The end of Time (defining time and space) Part II. Hajnal Andréka and István Németi

GOAL October 11, 2013.The End of Time 2Page: 2 a mathematical model in which time and space are derived, theoretical concepts

TOOLS October 11, 2013.The End of Time 2Page: 3 experimenters exploring space and time via signalling

October 11, 2013.The End of Time 2 Page: 4

October 11, 2013.The End of Time 2Page: 5 No clocksNo meter rods

EXAMPLES October 11, 2013.The End of Time 2Page: 6 we “put” the experimenters into various concrete space-times

First example Relativistic space-time October 11, 2013.The End of Time 2Page: 7 Later: Newtonian, Schwarzshild black hole, general relativistic, cosmological.

October 11, 2013.The End of Time 2Page: 8 Experimenters Signals Connects(e,s,e’) s e’ e Experimenters and signals in a concrete spacetime

October 11, 2013.The End of Time 2Page: 9 Signals Experimenters Connects(e,s,e’) s e’ e e e' s Experimenters know only this Goal to find out

What is SPACE? October 11, 2013.The End of Time 2Page: 10

October 11, 2013.The End of Time 2Page: 11 Locations for experimenter e are the experimenters motionless wrt him Compass and ruler Compass = meter-rod: Ed equidistance Ruler: 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 .

October 11, 2013.The End of Time 2Page: 12 By now experimenters have meter rods.

What is TIME? October 11, 2013.The End of Time 2Page: 13 Structure of events in which e participates Later, equi-duration, addition, multiplication

October 11, 2013.The End of Time 2Page: 14 Event = congruence class of sending and receiving signals “at the same time” z s s’ e e’ e sends s ≡ e’ sends z ≡ e receives s’ ≡ …

October 11, 2013.The End of Time 2Page: 15 Order of events Equi-duration of time-lapses Addition Multiplication.

October 11, 2013.The End of Time 2Page: 16 Multiplication

October 11, 2013.The End of Time 2Page: 17 By now experimenters have clocks.

October 11, 2013.The End of Time 2Page: 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.

October 11, 2013.The End of Time 2Page: 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.

October 11, 2013.The End of Time 2Page: 20

Second example Newtonian space and time October 11, 2013.The End of Time 2Page: 21 Variations, Schwarzshild black hole, general relativistic, cosmological,...

October 11, 2013.The End of TimePage: 22 Operational semantics for SpecRel Ontology of concepts Comparing theories Schwarzschild black hole Direction of time. Infinity of things to say yet:

October 11, 2013.The End of Time 2Page: 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

THE END October 11, 2013.The End of Time 2Page: 24 OF TIME as a basic notion