a logic based foundation and analysis of relativity theory and logicpage: 1

A Logic Based Foundation and Analysis of

Relativity Theory and Logic Page: 1

Special Relativity

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Bodies (test particles), Inertial Observers, Photons, Quantities, usual operations on it, Worldview

B

IOb

Ph

0

Q = number-line

W

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W(m, t x y z, b) body “b” is present at coordinates “t x y z” for observer “m”

BQIObW 4

t

x

y

b (worldline)

m

1. ( Q , + , ∙ ) is a field in the sense of abstract algebra (with 0 , −, 1 , / as derived operations)

2.

3. Ordering derived:

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AxField Usual properties of addition and multiplication on

Q :Q is an ordered Euclidean field.

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AxPhFor all inertial observers the speed of light is the same in all directions and is finite.In any direction it is possible to send out a photon.

t

x

y

ph1 ph

2ph

3

Formalization:

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AxEvAll inertial observers coordinatize the same events.

Formalization:

t

x

y

Wm

b1

b2

mt

x

y

Wk

b1

b2

k

))(,( txyzIObkm

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Formalization:

AxSelfAn inertial observer sees himself as standing still at the origin.t

x

y

t = worldline of the observer

m

Wm

))(( txyzIObm

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AxSymdAny two observers agree on the spatial distance between two events, if these two events are simultaneous for both of them, and |vm(ph)|=1.

Formalization:

k t

x

y

m

x

y

t

e1

e2

ph

is the event occurring at p in m’s worldview

vm(b)

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What is speed?m

b

ppt

ps

qs

qt q

1

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SpecRel = {AxField, AxPh, AxEv, AxSelf, AxSymd}

SpecRel Theorems

Proofs

…

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Thm2

Thm1

t

x

y

m

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Proof of Thm2 (NoFTL):

e1

k

ph

- AxSelf

- AxPh

e2

ph1

e3

e2

ph1

- AxEv

t

x

y

k

e1

- AxField

ph

p

q

Tangent plane

k asks m : where did ph and ph1 meet?

Conceptual analysis Which axioms are needed and why

Project: Find out the limits of NoFTL How can we weaken the axioms to make

NoFTL go away

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Igor D. Novikov, September 3 Budapest:

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It is an important research today to study spacetimes with more than one time dimensions

Thm3 SpecRel is consistent

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Thm4 No axioms of SpecRel is provable from the rest

Thm5 SpecRel is complete with respect to Minkowski

geometries (e.g. implies all the basic paradigmatic effects of Special Relativity - even quantitatively!)

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Thm6 SpecRel generates an undecidable theory.

Moreover, it enjoys both of Gödel’s incompleteness properties

Thm7 SpecRel has a decidable extension, and it also has

a hereditarily undecidable extension. Both extensions are physically natural.

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Moving clocks get out of synchronism. Captain claims that the two clocks show the same

time.

Thm8vk(m)

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Thm8 (formalization of clock asynchronism)

(1) Assume e, e’ are separated in the direction of motion of m in k’s worldview,

m

yk

k

xk

1t

v

1x

xm

ee’

|v|

Assume SpecRel. Assume m,k ϵ IOb and events e, e’ are simultaneous for m,

m

xm

e’e

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(2) e, e’ are simultaneous for k, too e, e’ are separated orthogonally to vk(m) in k’s

worldview

m

xm

yk

k

xk

ym

e

e’

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Thought-experiment for proving relativity of simultaneity.

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Thought-experiment for proving relativity of simultaneity.

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The Vienna Circle and Hungary, Vienna, May 19-20, 2008

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Moving clocks tick slowly Moving spaceships shrink

Thm9

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Thm9 (formalization of time-dilation)Assume SpecRel. Let m,k ϵ IOb and events e, e’ are on k’s lifeline.

m

xm

e

e’

k

1

v

k

xk

e

e’

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Thm10 (formalization of spaceship shrinking)Assume SpecRel. Let m,k,k’ ϵ IOb and assume

m

xm

k k’

e’e

k

x

k’

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Experiment for measuring distance (for m) by radar:

m’

m’’m

e

e’

Distm(e,e’)

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v = speed of spaceship

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t

x

y

Wm

b1

b2

mt

x

y

Wk

b1

b2

kwmk

evm evk

p q

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Thm11The worldview transformations wmk are

Lorentz transformations (composed perhaps with a translation).

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Minkowskian Geometry: Top-down approach to SR, observer-free.

Robert Goldblatt: complete FOL axiom system MinkGeo

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Interpretations

Definitional equivalence

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Minkowskian GeometryJames Ax: SignalsAlfred Robb: causalityPatrick Suppes: worldview transformations…

Connections between theories. Dynamics of theories. Interpretations between them. Theory morphisms. Definitional equivalence.

Tamás Füzessy, Judit Madarász and Gergely Székely began joint work in this

Definability theory of logic! (Tarski, Makkai)

Contribution of relativity to logic: definability theory with new objects definable (and not only with new relations definable). J. Madarász’ dissertation.

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SpecRel = {AxField, AxPh, AxEv, AxSelf, AxSymd}

SpecRel Theorems

Proofs

…

Conceptual analysis of SR goes on … on our homepage

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New theory is coming: