FOUNDATIONS OF SCIENCE N
A NDRÉKA –M ADARÁSZ –N ÉMETI –S ZÉKELY
On Logical Analysis of Relativity Theories
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• •
• •
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c
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d ≥ 2 d
{ B , IOb , Ph , Q , + , · , W } ,
B Q IOb
Ph B +
· Q W
2 + d B
Q IOb( k ) Ph( p ) k p
W( k, b, x
1, . . . , x
d−1, t ) k b
x
1, . . . , x
d−1, t x, . . . , x
d−1t
x = y
x y
· + Q
¬ ∧ ∨ →
↔ ∃ ∀
+ ·
AxFd Q , + , ·
• Q , + , ·
• ≤ x ≤ y ⇐⇒ ∃
dz x + z
2= y
• Q ∀ x ∃ y x = y
2∨ − x = y
22010-4.indd 208
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+ ·
0 1 − /
· · 0 + 0 1 − / √
AxFd
Cont
AxFd
¯
x, y ¯ ∈ Q
nx ¯ y ¯ n Q n ≥ 1
|¯ x | =
dx
21+ · · · + x
2n, x ¯ − y ¯ =
dx
1− y
1, . . . , x
n− y
n.
¯
x
s=
dx
1, . . . , x
d−1x
t=
dx
d¯
x = x
1, . . . , x
d∈ Q
dAxPh
∀ m ∃ c
m∀¯ x y ¯ IOb( m ) →
∃ p Ph( p ) ∧ W( m, p, x ¯ ) ∧ W( m, p, y ¯ )
↔ | y ¯
s− x ¯
s| = c
m· | y
t− x
t| . AxPh
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AxPh x ¯ y ¯
AxPh
c
m= 0 AxSm
m x ¯
m x ¯
ev
m(¯ x ) =
d{ b : W( m, b, x ¯ )} . AxEv
∀ mk IOb( m ) ∧ IOb( k ) → ∀¯ x ∃¯ y ∀ b W( m, b, x ¯ ) ↔ W( k, b, y ¯ ) .
AxPh AxFd AxEv
AxPh
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AxSf
∀ m IOb( m ) →
∀¯ x W( m, m, x ¯ ) ↔ x
1= 0 ∧ x
2= 0 ∧ x
3= 0 . AxSf
AxSf
AxSm
∀ mk IOb( m ) ∧ IOb( k ) → ∀¯ x y ¯ x ¯
y ¯
x
t= y
t∧ x
t= y
t∧
ev
m(¯ x ) = ev
k(¯ x
) ∧ ev
m(¯ y ) = ev
k(¯ y
) → | x ¯
s− y ¯
s| = |¯ x
s− y ¯
s| ,
∀ m IOb( m ) → ∃ p Ph( p ) ∧ W( m, p, 0 , 0 , 0 , 0) ∧ W( m, p, 1 , 0 , 0 , 1) . AxSm
AxSm
SpecRel =
d{AxFd , AxPh , AxEv , AxSf , AxSm} .
SpecRel
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SpecRel ∀ mk x ¯ y ¯ IOb( m ) ∧ IOb( k )
∧ W( m, k, x ¯ ) ∧ W( m, k, y ¯ ) ∧ x ¯ = ¯ y → |¯ y
s− x ¯
s| < | y
t− x
t| .
m k
w
mk(¯ x, y ¯ ) ⇐⇒
dev
m(¯ x ) = ev
k(¯ y ) . SpecRel
( y
t− x
t)
2− | y ¯
s− x ¯
s|
2d
¯ y, x ¯
SpecRel ∀ m, k IOb( m ) ∧ IOb( k ) → w
mkSpecRel
AxEv
AxEv AxEv
AxMeet
nn
∀ mkb
1. . . b
nx ¯ IOb( m ) ∧ IOb( k ) ∧ W( m, b
1, x ¯ ) ∧ . . . ∧ W( m, b
n, x ¯ )
→ ∃ y ¯ W( k, b
1, y ¯ ) ∧ . . . ∧ W( k, b
n, y ¯ ) . AxMeet
1Meet
ωAxMeet
nAxMeet
nAxMeet
n+1AxEv Meet
ω2010-4.indd 212
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AxEv AxMeet
n+1AxMeet
nAxMeet
nAxMeet
n+1Meet
ωAxEv
AxMeet
nAxMeet
n+1Q = {0 , 1 , . . . , n } B = { b
i: i ≤ n }
b
00 , . . . , 0
W( b
0, b
i, x ¯ ) x ¯ = 0 , . . . , 0
k = 0 b
kb
ii, . . . , i
i ≤ n W( b
k, b
i, x ¯ ) x ¯ = j, . . . , j i = j
n AxMeet
nn + 1 { b
0, . . . , b
n} b
0AxMeet
n+1Q Q
n n AxMeet
nn Meet
ωb
0{ b
1, b
2, . . . } AxEv
Ax(c = 0)
∀ mp x ¯ y ¯ IOb( m )∧Ph( p )∧W( m, p, x ¯ ) ∧W( m, p, y ¯ )∧ x
t= y
t→ x ¯
s= ¯ y
s.
AxMeet
3, AxFd , AxPh , Ax(c = 0) AxEv AxMeet
2, AxFd , AxPh , Ax(c = 0) AxEv Meet
ω, AxFd , AxPh AxEv
AxFd
Q
dc c = 0
c = 0
AxFd AxPh
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m c
mAxFd AxPh Ax(c = 0) m
AxFd c
m= 0
AxPh m
m k x ¯
AxEv x ¯
ev
m(¯ x ) = ev
k(¯ x
) x ¯
y ¯ = x
1+ c
m, x
2, . . . , x
d−1, x
t+ 1
¯
z = x
1− c
m, x
2, . . . , x
d−1, x
t+ 1 w ¯ = x
1, . . . , x
d−1, x
t+ 2
AxPh p
1p
2p
3p
1, p
2∈ ev
m(¯ x ) p
2, p
3∈ ev
m(¯ y ) p
1∈ ev
m(¯ z ) p
3∈ ev
m( ¯ w ) m
c
mp
1p
2¯ x
p
1p
3AxMeet
3AxMeet
2k
¯
x
k p
1p
2x ¯
k
p
1p
2k
3 m
¯
x
k p
1p
2p
∈ ev
k(¯ x
)
p
∈ ev
k(¯ x
) p
∈ ev
k(¯ x
) p
∈ ev
k(¯ x
) k
p
p
AxMeet
3m { p
1, p
2, p
}
{ p
1, p
2, p
} m x ¯ x ¯
m p
1p
2m
p
p
x ¯ AxMeet
3k p
p
k x ¯
k p
1p
2b W( m, b, x ¯ ) AxMeet
3k
p
1p
2b x ¯
p
1p
2¯
x
b ev
m(¯ x ) ⊆ ev
k(¯ x
)
ev
k(¯ x
) ⊆ ev
m(¯ x ) ev
m(¯ x ) = ev
k(¯ x
)
Q , + , · ω B = { m, k } ∪ { b
i: i ∈ ω } ∪ { p : p } m k
1 m k
p x ¯ x ¯ ∈ p m b
i¯
x x
t= 0 k b
0, . . . , b
n, . . . b
ix ¯ x
t= i
¯
x {¯ y ∈ Q
d: y
t= i }
AxFd AxPh Ax(c = 0)
m
d = 2
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m k
p
1p
1p
2p
2p
3p
3¯ x
¯
y ¯ z
¯ w
¯ x
b
b 1
1
c
mc
mk 2 AxMeet
2{ b
i: i ∈ ω } m AxEv
B = { m, k } ∪ { b
i: i ∈ ω } ∪ { p : p }
AxFd AxPh c = 0
m k n
Meet
ωm
{ b
i: i ∈ ω } AxEv
AxEv AxMeet
3AxSm AxEv AxMeet
3SpecRel AxMeet
3AxMeet
2d 3
d = 2
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SpecRel
Ob( m ) ⇐⇒ ∃
db x ¯ W( m, b, x ¯ ) .
AxCmv
AxCmv AxCmv
AxCmv
AxEv
−m k k
∀ m, k ∈ Ob W( m, k, x ¯ ) → ∃¯ y ev
m(¯ x ) = ev
k(¯ y ) . AxSf
−∀ m ∈ Ob ∀¯ x W( m, m, x ¯ ) → x
1= x
2= x
3= 0
∀¯ x y ¯ W( m, m, y ¯ ) ∧ W( m, m, x ¯ ) → ∀ t x
t< t < y
t→ W( m, m, 0 , 0 , 0 , t ) .
SpecRel
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AxDf
AxDf AxDf
AxDf
Cont Q
Cont AccRel
Cont Cont
AccRel Q Cont
Cont AccRel
Cont
SpecRel
AccRel =
dSpecRel ∪ {AxCmv , AxEv
−, AxSf
−, AxDf} ∪ Cont .
AccRel
TwP m
k e
1e
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e
1e
2∀ m ∈ IOb ∀ k ∈ Ob ∀¯ x x ¯
y ¯ y ¯
x
t< y
t∧ x
t< y
t∧
m, k ∈ ev
m(¯ x ) = ev
k(¯ x
) ∧ m, k ∈ ev
m(¯ y ) = ev
k(¯ y
) → y
t− x
t≤ y
t− x
t∧
y
t− x
t= y
t− x
t↔ enc
m(¯ x, y ¯ ) = enc
k(¯ y
, y ¯
) , enc
m(¯ x, y ¯ ) = {ev
m(¯ z ) : W( m, m, z ¯ ) ∧ x
t≤ z
t≤ y
t}
AccRel TwP AccRel − AxDf TwP AccRel − Cont TwP Th(R) ∪ AccRel − Cont TwP
Cont
AccRel TwP
AccRel
AccRel
GenRel AccRel
AccRel AxSf
−AxEv
−AxSf AxEv
AccRel → GenRel
AxSm AxPh
−AxSm
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AxPh
−AxSm
−GenRel =
d{AxFd , AxPh
−, AxEv
−, AxSf
−, AxSm
−, AxDf } ∪ Cont .
GenRel SpecRel
GenRel
GenRel
GenRel GenRel
GenRel
ϕ GenRel
ϕ GenRel GenRel ϕ ϕ
SpecRel SpecRel
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Compr
Compr Compr
GenRel
+=
dGenRel ∪ Compr . GenRel
+GenRel
+GenRel
+GenRel
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AccRel
∗
∗
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