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.
Counting numbers, and all numbers, can form an axiomatic theory {natural number theory}.
axioms
Start with null set and with two axioms. Numbers correspond to two sets of previously constructed numbers, with no member of left set greater than or equal to members of right set. Number is less than or equal to another number, if and only if no member of first-number left set is greater than or equal to second number and no member of second-number right set is less than or equal to first number.
axioms: procedure
First, make null set both right and left set, to obtain number zero. Then make right set null and left set contain number zero to obtain number one. Continue to build all counting numbers. All other numbers can derive from counting numbers.
equivalence
Equivalence or one-to-one correspondence can construct counting numbers. Start with null set. Define zero as cardinal number of elements in null set and equivalent sets. Define one as number of elements in set that contains only zero as element. Define two as number of elements in set that contains only elements zero and one. Define N as number of elements in set that contains only elements zero through N.
numbers in general
Axiomatizing natural numbers allows axiomatizing all number types. The number "one" belongs to a set. Set members have one and only one successor. If two successors are equal, then members are equal. The number "one" is not any number's successor. If subset contains "one" and another number, then number successor is in subset and subset is same as whole set.
integers
Positive integers can be an axiomatic system. Undefined terms are "one", "number", and "successor of". Dedekind-Peano postulates can construct the positive integers.
Geometries {non-Archimedian geometry} that violate Archimedes axiom can be axiomatic systems.
3-Mathematics-Axiomatic Theory
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Date Modified: 2022.0225