the end of time (defining time and space) part ii. hajnal andréka and istván németi

The end of Time(defining time and space)

Part II.Hajnal Andréka and István Németi

GOAL

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a mathematical model in which time and space are derived, theoretical concepts

TOOLS

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experimenters exploring space and timevia signalling

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No clocks No meter rods

EXAMPLES

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we “put” the experimenters into various concrete space-times

First exampleRelativistic space-time

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Later: Newtonian, Schwarzshild black hole, general relativistic, cosmological.

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Experimenters

Signals

Connects(e,s,e’)

s

e’

e

Experimenters and signals in a concrete spacetime

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Signals

Experimenters

Connects(e,s,e’)

se’

ee e'

s

Experimenters know only this

Goal to find out

What is SPACE?

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Locations for experimenter e are the experimenters motionless wrt him

Compass and rulerCompass = meter-rod: Ed equidistanceRuler: Collinearity Col, Betweenness Bw

Tarski’s elementary axiomatization of Euclidean space-time:

Bw(x,x,y), Bw(x,y,z)Bw(z,y,x), …Ed(xxyy), Ed(xyzw) Ed(zwxy),…

Let Th be their set.

Theorem 1: Mod(Th) = Sp(F) : F is a Euclidean field

Theorem 2: Mod(Th+) = Sp(F) : F is a real-closed field .

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By now experimenters have meter rods.

What is TIME?

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Structure of events in which e participatesLater, equi-duration, addition, multiplication

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Event = congruence class of sending and receiving signals “at the same time”

zs

s’

ee’

e sends s ≡

e’ sends z ≡

e receives s’ ≡ …

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Order of events

Equi-duration of time-lapses

Addition

Multiplication.

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Multiplication

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By now experimenters have clocks.

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Axioms connecting time and space, describing motion, …

Theorem 1: Mod(Sigma) = St(F) : F is a Euclidean field

Finally, they can get a finite set Sigma in the language of the “bare” models such that the models of Sigma are exactly “our bare models” with an arbitrary field F in place of the field of reals.

St(F) stands for “Spacetime over F” as defined on p.8.

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Theorem 2:

Th(Sigma+) = Th( St(R) ) where R is the field of the real numbers.

Mod(Sigma+) = St(F) : F is a real-closed field.

If they add the finite schema of axioms for a real-closed field, they can derive everything that is true in their world.

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Second exampleNewtonian space and time

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Variations, Schwarzshild black hole, general relativistic, cosmological, ...

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Operational semantics for SpecRelOntology of conceptsComparing theoriesSchwarzschild black holeDirection of time...

Infinity of things to say yet:

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Sean M Carroll on Origin of the Universe and the Arrow of Time. Youtube. (Talk at Perimeter Institute, Waterloo, October 17, 2009.)

Sean M Carroll, From eternity to here: the quest for the ultimate theory of time. 2010.

THE END

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OF TIMEas a basic notion