Geometry is an axiomatic system {geometry theory}.
primitives
Geometry can use two undefined ideas: point and order.
symbols
Geometry can use three symbols: point or line, segment, and motion or congruence. Points can be ordered-pairs. Line can be a ratio. Congruence can be translation, rotation, and reflection.
axioms
Points can form a set. At least one point exists. If there is a point, then there is another point. Given two points, there are lines. Given a line, there is another line. Given two lines, there is a space.
postulates
Axiomatized geometry has five postulates, similar to Euclid's five postulates. Lines can have no multiple points {Jordan curve, geometry}. Figures can have infinite perimeters.
In three-dimensional space, continuity axiom is true, but planes and surfaces do not need this axiom.
Dimension number is not necessarily coordinate number or point multiplicity.
theory
Geometry uses point, line, plane, how point "lies on" line, how point "lies on" plane, how point pairs are congruent, how angles are congruent, and/or how points order on lines {betweenness, geometry}.
consistency
Geometry is consistent if arithmetic is consistent.
Mathematical Sciences>Mathematics>Axiomatic Theory>Kinds
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Date Modified: 2022.0224