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September 27, 2013. The End of Time Page: 1
The end of Time(defining time and space)
Hajnal Andréka and István Németi
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We have experimenters communicating with each other. Observing the the laws of this communication, they create the concepts of time and space in order to act in this world more effectively.
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No clocks No meter rods
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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
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s
e’
e
e e'
s
Experimenters know only this
Bare, puritan modelRich , structured model
Getting familiar
sa’
a
a a’
z
z
What is SPACE?
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Motionless 1
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Motionless 2
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e e’
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Desargues’s theorem
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By now experimenters have meter rods.
What is TIME?
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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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Using these concepts, our experimenters can observe finitely many regularities between the events (they can state them to be physical laws) such that on the base of this they can know that their space-time is a 4-dimensional space over a field F .
Formally, this amounts to 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.
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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Direction of timeSchwarzschild black holeOperational semantics for SpecRelInterpretation between theoriesComparing theoriesOntology of concepts...
Infinity of things to say yet:
THE END
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OF TIMEas a basic notion
Thank you for your attention!
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