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
