Are all geometries complete/consisten/decidable?

Question

I just read this post on here, Is Euclidean Geometry complete and unique and after reading through Greenberg (2010) I was wondering does line of argument apply to all possible geometries?

In particular does this mean that the geometries of Special & General Relativity can be thought of as complete & decidable?

Answer 1

I am not sure how much trust I would put in this, but take a look at:

H. Andréka, J. Madarász, I. Németi, Decidability, undecidability, and Gödel's incompleteness in relativity theories. Parallel Process. Lett. 22 (2012), no. 3, 1240011, 14 pp

From the summary:

In this paper we investigate the logical decidability and undecidability properties of relativity theories. If we include into our theory the whole theory of the reals, then relativity theory still can be decidable. However, if we actually assume the structure of the quantities in our models to be the reals, or at least to be Archimedean, then we get possible predictions in the language of relativity theory which are independent of ZF set theory.

See also their survey paper:

Logical axiomatizations of space-time. Samples from the literature. Non-Euclidean geometries, 155–185, Math. Appl. (N. Y.), 581, Springer, New York, 2006.

Answer 2

This might not directly answer your question, but there's an interesting result about the undecidability of the spectral gap which can be found here: https://arxiv.org/abs/1502.04573

While this might not directly answer your question, it does shed some light on the decidability of physical theories.