Hajnal Andréka and István Németi

Logical Foundation of Spacetime

Foundation of Mathematics bulletinboard (FOM, December 2003 - February 2004) contains several postings by Harvey Friedman in which he proposes to export the methodology, spirit, and machinery of Foundational Thinking from the area of Foundations of Mathematics to a similar foundation of relativity which eventually would "cover" both special relativity and general relativity. Of course, one begins with special relativity (but keeping in mind the direction of general relativity). We will give an overview of this kind of research going on at various parts of the globe. Material on this:
http://www.math-inst.hu/pub/algebraic-logic/lstsamples.pdf,
http://www.math-inst.hu/pub/algebraic-logic/loc-mnt04.pdf,
http://www.math-inst.hu/pub/algebraic-logic/PartI.ps.gz.

Special relativity emerges as a rather transparent, simple and self-evident theory SPECREL of first-order logic (FOL). A natural, logical procedure called localization can be applied to SPECREL yielding the more flexible FOL-theory Loc(SPECREL). The latter is "local" in the sense of general relativity. Loc(SPECREL) can be regarded as a starting point for building up a FOL-theory of general relativity. Looking at this with the logician's eye, we note that obtaining such FOL theories (and proving their adequateness etc) is only the beginning of a logic-based conceptual analysis (and, in general, foundational thinking for) relativity and not the end.

Logic, relativity theory, closed timelike curves ("time-travel")

No familiarity with general relativity will be assumed in the talk.

Besides FOM |--> Foundation of Relativity mentioned in the previous talk we mention 3 further kinds of connection between logic and relativity.

(1) Beginning with Kurt Gödel's spacetime, there is a family of spacetimes in which closed timelike curves (CTC's) occur. Intuitively, CTC's make time-travel to the past possible. These lead to interesting logical puzzles (since they invoke something like self-reference). However, in logic we have a ready-made machinery for handling self-reference. We will conclude that the alleged logical "paradoxes of time-travel" can be resolved by logical means showing that there are no logical reasons excluding time travel. See:
http://aardvark.ucsd.edu/grad_conference/wuthrich.pdf,
http://philsci-archive.pitt.edu/archive/00000965/00/TMdraft108.pdf.

(2) Examples of spacetimes milder than the ones in item (1) above make it possible to design relativistic computers which can compute non-Turing computable functions. Cf. Etesi-Németi and/or Hogarth. Tipler in his 1974 book, in section G, arrives at similar conclusions, using general relativity in a similar fashion. This is a connection in the Logic <--| Relativity direction. One of the key ideas of these relativistic computers is that they do NOT involve knowing things like in what state my computer is after having performed an infinite number of steps.

(3) The spacetimes in (1) include Gödel's one, Kerr one, Kerr-Newman one, Gott's one, van Stockum - Tipler one, to mention a few. All of these involve rotation of that gravitating matter which is the source of the gravitational field we are studying. The time-orientation of the CTC's under discussion also can be regarded as a kind of "rotation". We will look at the question of whether these two kinds of rotation are connected to each other, e.g. whether it is a logical necessity that they "move" in opposite directions.

On definability theory of first-order logic

We will see two first-order logic (FOL) theories T1 and T2 such that T1 is definable over T2 and T2 is definable over T1, but they are not definitionally equivalent.

We use the following definitions: T1 is said to be definable over T2 if Mod(T1) is a reduct of a definitional expansion of Mod(T2). Further, T1 and T2 are said to be definitionally equivalent if they have a common definitional expansion.