a=
F A -G+F
bv a~ d .
4. Free and bound variables ‘x’
‘(Vx)F’, ‘ x ~ F ’ ,
‘(?x)(F,a)’; bound variable;
free variable
‘ F ( x , y , ...)’ ‘x’, ‘y’, .. ., ‘F(x,y , . ..)’ sentential function x , y , .... ‘F’ ‘x’ bound ‘F’. ‘ F ( x , y,... )’, ‘F(a,b,...)’ F ( x , y , ...) ‘x’ by ‘a’, ‘y’ by ‘b’,... ‘F’. ‘F(a)’ original x original ‘F(x)’. ‘F(x)’ ‘ x + a = x ’ , ‘F(a)’ ‘a + a = a’. ‘F(b)’ ‘ b+ a = b’ ‘ b+ b = b’, by ‘b’ ‘a’ ‘F(a)’. ‘ x does not occur free F’ by ‘xbF’ ‘ x free F’ by ‘xfF’. ‘xbF’ ‘F’,
‘b’
‘f’
5. Definitions book.
D1.
-F
for FIF.
-
117, ‘ F’ D2.
F A G for 117
‘F -(FIG). ‘FA
‘F and G ’ .
D3.
F v G for
‘not F’.
-FI-G.
‘F
G
A
51
135
DEFINITIONS
‘FV
‘For
D4.
G+H
GI-H.
‘G-PH’
IS
G
‘G+H’ then antecedent
D5.
F-G
H’,
‘if
‘FctG’ equivalent D6.
for
sufficient H’, H consequent.
F+G,A.G+F.
F ‘Fifandonly ifG’, FvC
for
Fis ‘ F o r G, but not F and G’.
x D10.
F
(3x)F 118
D9.
necessary andsuficient
T,
D1, ‘(3x)F’
F’.
D8.
‘F
-(F*G).
‘ F v G’ D7.
‘G+H’ implies necessary
at least one x existential quantij5er.
(Vx,, ...,xn)F for (Vxl) ...(Vx,)F, (3 x,, ..., x,)F for X J ...(3 xn)F. ( 3 ‘ x ) F ( x ) for ( V x , y ) . F ( x ) ~ F ( y ) + x = y .
118,15 F’.
D2,4, ‘(3’x)F(x)’
( 3 ! x ) ~ for
D7,9,
at most one
(~~)F.A(~’x)F.
!x)F’
exactly one x
F’.
D11.
( ? x ) F for 17,
D12.
(?x)(F,0).
!x)F
{u} for
(?x)F
x;
(1x)F=0.
+a,
{ u , b } for {~~,...,a,+~ f o}r
13, { a l ,...,a,)
a,, ...,a,.
136
FOUNDATION
D13.
(a> f o r a , >.
‘(al, ...,a,)’ 014.
ordered n-tuple.
V for x 3 x = x .
x=x Dt5.
A for
x#x, A
no x no D16-23 D16.
‘A’
A“
for
D17.
AuB i s union A D18.
AnB section of A D19.
for
X~XEAVXEB.
‘ A = B’ B’.
B,
B. A n B for
~ ~ ~ E A A X E B .
is
A and B,
inter-
B. A c B for
(Vx).xeA+x~B.
A
‘A
B’,
B’. A c B for
AEBA-BGA.
‘AcB’ ‘A proper subclass D21.
A.
complement
A or
‘AcB’ subclass D20.
‘B’
A,
not AuB
null class,
x3,-x~A.
A”
D22.
universal class,
113
x,
A E B for
‘A
‘A
B
B’. AGBAEGA. ‘A
B P6,7
(xl, ..., x , ) 3 F for y 3 (3 xl, ...,x,) .y = (xl, . ..,x , )
A
F.
8
equal
137
DEFINITIONS
...,x,)3F’
y
F.
‘{xl,...,x,lF}’. D23.
‘Ji& for (V y ) .y E d + ( 3
XI,
...,x,) y = ( X I , . ..,x,).
‘%,,d’ ‘d
‘d n=2
relation’. ternary
n=3, a?
d
is D24. D25.
a 2 b for ( a , b ) E d . (W reflexive d)for
(Vx). X E d g _ t X W X . D26.
D32.
(9 irrej’exive d) for (V x). x E d-+ X W X . (92 symmetric d)for ( v X , y ) .X , y € . d A X ~ ~ + Y ~ ’ x - . (9 asyrnmelric d) for (vX,y). X,YE d A XBY+ wY9.X. antisymmetric d) for ( V X , ~ )~. , Y E . ~ A x ~ Y A ~ ~ x - + x = ~ . (92 transitive d) for (vX,y,Z). X , y , Z € d A X % y A y & ? Z + X ~ Z . semitransitive d)for (vX,y,Z). X , y , Z € d A X . ~ y A X ~ Z ~ y ~ Z . (92 equivalence d) for
D33.
9 (2
N
D27. D28. D29. D30. D31.
A’ D34.
d, strict partial ordering
&‘)
for
d.
(W partial ordering d) for 3 a2 D35. %M-1 for 9 (Ji2.% A (v X,y, Z). X B y A X.%Z-+y= Z . ‘!RIM‘92 many-one relation, function’. D36. for g 2 2A (v X , y , Z ) . YWX A Z.!%X+y=Z . -,&’ ‘92 one-many relation’.
n-place binary
9
138
[A
FOUNDATION
D37.
%1-1 for9 % M - 1 9 2 ~ % l - M 2 . ‘2 one-one relation, correspondence’.
-
6. Axioms
1,
I
28.
0
I1 8.
null 7. Postulates of classification Yariabfes: -
‘4’
Bxl
!€33x
23,
Bll I I .. . ;
.. .
23x4 23DX.
...,
‘x’ by
‘BD,~’, ....
uII,...
18,
B, ,,B, [1950] p. 187).
Terms: -
Bx4Zx. 2011,....
by
D
5-7
A
139
POSTULATES OF CLASSIFICATION
71
Za, Zb+Z(aub).
PT2.
A2
D12,13,
( a } , (a,b), ( a ) , ...,a,)
a
b
...,un}
(a,b).
a x , %a, g F + 5 ( ( 7 x ) ( F , a ) ) .
PT3.
‘a’
A2
by
:
Bx, PT4.
x)F).
(1)
(Ld,t-U Ed-+Za.
‘Cd’ Formulas: -
PF1. PF2. PF3.
Za,Cd4g(aEd).
PF4.
SF,S G + S ( W ) . 01-6, -F, F A G , F v G, F-rG, F-G,
PF5.
B&
F F v G.
SF+?i((Vx)F). x
‘(3x)F’, ‘(3’x)F’, Classes: -
PC1.
Bx, 3F4C(X 3 F ) . %at”.
PC1:
G
‘(I !x)F’
F
140
FOUNDATION
T1. Proof.
3
=a).
Al, PV2, PT1 P F I :ZD1, ZU-D
=a).
PC1: BD,, %(Dl
TJ,
(14).
a.
no
d, .d,
a
D21).
a-.d
8. Postulates about true formulas Propositions: P1.
I-F+G4I-G.
modus yonens. F
‘F
[ 19251 P1.
P1
‘1’) Quantrjiers: FVI,
upon P3. P4. P5.
pp. 93-94):
%x,
Bx, Z a , S F (x) --o I- (V x) F (x). +F (u). !Ex, B F ,
P3
by P4 P3,
F
x, F
P2 D4,l.
A
81
141
POSTULATES ABOUT TRUE FORMULAS
‘x’ bound xfF
4).
‘F’
‘(Vx)F’
‘(Vx)F’ P4 x,
F.
go
F
a. ‘2x,S F , 3 G
F P5
‘x’
‘F’(‘xbF’
x,F
F
x,
Identity and ixembership: P6. P7.
Z U , ~ ~ , ~ E ~ ~ U = ~ ~ . U E ~ ~ ~ E E . ZU,Z b 4 k a 3 b + a = b. equality [1958] pp. 52-53). P6
extettsionalily, a=b,
P12
bed;
ued
a=b, 6.
P7, D21),
u,b
a= 6.
Description: P8.
B x , %a, Z b , S F ( x ) - + tF(a)/\(t/x).F(x)->x=a:-+(? x ) ( F ( x ) , b ) = a . B x , Zb, 5 F ( x ) 4 t (3 ! x ) F ( x ) .-+(7 x ) ( F ( x ) , b )= b.
-
P9.
([1958] P8,
by p. 49),
1
no
x
Sets and classes:
P10. P11. P12.
( ? x ) ( F ( x ) , b ) = a ;P9 (ix)(F(x),b)=b.
F, F,
a
17.
-
€0.
Za+k - a
ZU,2 6 , Z C + ~ C E ( U ~ ~u) v+cW = b. E Bx,Za, SF(x)4ta E.X3F(x):-F(a).
0, u, 12,3,13, P10
no
aub
CEU
x
3
([1958] p, 65). P11 c
b; F’
P12
‘a ‘a
F’. [1958]).
book.
142 9. Application of symbolic logic
SL by upon Concepts : N,0, ‘, Interpretations: JV n’ successor Axionzs: K N , I-OEJV. Postulate : t n EN-+ I-n’E M .
0
zero,
n.
(2)
6,1,
SL.
E
by
.A’4xo.
~:J$-,~OE
20
0 O‘,O”
(3)
‘2’, ...). SL book.
...
B. LOGICAL THEOR E MS
[1925]
[1958].
As
1. Rules of inference T1.
t F , I-
t
k T2.
tF+ tF-+
t
H 4 t F-+ H . tF-+H. i-
-+
T3.
T4. T5.
T7.
. T9. T10. T11. T12. T13.
+ H , H + Z 4 tF+ . tF4.G-+H,tI-+G4tF-+.I-tH. EF,, .._,kF,-+!-F, A ... AF,. t F 4 G l , ..., tF-+G,+tF-+G, A ... A ..., v ... v F,-+G. A
wF4t-F v t(Vx)F,t(Vx).F-+G-+t(Vx)G. t(V x). F-+ t(V x).F+ H . s,t
t
t sFt
-+
t sG t. 143
144
LOGICAL THEOREMS
2. Propositions
T1. T2. T3. T4. T5. T6. T7. T8. T9. T10. T11. T12. T13. T14. T15. T16. T17. T18. T19. T20. T21. T22. T23. T24. T25. T26. T27. T28. T29. T30.
- . F A -F. F V -F. F V -F. FAG.v.FA-G.v.-FAG.v.-FA v .-F F+.G+F. -F+.F+G. F A .F+G:+G. A.F+G:+-F. F + G A -G.+-F. F A.F v - F A .F v F+G.+.F AH+G A H . - + . F v H - + G vH . F - t G . A .H+I : + . F v H+G v I . F , A ... A F,+F,(i= 1, ..., n ) . F,+F,v ... v F , ( i = l ...., u). F-F. -F-F. F A F-F. F v F-F. F+G.t+.-G+-F. F-+G.++.-FvG. F+G.-. -(F A G). F-G.-.G-F. F-G.++. F. FAG--(F+-G). F V G++.-F+G. F V G c t : F v G . A . - ( F A G). F , A ... A Fn-Ftl A ... A F,,,, F, v ... v Fn-Fi, v . .. v F F , v ... v Fn-Fil v ... v F,,,
-
-
- -
( i l ,..., in) T31.
F-+.G+H:-:G+.F+H.
(1,. . .,n).
A
-GAH. -GA - H .
B
2,31
QUANTIFIERS
T41.
F A G+H.++.F-+.G+H ++.FA -.H-+-G. F+G. A . H - + I : + + . F A H - + G A I . F+G1 A ... A G,.tt:F-+G,. A ... A .F-+G,. F+G, v ... v G , . o : F - + G , . v ... v .F+G,. F , A ... AF,-+G.++:F,-+G. v ... v .F,,-tG. Fl v ... v Fn-+G.++:Fl-+G.A .., A .F,-+G. -(Fl A _..A Fn)++~F1v ... v -F,. -(Fl v ... vF,)++*F~ A ... A -F,. FA v ... v G,:+-+:F A v ... v . F A G , . F v . G , A ... A G , : ~ - , : F V G , . A... A.FvG,.
T42.
FvG.A.HvI:++:FAH.v.FAI.v.GAH.v.GAI.
T32. T33. T34. T35. T36.
T37. T38. T39. T40.
3. Quantifiers T1.
T2. T3. T4.
T5. T6. T7. T8.
T9. TlO. T11. T12. T13. T14. T15. T16. T17.
T18. T19. T20.
(V X)F(X).++(~’)F(L’). (V X ) ( F -+ :(V x) F . +(V x)G. (V x)(FctC).-+ :(V x) F.++(V x)G.
( V X ) F A G . + + : ( V X ) FA. ( V X ) G . (Vx)F.v(Vx)G:+(Vx)FvG. xbF -+k( V x )F.++F. xbF+ k(V x)F A G.-F A (V x)G. xbF+k(Vx)F v G.*F v ( V x ) G . x)F(x).++(3Y ) F ( Y ) . ( 3 x)F.-+(3 x)G:-+(3 x).F+G. x) F A -+: x) F. A x)G. (3x)FvG.++:(3x)F. v ( 3 x ) G . xbF-+ k(3 x)F.++F. xbF-+t-(3 x)(F+G).++.F-+(3 x)G. x b F 4 k( 3 x)F A G.++F A ( 3 x ) G. x b F 4 k(3 x ) F v G.-F v ( 3 x) G . ( 3 X) F.-(V x)F, ( 3 x)-F.++-(Vx)F, x)F.++(Vx)-F, x)F.++ (V x) F . (Vx)(F-+G).++-(3X).FA -G. xbG+ k(V x)(F-+G).++: x )F. +G. x)(VY)F.-+(V’F x)F.
- -
- -
145
146
T21. T22.
[B
LOGICAL THEOREMS
( 3 !x)F(x).-(3 y)(Vx).F(x)++x=y. ( 3 x)x y 3 F ( y ) . +(3 ! x) x = y 3 F ( y ) .
4. Descriptions
T1. T2. T3. T4.
11 '(7
s)(x = a).
(V x) (F-G). +( X) = ( X) (3 ! x)F(x).+ : a =( 1 x)F(x).++F(a). (V s).F(x)++x = a :-t :a = ( 1 x) F(x).-F
5. Identity
Tl. T2. T3. T4. T5. T6.
=
= 6 -+ .F(a)c*F(b). a = 6 A F(a)-a = 6 A F ( 6 ) . u = 6++(V X).X = UHS = 6. F (a)-(V x). x = a F (x). F(U)++(3 X).X = U A F(X). --f
6. Set or class relations
Tl. T2. T3. T4.
T5. T6. T7. T8. T9. T10. T11.
T12. T13. T14.
T15.
A s A. AEBABGC-+AGC. AgB-AcBvA=B. -AcA, A C B-t B C A , A cB A B c C + A c C . a = b-a = b.
-
3
AuA-A. AuBEBuA. A u( BuC )= ( Au B )u C. AnAEA. AnBA. An(BnC) (AnB)nC. ( A u B ) " =A " n B - . (AnB)" =A" u B " . A u( B nC )=( Au B ) n ( Au C ) .
-
(a).
3-6
B
671
147
RELATIONS
T16. T17. T18. T19. T20. T21. T22. T23. T24. T25. T26. T27.
( i l ,...,in) T28. T29.
..,n).
..., n } + a i ~ { a , ,..., a,,}. (a,,
..., u n ) = ( b , , ..., b n ) ~ =ub ~, A ... ~ u , , = b , , .
7. Relations Tl.
2
T2. T3.
-92 99 3 -92
d
d. d.
d-3
d d.
REFERENCES
37 A.
288. Axiomatic Set Theory
Theory and Applicaiions of Distance Geometry
Co.,
Foundations of Geometry
K., Statistical Theory and Methodology in Science and Engineering The Geometry of Geodesics Introduction to Symbolic Logic and its Applications Legons sur la GPome'irie des Espaces de Riemann Methods of Mathemaiical Physics, Vol. 1 Mathematical Methods of Statisiics Gravitation and Relativity L., Riemannian Geometry A., 20 1079. The Foundations of Euclidean Geometry A., Intrinsic Geometry of Ideal Space,
I
I1 Co.,
C., Introduction to Metamaihemutics Rev. 129 2371. Appl. 2
W., W. W.
J.,
67. 68 D The Rotation of the Earth 19
523.
32.
4 337. Axiomatik der Relativistischen Rairm-Zeit-Lehre
The Philosophy of Space and Time A., A Theory of Time and Spare The Elements of Muthematical Logic
149
150
REFERENCES
RUSSELL, B., The Analysis of Matter (Dover Publications, New York; Republication 1927 ed.). SAGNAC,G., Compt. Rend. 157 (1913) 708, 1410. H., Trans. Amer. Math. SOC.14 (1913) 481. SCHEFFER, SCHNELL, Eine Topologie der Zeit in Logisrischer Darstellung (Munster Westfallen, 1938). D., The Elements of Non-Euclidean Geometry (Dover Publications, New SOMMERVILLE, York, 1958). SUPPES,P., Introduction to Logic (Van Nostrand Co., Princeton, 1957). SYNGE,J., Relativity: The Special Theory (North-Holland Publishing Co., Amsterdam, 1956). TARSKI,A., Introduction to Logic and to the Methodology of Deductive Sciences (Oxford University Press, New York, 1959). WHITEHEAD, A. and RUSSELL, Principia Mathemafiea, Vol. 1 , 2nd ed. (Cambridge University Press, Cambridge, 1925).
SYMBOLS
1. Latin 2.5 2.1 5.4
C
29 29 21 11
D V 3.2
41
c
31 75 52 56 68 11
3.7 5.2 2.2 3.2 VII 2.3
C
2.6 2.2
D
2.3 2.4 2.7 2.8 D Vl 4.4 4.1
6.1 151
30 30
31 31 70 70 58 42
76
152
SYMBOLS
c I1 2.1
12 52
9
c I1 4.3 11 4.1 11 4.2
c I1 5.4 I1 5.1 5.3 4.1 4.2 2.6 7.1 3.4 3.3 D VLIT 6.1 7.5 7.4 1.1 3.2
D It 5.6
....HJ
...>
I1 5.5 I1 5.7
6 111 3.1 D I11 3.2
I1 4.5 I1 4.4
14 14 17 19 19 21 84 85 69 70 77 71 71 87 78 78 90 113 23 22 23 54 26 27 18 18
153
SYMBOLS
D 3.2 D 4.3 D 4.4 D 6.1 D 6.2 D IX 6.3
56 73 73 104 105 105
c v 3.8 D D D D D D
2.1 3.5 3.4
40 41 54 57 57 57 52 114 113
D D D
4.3 3.3 7.1
11 35 113 121
D
3.1 Eq. 3.2
D
v
4.2
2.1
67
3.1
V
D D
53 62
w
D
2.2
39
D
5.1
85
D
2.4
69
C II 3.2 3.1 3.2
D D
13 13 13
154
SYMBOLS
2. German
IA 1 IA D IA
D
2.9
Eq. 3.11 D A 5.23 D 4.1 D VII 2.5 A 2.1 Eq. 3.9 I A 2.8 I A 2.10
132 134 132 114 132 134 55 137 57 69 131 55 132 132
D 5.3 Eq. 7.5 D 5.2 Eq. 7.6 4 Eq. 7.4 5 Eq. 8.1 D VII 7.2 D 7.3 D X 4.1 3.6 D 3.1 D 3.2 D D 4.2 3.1
117 122 116 122 120 122 120 125 77 77 114 54 70 71 72 72 54
2.12 4.2 2.14
3. Greek
SYMBOLS
155
Page Eq. VI 3.7 D VI 4.1 D VII 2.5 D VI 5.1 D VI 5.2 D VZ 6.1 Eq. VI 3.10
54 57 69 58 59 62 55
Eq. 12.2 D I1 5.2 D I11 1.1 D I11 1.2 D TI1 1.3 D 111 2.1 D I11 2.2 D IV 3.1 D IV 4.1 D IV 4.2 D V 1.1 D V 1.2 D V 1.3 D V 1.4 D V 2.1
5 20 25 25 25 26 26 32 33 34 36 37 37 37 38
D VI 6.3 D VI 6.4 D VII 2.2 D VII 5.1 D VIII 2.1 D VIII 2.2 D IX 3.1 D IX 3.2 D IX 4.1 D IX 4.2
63 63 68 73 79 81 94 95 97 98
4. Miscellaneous
156
SYMBOLS
.P
D 4.3 D 3.1 DX
100 111
A 2.2 I A 2.3 D A 5.11 A 2.13 D A 5.22 I A 2.15 A 2.16 A 2.18 I A 2.19 D A 5.1 D A 5.2 D A 5.3 D A 5.4 D A 5.5 D A 5.6 D A 5.7 D A 5.9 D A 5.10 D A 5.12 D A 5.13 D A 5.17 D A 5.18 D A 5.19 D A 5.20 D A 5.21
131 131 135 132 136 132 132 133 133 134 134 134 135 135 135 135 135 135 135 136 136 136 136 136 136
5. Logical
E
CONVENTIONS
book. a, b, c, d, e .
F,
H, I, J . i, j , k , I, m, n,p , q, r, s. A , C, D , I , P, T. A, I, P, T. it.
A , B, D. F, H. Q , R, S , T, U . y , q, Y, s, t , (5, R , S, T. x, 9,3. a, b, c, d.
E , 0.
: U , V , W, X , Y , Z . u, u,
157
II',
x, y , z.
ABBREVIATIONS
102 6 106 69 77 5 5 54 57 76 9
GR
ML
SG
159
INDEX
65, 66
011112.1 24
5
24 4 1x2
4, 4
55
13
33 DV 1 . 1
1x5 DV1.
131,
1
DVI3.
52
107
67
56, 56
57 16
.I
65, 66 3, 4
28,
55 55
3
161
162
INDEX
11
3 54 54 for 6
1x6 65
-
1
93
93, 115
10 5 11
VI
11
go
1115.4
. . ., 5
5 4
49, 52
24, 26 107
19 26 108
94 3 10
54
3 3
5
3, 5
58 - 58
- 116
3, 4 54 5
5
INDEX
14, 11
50 54
107 4
-5 5
15, 07 62 14 56
57 69 66
65
62 62
67 1 1113.2
131, 133,
44 65
bound I , Sec.
