## Hajnal Andréka, Judit X. Madarász, and
István Németi

Alfréd Rényi Mathematical Institute

P.O.Box 127

Budapest

H-1364 Hungary

andreka@math-inst.hu,
madarasz@math-inst.hu,
nemeti@math-inst.hu

### Abstract

Here we outline an approach to a logical analysis of relativity theory
conducted purely in first order logic (for methodological reasons). Here
we concentrate on special relativity, but in a more extensive work
referred to here as (AMNSS 1998), some steps are made in the direction of
generalizing the present approach towards general relativity. In (AMNSS
1998) we build up variants of relativity theory as ``competing'' axiom
systems formalized in first order logic. The reason for having several
versions for the theory, i.e. several axiom systems, is that this way we
can study the consequences of the various axioms, enabling us to find out
which axiom is responsible for some interesting or ``exotic'' prediction
of relativity theory. Among others, this enables us to refine the
conceptual analysis in Friedman's and Rindler's books, or compare the
Reichenbach-Grunbaum approach to relativity with the standard one.
After having formalized relativity in first order logic, we use the well
developed machinery of first order logic for studying properties of the
theory (e.g. the number of non-elementarily equivalent models, or its
relationships with Goedel's incompleteness theorems, independence issues
etc).

In the present paper, first we recall one of our axiom systems
``*Specrel*'' for relativity from (AMNSS 1998). Then we present
some of the typically logical investigations of *Specrel*, e.g.
independence of the axioms, consistency properties of models. We also
discuss the intuitive contents (and consequences) of some of the axioms.
We also present a little bit of the conceptual analysis part of (AMNSS
1998) using the example of *Specrel* and ``FTL observers''.
Finally, we study how much Goedel's incompleteness theorems apply to
*Specrel*: *Specrel* is undecidable, and it admits natural
extensions some of which are decidable, while to others the full strength
of both of Goedel's incompleteness theorems applies. E.g. there is an
extension *Specrel+* in which its own consistency is formalizable
(and is neither provable nor refutable). *Specrel+* is hereditarily
undecidable.

We deliberately try to keep the number of axioms in *Specrel* small,
and their intuitive contents simple, transparent, and tangible from the
logical point of view.

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