Tarskian Geometry as a First-order Theory

Tarskis’ axioms provide a system for Euclidean Geometry as a direct first-order theory, not having an underlying set theory. While this does have some limitations it provides a lot of advantages as well, particularly that we don’t need any understanding of set theory to begin!

Primitive notions and Signature

We shall start by looking at the primitive notions and signature of Tarski’s geometry, of which we have only three.

The first are points. In Tarskis’ axiomatization, the only objects in the universe are points. This is similar to how every object in set theory is a set.

We also have two primitive relations,

• Betweenness, a triadic (3-ary) relation denoted Babc, and

• Congruence, a tetradic (4-ary) relation denoted ab \equiv cd.

The betweenness relation Babc says that the point b is ‘between’ a and c. This relation is used to make affine statements, like statements concerning parallelism.

The congruence relation ab \equiv cd says that the ‘distance’ between points a and b and the distance between points c and d are equal. This relation captures the metric aspect of geometry, and is used to make statements of angels and distance.

We may now begin examining the axioms!