![Page 1: The end of Time (defining time and space) Part II. Hajnal Andréka and István Németi](https://reader035.vdocuments.us/reader035/viewer/2022062309/56649ec05503460f94bcc345/html5/thumbnails/1.jpg)
The end of Time(defining time and space)
Part II.Hajnal Andréka and István Németi
![Page 2: The end of Time (defining time and space) Part II. Hajnal Andréka and István Németi](https://reader035.vdocuments.us/reader035/viewer/2022062309/56649ec05503460f94bcc345/html5/thumbnails/2.jpg)
GOAL
October 11, 2013. The End of Time 2 Page: 2
a mathematical model in which time and space are derived, theoretical concepts
![Page 3: The end of Time (defining time and space) Part II. Hajnal Andréka and István Németi](https://reader035.vdocuments.us/reader035/viewer/2022062309/56649ec05503460f94bcc345/html5/thumbnails/3.jpg)
TOOLS
October 11, 2013. The End of Time 2 Page: 3
experimenters exploring space and timevia signalling
![Page 4: The end of Time (defining time and space) Part II. Hajnal Andréka and István Németi](https://reader035.vdocuments.us/reader035/viewer/2022062309/56649ec05503460f94bcc345/html5/thumbnails/4.jpg)
October 11, 2013. The End of Time 2 Page: 4
![Page 5: The end of Time (defining time and space) Part II. Hajnal Andréka and István Németi](https://reader035.vdocuments.us/reader035/viewer/2022062309/56649ec05503460f94bcc345/html5/thumbnails/5.jpg)
October 11, 2013. The End of Time 2 Page: 5
No clocks No meter rods
![Page 6: The end of Time (defining time and space) Part II. Hajnal Andréka and István Németi](https://reader035.vdocuments.us/reader035/viewer/2022062309/56649ec05503460f94bcc345/html5/thumbnails/6.jpg)
EXAMPLES
October 11, 2013. The End of Time 2 Page: 6
we “put” the experimenters into various concrete space-times
![Page 7: The end of Time (defining time and space) Part II. Hajnal Andréka and István Németi](https://reader035.vdocuments.us/reader035/viewer/2022062309/56649ec05503460f94bcc345/html5/thumbnails/7.jpg)
First exampleRelativistic space-time
October 11, 2013. The End of Time 2 Page: 7
Later: Newtonian, Schwarzshild black hole, general relativistic, cosmological.
![Page 8: The end of Time (defining time and space) Part II. Hajnal Andréka and István Németi](https://reader035.vdocuments.us/reader035/viewer/2022062309/56649ec05503460f94bcc345/html5/thumbnails/8.jpg)
October 11, 2013. The End of Time 2 Page: 8
Experimenters
Signals
Connects(e,s,e’)
s
e’
e
Experimenters and signals in a concrete spacetime
![Page 9: The end of Time (defining time and space) Part II. Hajnal Andréka and István Németi](https://reader035.vdocuments.us/reader035/viewer/2022062309/56649ec05503460f94bcc345/html5/thumbnails/9.jpg)
October 11, 2013. The End of Time 2 Page: 9
Signals
Experimenters
Connects(e,s,e’)
se’
ee e'
s
Experimenters know only this
Goal to find out
![Page 10: The end of Time (defining time and space) Part II. Hajnal Andréka and István Németi](https://reader035.vdocuments.us/reader035/viewer/2022062309/56649ec05503460f94bcc345/html5/thumbnails/10.jpg)
What is SPACE?
October 11, 2013. The End of Time 2 Page: 10
![Page 11: The end of Time (defining time and space) Part II. Hajnal Andréka and István Németi](https://reader035.vdocuments.us/reader035/viewer/2022062309/56649ec05503460f94bcc345/html5/thumbnails/11.jpg)
October 11, 2013. The End of Time 2 Page: 11
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 .
![Page 12: The end of Time (defining time and space) Part II. Hajnal Andréka and István Németi](https://reader035.vdocuments.us/reader035/viewer/2022062309/56649ec05503460f94bcc345/html5/thumbnails/12.jpg)
October 11, 2013. The End of Time 2 Page: 12
By now experimenters have meter rods.
![Page 13: The end of Time (defining time and space) Part II. Hajnal Andréka and István Németi](https://reader035.vdocuments.us/reader035/viewer/2022062309/56649ec05503460f94bcc345/html5/thumbnails/13.jpg)
What is TIME?
October 11, 2013. The End of Time 2 Page: 13
Structure of events in which e participatesLater, equi-duration, addition, multiplication
![Page 14: The end of Time (defining time and space) Part II. Hajnal Andréka and István Németi](https://reader035.vdocuments.us/reader035/viewer/2022062309/56649ec05503460f94bcc345/html5/thumbnails/14.jpg)
October 11, 2013. The End of Time 2 Page: 14
Event = congruence class of sending and receiving signals “at the same time”
zs
s’
ee’
e sends s ≡
e’ sends z ≡
e receives s’ ≡ …
![Page 15: The end of Time (defining time and space) Part II. Hajnal Andréka and István Németi](https://reader035.vdocuments.us/reader035/viewer/2022062309/56649ec05503460f94bcc345/html5/thumbnails/15.jpg)
October 11, 2013. The End of Time 2 Page: 15
Order of events
Equi-duration of time-lapses
Addition
Multiplication.
![Page 16: The end of Time (defining time and space) Part II. Hajnal Andréka and István Németi](https://reader035.vdocuments.us/reader035/viewer/2022062309/56649ec05503460f94bcc345/html5/thumbnails/16.jpg)
October 11, 2013. The End of Time 2 Page: 16
Multiplication
![Page 17: The end of Time (defining time and space) Part II. Hajnal Andréka and István Németi](https://reader035.vdocuments.us/reader035/viewer/2022062309/56649ec05503460f94bcc345/html5/thumbnails/17.jpg)
October 11, 2013. The End of Time 2 Page: 17
By now experimenters have clocks.
![Page 18: The end of Time (defining time and space) Part II. Hajnal Andréka and István Németi](https://reader035.vdocuments.us/reader035/viewer/2022062309/56649ec05503460f94bcc345/html5/thumbnails/18.jpg)
October 11, 2013. The End of Time 2 Page: 18
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.
![Page 19: The end of Time (defining time and space) Part II. Hajnal Andréka and István Németi](https://reader035.vdocuments.us/reader035/viewer/2022062309/56649ec05503460f94bcc345/html5/thumbnails/19.jpg)
October 11, 2013. The End of Time 2 Page: 19
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.
![Page 20: The end of Time (defining time and space) Part II. Hajnal Andréka and István Németi](https://reader035.vdocuments.us/reader035/viewer/2022062309/56649ec05503460f94bcc345/html5/thumbnails/20.jpg)
October 11, 2013. The End of Time 2 Page: 20
![Page 21: The end of Time (defining time and space) Part II. Hajnal Andréka and István Németi](https://reader035.vdocuments.us/reader035/viewer/2022062309/56649ec05503460f94bcc345/html5/thumbnails/21.jpg)
Second exampleNewtonian space and time
October 11, 2013. The End of Time 2 Page: 21
Variations, Schwarzshild black hole, general relativistic, cosmological, ...
![Page 22: The end of Time (defining time and space) Part II. Hajnal Andréka and István Németi](https://reader035.vdocuments.us/reader035/viewer/2022062309/56649ec05503460f94bcc345/html5/thumbnails/22.jpg)
October 11, 2013. The End of Time Page: 22
Operational semantics for SpecRelOntology of conceptsComparing theoriesSchwarzschild black holeDirection of time...
Infinity of things to say yet:
![Page 23: The end of Time (defining time and space) Part II. Hajnal Andréka and István Németi](https://reader035.vdocuments.us/reader035/viewer/2022062309/56649ec05503460f94bcc345/html5/thumbnails/23.jpg)
October 11, 2013. The End of Time 2 Page: 23
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.
![Page 24: The end of Time (defining time and space) Part II. Hajnal Andréka and István Németi](https://reader035.vdocuments.us/reader035/viewer/2022062309/56649ec05503460f94bcc345/html5/thumbnails/24.jpg)
THE END
October 11, 2013. The End of Time 2 Page: 24
OF TIMEas a basic notion