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MAT 360 Lecture 7

MAT 360 Lecture 7. Hilbert Axioms Continuity 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.

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MAT 360 Lecture 7

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  1. MAT 360 Lecture 7 • Hilbert Axioms • Continuity and Parallelism

  2. Exercise: Prove the following • Given any segment s, there exists an equilateral triangle have s as one of its sides.

  3. 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.

  4. Circular continuity principle • If 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.

  5. Elementary continuity principle • If one endpoint of a segment is inside a circle and the other outside, then the segment intersects the circle.

  6. 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.

  7. Dedekind’s Axiom • Suppose that the set of all points on a line is the disjoint union of S and T, • S U T • where 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.

  8. 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.

  9. Definition • An Euclidean plane is a model for the axioms. • Incidence • Congruence • Continuity (Dedekind’s Axiom) • Parallelism

  10. Example: An Euclidean plane • Point (x,y), x and y are real numbers • Auxiliary 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.

  11. 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.

  12. 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?

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