[FOM] Re: On foundation of Special Relativistic Kinematics 4

[Istvan Nemeti]

In his posting #212 Feb 7 18:28:19 (on FoSRK4), Harvey presents the
following axioms for SRK (math. version):

>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. 
>
>6. MAXIMALITY axiom. No proper extension of this three sorted system,
>preserving the real numbers, exists.
>
>This completes the mathematical axiomatization for special relativistic
>kinematics, SRK(math).

Then he writes

>Note that the "real physics", whatever that means, is contained in
axioms 4 >and 5. Presumably, a physicist might not even notice the
remaining axioms.

Without the coordinate axiom, axioms 4 and 5 do not imply anything
(practically). Actually, one of the key differences between special
relativity and general relativity is that in generalizing our FOL
theories towards general relativity (in e.g.[MNTBerlin]), we have to
weaken the coordinate axiom. We explain this in our papers [MNTBerlin],
[Samples], [ANBerlin].

We suggest saying that the "real physics" is in axioms 1,4,5 and c\ne 0
(coordinate-, lightpath-, and simultaneous distance axioms). Indeed,
from axioms 1,4,5, c\ne 0 one can prove practically all the interesting
predictions of SRK, e.g. the Twin Paradox.

Though the present notation of SRK4 does not make it possible, a "real
physics" axiom is  c  is finite nonzero , i.e. 0 \ne c \ne infinity". An
example where this is done is the prestigeous physics book of Landau and
Lifsic [LL].  Newtonian kinematics can be reconstructed as a special
case of  SRK  by setting  c=infinity if that is permitted in the
language, we did this in [AMNbook] section 4.1. In the more advanced
language of our version of SRK, we do allow c=infinity, cf. [AMNbook]
sections 4.1, 4.4  and [MD] section 3 (More general ... Specrel).

References

[LL] Landau, L. D. and Lifsic, E. M., Classical Theory of Fields,
Pergamon Press, Oxford, 1975. 

[MD] Madarasz, J. X., Logic and Relativity (in the light of definability
theory), PhD Dissertation, Budapest, 2002.

[MNTBerlin] Madarasz-Nemeti-Toke: Generalizing the logic approach to
space-time towards general relativity: first steps. In: First-order
Logic Revisited (Proc. FOL75 Berlin), Kluwer, to appear.

[Samples] Andreka-Madarasz-Nemeti: Logical axiomatizations of
space-time. Samples from the literature. In: Non-Euclidean Geometries,
Kluwer, to appear.

[AMNBerlin] Andreka-Madarasz-Nemeti: Logical analysis of relativity
theories. In: First-order Logic Revisited (Proc. FOL75 Berlin), Kluwer,
to appear.

[AMNbook] Andreka-Madarasz-Nemeti: On the logicalstructure of relativity
theories. http://www.math-inst.hu/pub/algebraic-logic/Contents.html

(all of the above references, except for [LL] can be found in
http://www.math-inst.hu/pub/algebraic-logic/Contents.html   )

Hajnal Andreka and Istvan Nemeti