Module 3, continued

by Leigh Hollyer

dog@uh.edu

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Table of Contents

Module 3, continued

A3: Every two distinct lines have a least one point on them both.A4:  Every line has at least three points on it.

Read and format the information create models look for familiar objects and look for any differences…can you “bend” familiar definitions (triangle, parallel, quadrilateral) to fit the new space?

Sketch and Model time

More Definitions:

Module 4

Model

Axiomatic Systems

Metamathematics

Metamathematics provides the framework upon which we have built all of the structures that allow mathematics

Properties of Axiomatic Systems

Consistency

Independent

Completeness

More Metamath

Euclidean Geometry

From algebra

A function, f, is said to be one-to-one if, for any choice of two distinct domain set elements, the image of these is two distinct set elements from the range.

Definition

All models of Euclidean geometry using modern axioms are isomorphic.

Geometry of Example 1, p30

Axioms:  A1: There is at least one point   A2: Every point is on exactly two lines  A3: Every line is on exactly three points  Explore the geometry with sketching and models. Check to see which models are isomorphic and which are not.

Definition

Taxicab distance:

The taxicab distance shows one very subtle flaw in the Euclid’s Postulates…he assumed that the traditional definition of distance was the only feasible one…it’s not.

Example 4

Construct a spherical line segment

To draw a straight line from any point to any point.

Use the construction from Proposition I-1:

Independent Statements

I-1 is in the section of propositions that use only the first four postulates and, if you use all of Euclid’s assumptions, it does hold.

Notebook Problems

In-class problems

Sketchpad Demo