Knowra Tarski's axioms Tarski's axioms Tarski's axioms are a first-order system for Euclidean geometry whose primitive relations express betweenness and congruence between points. They characterize Euclidean geometry without taking lines or angles as primitive objects.
First-order logic : A formal language and deductive system using objects, predicates, quantifiers, and variables. Tarski's axioms quantify over points and state properties of their betweenness and congruence.
Segment construction axiom : An axiom guaranteeing a point on a ray that forms a segment congruent to a specified segment. It ensures segments can be extended to match any prescribed length.
Alfred Tarski : A Polish-American logician and mathematician known for work in logic, semantics, and geometry. He developed the point-based axiom system bearing his name.
Decidability of elementary geometry : The result that a decision procedure can determine whether sentences in elementary Euclidean geometry follow from its axioms. Tarski's axiomatization supports a celebrated algorithmic decision procedure for geometric statements.
Hilbert's axioms : An axiomatic system for Euclidean geometry using points, lines, planes, incidence, order, congruence, and continuity. Unlike Tarski's system, Hilbert's takes lines and planes as primitive objects.
Betweenness : A relation stating that one point lies between two other points on a line. This primitive relation supplies the system's account of order and collinearity.
Five-segment axiom : An axiom relating congruent segments in two configurations of four points, yielding a corresponding fifth congruence. It provides the system's central criterion for transferring congruence between triangle-like configurations.
David Hilbert : A German mathematician whose foundational work shaped modern axiomatic geometry and mathematics. Hilbert's axiomatization provided a major predecessor and comparison for Tarski's approach.
Quantifier elimination : A method for replacing quantified formulas by equivalent formulas without quantifiers in a specified theory. It underlies decision and structural results for suitable formulations of Euclidean geometry.
Birkhoff's axioms : A system for Euclidean geometry that uses real numbers and distance and angle measurement as foundations. It introduces numerical measurement directly, unlike Tarski's relational primitives.
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