MAT 360 Lecture 7

Hilbert AxiomsContinuity and Parallelism

Exercise: Prove the following Given any segment s, there exists an

equilateral triangle have s as one of its sides.

Definition A point P is inside of a circle of radius

AB and center A if AP<AB. A point P is outside of a circle of

radius AB and center A if AP>AB.

Circular continuity principleIf a circle C1 has one point inside the circle

C2 and one point outside of the circle C2, then the C1 and C2 intersect in two points.

Elementary continuity principle If one endpoint of a segment is inside a

circle and the other outside, then the segment intersects the circle.

Archimedes’ Axiom If CD is any segment, A any point and, r

any ray with vertex A, then for every point B≠A on r there exists a number n such that when CD is laid of n times starting at A, a point E (on r) is reached such that

n . CD~ AE

and either B=E or B is between A and E.

Dedekind’s AxiomSuppose that the set of all points on a line is

the disjoint union of S and T, S U Twhere S and T are of two non-empty subsets

of l such that no point of either subsets is between two points of the other. Then there exists a unique point O on l such that one of the subsets is equal to a ray of l with vertex O and the other subset is equal to the complement.

Axiom of Parallelism For every line l and for every point P not

lying in l, there is at most one line m through P such that m is parallel to l.

Definition An Euclidean plane is a model for the

axioms. Incidence Congruence Continuity (Dedekind’s Axiom) Parallelism

Example: An Euclidean plane Point (x,y), x and y are real numbersAuxiliary definition: (u,v,w) is a good triple,

if u, v and w are real numbers and u≠0 or v ≠0.

Line: Set of points (x,y) for which there exist a good triple (u,v,w) such that ux+vy+w=0.

Incidence: Set membership.

Example and Exercise: Euclidean plane Incidence: Usual Distance between points: Usual

Pythagorean formula. This gives congruence between segments

A*B*C holds if d(A,B)+d(B,C)=d(A,C) <ABC ~ <DEF if A, C, D and F can be

chosen on the sides of these angles, so that AB ~DE, BC ~ EF, and AC ~ CF.

With this interpretations, it is possible to verify the Hilbert’s axioms.

Another interpretation Points (x,y), x and y rational numbers Lines: Determined by “good pairs” (u,v,w)

where u, v and w are rational numbers. Congruence, betweennes, as in the

previous axioms

Study which of Hilbert axioms hold.What about Dedekind’s?