[FOM] 214:On foundations of special relativistic kinematics 6
Harvey Friedman
friedman at math.ohio-state.edu
Sat Feb 14 09:43:12 EST 2004
This brief continuation will touch on some alternatives to the SIMULTANEOUS
DISTANCE axiom. I plan to continue with some additional material after I
finish some urgent business in f.o.m. I'll be back. In the meantime, those
who are hungry for more foundations of special relativity should consult the
FOM postings of Andreka and Nemeti, and the various references given there.
I will repeat my particular axiomatic setup, this time merged with some
alternatives to the SIMULTANEOUS DISTANCE axiom.
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We declare three sorts of objects. One sort is for observers. Another sort
is for events. The third sort is for real numbers. We have the usual
<,0,1,+,-,x on the real numbers.
We have equality between observers and equality between events and equality
between real numbers.
We have four binary function symbols, TC,XC,YC,ZC. For each of the four, the
first argument must be an observer, the second argument must be an event,
and the value must be a real number.
Finally, we have a constant symbol c for a real number that represents the
speed of light in a vacuum.
We read
TC(O,E) for the time coordinate of the event E according to observer O.
XC(O,E) for the x-coordinate of the event E according to observer O.
YC(O,E) for the y-coordinate of the event E according to observer O.
ZC(O,E) for the z-coordinate of the event E according to observer O.
The position of E according to O is the list of real numbers
(TC(O,E),XC(O,E),YC(O,E),ZC(O,E)).
The vector from E to E' according to O is obtained by subtracting the
position of E according to 0 from the position of E' according to O,
coordinatewise.
The time interval from E to E' according to O is the real number
TC(O,E') - TC(O,E).
Note that time intervals can be positive, negative, or zero.
The distance from E to E' according to O is the ordinary Euclidean distance
between the triples
(XC(O,E), YC(O,E), ZC(O,E)) and (XC(O,E'), YC(O,E'), ZC(O,E'))
as points in Euclidean three space. Thus the distance from E to E' according
to O must be a nonnegative real number.
1. COORDINATE axiom. According to any observer, every list of four real
numbers are the time, x-, y-, z- coordinates of some unique event.
2. IDENTIFICATION axiom. If two observers agree on the coordinates of every
event, then the two observers are the same.
3. LIGHT SPEED axiom. c > 0.
4. LIGHT PATH axiom. Let two events be given. If some observer thinks that
the Euclidean distance between the two events is c times the time interval
from the first event to the second event, then every observer thinks so.
5. SIMULTANEOUS DISTANCE axiom. Let two events be given. Any two observers
that agree that the two events are simultaneous, agree on their Euclidean
distance.
5a. TIME DISTANCE axiom. Let two events be given. Any two observers that
agree on the time interval between the two events, agree on their Euclidean
distance.
5b. DISTANCE TIME MAGNITUDE axiom. Let two events be given. Any two
observers that agree on the Euclidean distance between the two events, agree
on the magnitude of the time interval between the two events.
5c. AGREEMENT axiom. There exist two distinct observers and two distinct
events such that the observers agree on the time interval and the distance
between the two events.
5d. DISTANCE TIME axiom. Let two events be given. Any two observers that
agree on the Euclidean distance between the two events, agree on the time
interval between the two events.
6. MAXIMALITY axiom. No proper extension of this three sorted system,
preserving the real numbers, exists. (This refers to 1-5 above with 5, 5a,
5b, 5c, or 5d, whatever choice was made).
Note that the TIME DISTANCE axiom is a strong version of the SIMULTANEOUS
DISTANCE axiom. Note that the DISTANCE TIME axiom is a strong version of the
DISTANCE TIME MAGNITUDE axiom.
The choice of 5, 5a,5b, or 5c leads to the same models - models built out of
the c-Poincare transformations.
However, the choice of 5d leads to the (relativistically) trivial model
based on the functions that are time interval preserving and Euclidean
distance preserving. The DISTANCE TIME axiom is too strong.
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
Of course, we haven't broken any serious new ground in these postings yet,
in light of the long literature on this topic, including the Aleksandrov
Zeeman theorem and various work up to and including that of Andreka and
Nemeti and associates (see their FOM postings and reference therein).
However, I do see some clear opportunities to break serious new ground in
this small corner of physics - elementary special relativity, even in its
pristine form without mechanics. This new research will have to wait until I
finish up some urgent f.o.m. matters.
I'll be back.
In any case, this brief experiment into foundational thinking outside f.o.m.
convinces me just how effective f.o.m. style foundational thinking is - how
powerful it is in gaining a deeper insight into perhaps any area of
systematic thought(!).
As this projected explosion in foundational thinking really takes hold
across all systematic knowledge, there will be a profound expansion in the
accessibility of one subtle subject after another. This will be the result
of an emerging incredibly effective unified exposition of all systematic
knowledge. This will form the basis of a great expansion of
interdisciplinary research activity, as the usual divisions of academia into
counterproductive disciplinary subdivisions are seen to be irrelevant,
obstructive, and baroque.
*********************************************
I use http://www.mathpreprints.com/math/Preprint/show/ for manuscripts with
proofs. Type Harvey Friedman in the window.
This is the 214th in a series of self contained numbered postings to
FOM covering a wide range of topics in f.o.m. The list of previous
numbered postings #1-149 can be found at
http://www.cs.nyu.edu/pipermail/fom/2003-May/006563.html in the FOM
archives, 5/8/03 8:46AM. Previous ones counting from #150 are:
150:Finite obstruction/statistics 8:55AM 6/1/02
151:Finite forms by bounding 4:35AM 6/5/02
152:sin 10:35PM 6/8/02
153:Large cardinals as general algebra 1:21PM 6/17/02
154:Orderings on theories 5:28AM 6/25/02
155:A way out 8/13/02 6:56PM
156:Societies 8/13/02 6:56PM
157:Finite Societies 8/13/02 6:56PM
158:Sentential Reflection 3/31/03 12:17AM
159.Elemental Sentential Reflection 3/31/03 12:17AM
160.Similar Subclasses 3/31/03 12:17AM
161:Restrictions and Extensions 3/31/03 12:18AM
162:Two Quantifier Blocks 3/31/03 12:28PM
163:Ouch! 4/20/03 3:08AM
164:Foundations with (almost) no axioms, 4/22/0 5:31PM
165:Incompleteness Reformulated 4/29/03 1:42PM
166:Clean Godel Incompleteness 5/6/03 11:06AM
167:Incompleteness Reformulated/More 5/6/03 11:57AM
168:Incompleteness Reformulated/Again 5/8/03 12:30PM
169:New PA Independence 5:11PM 8:35PM
170:New Borel Independence 5/18/03 11:53PM
171:Coordinate Free Borel Statements 5/22/03 2:27PM
172:Ordered Fields/Countable DST/PD/Large Cardinals 5/34/03 1:55AM
173:Borel/DST/PD 5/25/03 2:11AM
174:Directly Honest Second Incompleteness 6/3/03 1:39PM
175:Maximal Principle/Hilbert's Program 6/8/03 11:59PM
176:Count Arithmetic 6/10/03 8:54AM
177:Strict Reverse Mathematics 1 6/10/03 8:27PM
178:Diophantine Shift Sequences 6/14/03 6:34PM
179:Polynomial Shift Sequences/Correction 6/15/03 2:24PM
180:Provable Functions of PA 6/16/03 12:42AM
181:Strict Reverse Mathematics 2:06/19/03 2:06AM
182:Ideas in Proof Checking 1 6/21/03 10:50PM
183:Ideas in Proof Checking 2 6/22/03 5:48PM
184:Ideas in Proof Checking 3 6/23/03 5:58PM
185:Ideas in Proof Checking 4 6/25/03 3:25AM
186:Grand Unification 1 7/2/03 10:39AM
187:Grand Unification 2 - saving human lives 7/2/03 10:39AM
188:Applications of Hilbert's 10-th 7/6/03 4:43AM
189:Some Model theoretic Pi-0-1 statements 9/25/03 11:04AM
190:Diagrammatic BRT 10/6/03 8:36PM
191:Boolean Roots 10/7/03 11:03 AM
192:Order Invariant Statement 10/27/03 10:05AM
193:Piecewise Linear Statement 11/2/03 4:42PM
194:PL Statement/clarification 11/2/03 8:10PM
195:The axiom of choice 11/3/03 1:11PM
196:Quantifier complexity in set theory 11/6/03 3:18AM
197:PL and primes 11/12/03 7:46AM
198:Strong Thematic Propositions 12/18/03 10:54AM
199:Radical Polynomial Behavior Theorems
200:Advances in Sentential Reflection 12/22/03 11:17PM
201:Algebraic Treatment of First Order Notions 1/11/04 11:26PM
202:Proof(?) of Church's Thesis 1/12/04 2:41PM
203:Proof(?) of Church's Thesis - Restatement 1/13/04 12:23AM
204:Finite Extrapolation 1/18/04 8:18AM
205:First Order Extremal Clauses 1/18/04 2:25PM
206:On foundations of special relativistic kinematics 1 1/21/04 5:50PM
207:On foundations of special relativistic kinematics 2 1/26/04 12:18AM
208:On foundations of special relativistic kinematics 3 1/26/04 12:19AAM
209:Faithful Representation in Set Theory with Atoms 1/31/04 7:18AM
210:Coding in Reverse Mathematics 1 2/2/04 12:47AM
211:Coding in Reverse Mathematics 2 2/4/04 10:52AM
212:On foundations of special relativistic kinematics 4 2/7/04 6:28PM
213:On foundations of special relativistic kinematics 5 2/8/04 9:33PM
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