Tarski developed an axiomatic description of Euclidean geometry in first order logic. Its primitive notions are points and its primitive relations are betweeness and congruence of points. The Parallel axiom is stated using betweeness.


He proved it


1. Consistent. That is, it does not prove both any sentence and its negation.
2. Complete. That is, it proves any sentence or its negation to be true.
3. Decidable. There is an algorithm assigning a truth value to every sentence.


Obviously, we can negate the Parallel axiom to obtain Tarskian hyperbolic geometry.


**Q. Is this geometry also consistent, complete and decidable?**



Is Tarskian hyperbolic geometry consistent, complete & decidable?
The canonical reference for Tarski-style elementary geometry is the monograph Schwabhäuser, Szmielew, Tarski [1]. This includes a treatment of hyperbolic geometry in parallel with Euclidean geometry; in particular, the consistency, completeness, and decidability of $n$-dimensional hyperbolic geometry for any $n\ge2$ is proved in Satz II.3.65.


**Reference:**


[1] Wolfram Schwabhäuser, Wanda Szmielew, Alfred Tarski: *Metamathematische Methoden in der Geometrie*, Springer-Verlag, 1983.


