Lecture Notes 1

Introduction

Axiomatic Systems

The need for undefined terms
Vish (Vicious Circle) : Start with any word in a dictionary and continue to look up words used in the definition until some word gets repeated for the first time.
" Vish illustrates the important principle that any definition of a word must inevitably involve other words, which require further definitions. The only way to avoid a vicious circle is to regard certain primitive concepts as being so simple and obvious that we agree to leave them undefined. Similarly, the proof of any statement uses other statements; and since we must begin somewhere, we agree to leave a few simple statements unproved. These primitive statements are called axioms." - Coxeter, Projective Geometry, pg. 6.
Example: Vish using The American Heritage Dictionary

Finite Geometries

Can be traced back to Gino Fano (1892) with some ideas going back to von Staudt (1852).

There are two undefined terms : points, lines. There is also a relation between them called on. This relationship is symmetric so we speak of points being on lines and lines being on points.

A Three-Point Geometry:
  1. There exist exactly 3 points in this geometry.
  2. Two distinct points are on exactly one line.
  3. Not all the points of the geometry are on the same line.
  4. Two distinct lines are on at least one point.
Theorem 1.1 : Two distinct lines are on exactly one point.

Theorem 1.2 : The three point geometry has exactly three lines.

A Four-Line Geometry:
  1. There exist exactly 4 lines.
  2. Any two distinct lines have exactly one point on both of them.
  3. Each point is on exactly two lines.
Theorem 1.3 : The four line geometry has exactly six points.

Theorem 1.4 : Each line of the four-line geometry has exactly 3 points on it.

The plane dual of a statement is the statement obtained by interchanging the terms point and line.

The plane duals of the axioms for the four-line geometry will give the axioms for the four- point geometry. And the plane duals of Theorems 1.3 and 1.4 will give valid theorems in the four-point geometry.

Fano's Geometry:
  1. There exists at least one line.
  2. Every line of the geometry has exactly 3 points on it.
  3. Not all points of the geometry are on the same line.
  4. For two distinct points, there exists exactly one line on both of them.
  5. Each two lines have at least one point on both of them.
Theorem 1.7 : Each two lines have exactly one point in common.

Theorem 1.8 : Fano's geometry consists of exactly seven points and seven lines.

Young's Geometry:
1-4 above
5'. If a point does not lie on a given line, then there exists exactly one line on that point that does not intersect the given line.