Comparison With Hilbert
Hilbert's axioms for plane geometry number 16, and include Transitivity of Congruence and a variant of the Axiom of Pasch. The only notion from intuitive geometry invoked in the remarks to Tarski's axioms is triangle. (Versions B and C of the Axiom of Euclid refer to '"circle" and "angle," respectively.) Hilbert's axioms also require "ray," "angle," and the notion of a triangle "including" an angle. In addition to betweenness and congruence, Hilbert's axioms require a primitive binary relation "on," linking a point and a line. The Axiom schema of Continuity plays a role similar to Hilbert's two axioms of Continuity. This schema is indispensable; Euclidean geometry in Tarski's (or equivalent) language cannot be finitely axiomatized as a first-order theory. Hilbert's axioms do not constitute a first-order theory because his continuity axioms require second-order logic.
The first four groups of axioms of Hilbert's axioms for plane geometry can be proved using Tarski's axioms.
Read more about this topic: Tarski's Axioms
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