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Home > Hilbert's axioms

1 Postulates

1.1 I. Axioms of Incidence

1.1.1 Postulate I.1

For any two points A, B, there exists a line a that contains each of the points A, B

1.1.2 Postulate I.2

For any two points A, B there exists no more than one line containing both A and B

1.1.3 Postulate I.3

There exist at least two points on any given line. There exist at least three points that do not lie on a given line

1.1.4 Postulate I.4

For a set of three points {A, B, C} that do not lie on the same line, there exists a plane α that contains each of the points in the set. For every plane there exists at least one point which it contains.

1.1.5 Postulate I.5

For a set of three points {A, B, C} that do not all lie on the same line, there exists only one plane that contains each of the points in the set.

1.1.6 Postulate I.6

If two points {A, B} of a line, a, lie in a plane, α, then every point in a lies in α

1.1.7 Postulate I.7

If two planes {α, β} have a point A in common, then they have at least one other point, B, in common

1.1.8 Postulate 1.8

There exist at least four points which do not lie in a plane

1.2 II. Axioms of Order

1.2.1 Postulate II.1

If a point B lies between points A and C, then the points {A, B, C} are three distinct points on the same line and B lies between C and A

1.2.2 Postulate II.2

Given two points {A, C}, a point B exists on the line AC such that C lies between A and B

1.2.3 Postulate II.3

Given any three points {A, B, C} of a line, one and only one of the points is between the other two

1.2.4 Postulate II.4

Given three points {A, B, C} that do not lie on a line and given a line, a, that lies in the plane ABC which does not not intersect any of the points A, B, C: if the line a passes through a point of the segment AB, it also passes through a point in the segment AC or through a point in the segment BC

1.3 III. Axioms of Congruence

1.3.1 Postulate III.1

Given two points {A, B} on a line a and given a point A' on a or another line a', there exists a point B' on a side of the line a' such that ABA'B' are congruent

1.3.2 Postulate III.2

Given segments A'B' and AB such that both are congruent to the same segment AB, then A'B'AB

1.3.3 Postulate III.3

Given a line a with segments AB and BC such that the point B is the only intersection of the two points and on the same line or a line a' with segments A'B' and B'C' such that the point B' is the only intersection: if ABA'B' and BCB'C' then ACA'C'

1.3.4 Postulate III.4

If ABC is an angle and B'C' is a ray, then there is one and only one ray B'A' on each side of the line B'C' such that A'B'C'ABC Corollary: Every angle is congruent to itself

1.3.5 Postulate III.5

Given two triangles ABC and A'B'C' with congruences such that ABA'B', ACA'C' and BACB'A'C' then ABCA'B'C'.

1.4 IV. Axiom of Parallels

1.4.1 Postulate IV.1

Given a line a and a point A not on a, there is at most one line in the plane that contains a and A that passes through A and does not intersect a

1.5 V. Axioms of Continuity

1.5.1 Postulate V.1 ( Archimedes Axiom)

Given segments AB and CD, there exists a number n such that n copies of CD constructed contiguously from A along the ray AB will pass beyond the point B

1.5.2 Postulate V.2 (Line Completeness)

There exists no extension of a set of points on a line with order and congruence relations that would preserve the relations existing among the original elements as well as preserving line order and congruence, i.e., Axioms I-III and V.1.

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Hilbert spaceIn mathematics, a Hilbert space is an inner product space that is complete with respect to the norm defined by the inner product. Hilbert spaces serve to clarify and generalize the concept of Fourier expansion, certain linear transformations such as the F Hilbert's basis theoremCommutative algebra Ring theory Theorems In mathematics, Hilbert's basis theorem first proved by David Hilbert in 1888, states that, if k is a field, then every ideal in the ring of multivariate polynomials k ''x x . x is finitely generated. This can be t Hilbert's paradox of the Grand HotelSet theory In mathematics, the German mathematician David Hilbert ( 1862 1943) presented the following paradox about infinity. The paradox of the Grand Hotel In a hotel with a finite number of rooms, once it is full, no more guests can be accommodated. Hilbert, WisconsinHilbert is a village located in Calumet County in the U. state of Wisconsin. As of the 2000 census, the village had a total population of 1,089. Geography Hilbert is located at 44°8'24" North, 88°9'44" West (44. 140040, -88. According to the United States Hilbert's NullstellensatzHilbert's Nullstellensatz ( German: "theorem of zeros") is a theorem in algebraic geometry that relates varieties and ideals in polynomial rings over algebraically closed fields. It was first proved by David Hilbert. Let K be an algebraically closed field Hilbert's third problemThe third on Hilbert's list of mathematical problems, presented in 1900, is the easiest one. The problem is related to the following question: given any two polyhedra of equal volume, is it always possible to cut the first into finitely many polyhedral pi Hilbert's problems Hilbert cube Hilbert matrix Hilbert's sixteenth problem Hilbert's seventh problem Hilbert (crater) Hilbert College Hilbert transform Hilbert's program

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