FOUNDATIONS OF SCIENCE NANDRÉKA–MADARÁSZ–NÉMETI–SZÉKELY

On Logical Analysis of Relativity Theories

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••••

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ANDRÉKA–MADARÁSZ--NÉMETI–SZÉKELY: ON LOGICAL ANALYSIS… 207

c

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d ≥ 2d

{B , IOb,Ph, Q ,+, ·, W },

B Q IOb

Ph B +· Q W

2 + d B

Q

IOb(k) Ph(p) kp

W(k, b, x1, . . . , xd−1, t) k b〈x1, . . . , xd−1, t〉 〈x, . . . , xd−1〉 t

x = yx y

Q

· +¬ ∧ ∨ →

↔ ∃ ∀

+ ·

AxFd 〈Q ,+, ·〉• 〈Q ,+, ·〉

• ≤ x ≤ yd

⇐⇒ ∃z x + z2 = yQ

• ∀x ∃y x = y2 ∨ −x = y2

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ANDRÉKA–MADARÁSZ--NÉMETI–SZÉKELY: ON LOGICAL ANALYSIS… 209

+ ·0 1 − /

0· · + 0 1 − /

√

AxFd

Cont

AxFd

x, y ∈ Qn x y n Q n ≥ 1

|x|d

=√

x2

1+ · · · + x2

n, x − y

d

= 〈x1 − y1, . . . , xn − yn〉.

xs

d

= 〈x1, . . . , xd−1〉 xt

d

= xd

x = 〈x1, . . . , xd〉 ∈ Qd

AxPh

∀m ∃cm ∀xy IOb(m) →(∃p Ph(p) ∧ W(m, p, x) ∧ W(m, p, y)

)↔ |ys − xs| = cm · |yt − xt|.

AxPh

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AxPh x y

AxPh

cm �= 0 AxSm

m xm x

evm(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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ANDRÉKA–MADARÁSZ--NÉMETI–SZÉKELY: ON LOGICAL ANALYSIS… 211

AxSf

∀m IOb(m) →(∀x W(m,m, x) ↔ x1 = 0 ∧ x2 = 0 ∧ x3 = 0

).

AxSf

AxSf

AxSm

∀mk IOb(m) ∧ IOb(k) → ∀xyx′y′ xt = yt ∧ x′

t= y′

t∧

evm(x) = evk(x′) ∧ evm(y) = evk(y

′) → |xs − ys| = |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

SpecReld

= {AxFd,AxPh,AxEv,AxSf,AxSm}.

SpecRel

�

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SpecRel � ∀mkxy IOb(m) ∧ IOb(k)

∧ W(m,k, x) ∧ W(m,k, y) ∧ x �= y → |ys − xs| < |yt − xt|.

m k

wmk(x, y)d

⇐⇒ evm(x) = evk(y).

SpecRel

(yt − xt)2 − |ys − xs|

2 dy, x

SpecRel � ∀m,k IOb(m) ∧ IOb(k) → wmk

SpecRel

AxEv

AxEv

AxEv

AxMeetn n

∀mkb1 . . . bnx IOb(m) ∧ IOb(k) ∧ W(m, b1, x) ∧ . . . ∧ W(m, bn, x)

→ ∃y W(k, b1, y) ∧ . . . ∧ W(k, bn, y).

AxMeet1

Meetω

AxMeetn AxMeetn AxMeetn+1

AxEv Meetω

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ANDRÉKA–MADARÁSZ--NÉMETI–SZÉKELY: ON LOGICAL ANALYSIS… 213

AxEv � AxMeetn+1 � AxMeetn

AxMeetn � AxMeetn+1

Meetω � AxEv

AxMeetn

AxMeetn+1 Q = {0, 1, . . . , n} B = {bi : i ≤ n}b0 〈0, . . . , 0〉

W(b0, bi, x) x = 〈0, . . . , 0〉k �= 0 bk bi 〈i, . . . , i〉

i ≤ n W(bk, bi, x) x = 〈j, . . . , j〉 i �= jn AxMeetn

n + 1{b0, . . . , bn} b0 AxMeetn+1

Q Q

nn AxMeetn

n Meetω b0

{b1, b2, . . .} AxEv �

Ax(c �= 0)

∀mpxy IOb(m)∧Ph(p)∧W(m, p, x)∧W(m, p, y)∧xt �= yt → xs �= ys.

AxMeet3,AxFd,AxPh,Ax(c �= 0) � AxEv

AxMeet2,AxFd,AxPh,Ax(c �= 0) � AxEv

Meetω,AxFd,AxPh � AxEv

AxFd

Qd c c = 0c �= 0

AxFd AxPh

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mcm

AxFd AxPh Ax(c �= 0) mAxFd

cm �= 0AxPh m

m k xAxEv x′

evm(x) = evk(x′) x′ y = 〈x1 + cm, x2, . . . , xd−1, xt + 1〉

z = 〈x1 − cm, x2, . . . , xd−1, xt + 1〉 w = 〈x1, . . . , xd−1, xt + 2〉AxPh p1 p2 p3 p1, p2 ∈ evm(x)

p2, p3 ∈ evm(y) p1 ∈ evm(z) p3 ∈ evm(w) mcm p1 p2 xp1 p3

AxMeet3 AxMeet2 kx′ k p1 p2 x′ k

p1 p2 k3 m

x′′ k p1 p2 p′ ∈ evk(x′)

p′′ ∈ evk(x′′) p′ �∈ evk(x

′′) p′′ �∈ evk(x′) k

p′ p′′AxMeet3 m {p1, p2, p

′}{p1, p2, p

′′} m x xm p1 p2 m

p′ p′′ x AxMeet3 k p′ p′′

k x′ k p1 p2

b W(m, b, x) AxMeet3 kp1 p2 b x′ p1 p2

x′ b evm(x) ⊆ evk(x′)

evk(x′) ⊆ evm(x) evm(x) = evk(x

′)

〈Q ,+, ·〉ω

B = {m,k} ∪ {bi : i ∈ ω} ∪ {p : p } m k1 m k

p x x ∈ p m bi

x xt = 0 k b0, . . . , bn, . . . bi x xt = ix {y ∈ Qd : yt = i}

AxFd AxPh Ax(c �= 0)m

d = 2

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ANDRÉKA–MADARÁSZ--NÉMETI–SZÉKELY: ON LOGICAL ANALYSIS… 215

m k

p1

p1p2

p2

p3

p3

x

y z

w

x′b

b

1

1

cm cm

k 2 AxMeet2

{bi : i ∈ ω} mAxEv

B = {m,k} ∪ {bi : i ∈ ω} ∪ {p : p }

AxFd AxPh c = 0m k n

Meetω m{bi : i ∈ ω} AxEv �

AxEv AxMeet3

AxSm

AxEv AxMeet3 SpecRel

AxMeet3 AxMeet2

d 3d = 2

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SpecRel

Ob(m)d

⇐⇒ ∃bx W(m, b, x).

AxCmv

AxCmv

AxCmv

AxCmv

AxEv− m k k

∀m,k ∈ Ob W(m,k, x) → ∃y evm(x) = evk(y).

AxSf−

∀m ∈ Ob ∀x W(m,m, x) → x1 = x2 = x3 = 0

∀xy W(m,m, y) ∧ W(m,m, x) → ∀t xt < t < yt → W(m,m, 0, 0, 0, t).

SpecRel

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ANDRÉKA–MADARÁSZ--NÉMETI–SZÉKELY: ON LOGICAL ANALYSIS… 217

AxDf

AxDf

AxDf

AxDf

Cont Q

Cont AccRel

Cont

Cont

AccRel Q

Cont

Cont

AccRel

Cont

SpecRel

AccReld

= SpecRel ∪ {AxCmv,AxEv−,AxSf

−,AxDf} ∪ Cont.

AccRel

TwP mk e1 e2

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e1 e2

∀m ∈ IOb ∀k ∈ Ob ∀xx′yy′ xt < yt ∧ x′

t< y′

t∧

m,k ∈ evm(x) = evk(x′) ∧ m,k ∈ evm(y) = evk(y

′) → y′

t− x′

t≤ yt − xt

∧(y′

t− x′

t= yt − xt ↔ encm(x, y) = enck(y

′, y′)),

encm(x, y) = {evm(z) : W(m,m, z) ∧ xt ≤ zt ≤ yt}

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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ANDRÉKA–MADARÁSZ--NÉMETI–SZÉKELY: ON LOGICAL ANALYSIS… 219

AxPh−

AxSm−

GenReld

= {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+ d

= GenRel ∪ Compr.

GenRel+

GenRel+

GenRel+

GenRel+

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ANDRÉKA–MADARÁSZ--NÉMETI–SZÉKELY: ON LOGICAL ANALYSIS… 221

AccRel

∗

∗

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