The Axiomatic Method

The Axiomatic Method

• Any axiomatic system must contain technical terms that are deliberately chosen as undefined terms.
• Undefined terms are subject to the reader's interpretation.
• Other technical terms are defined using the undefined terms.
• The definitions of the system are these terms.
• The axiomatic system has statements dealing with undefined terms and definitions.
• These statements are chosen to remain unproven, and are the axioms of the system.
• All other statements of the system must be logical consequences of the axioms.
• Theorems of the axiomatic system are the derived statements.

Simple Abstract Axiomatic System Example

• Undefined terms include Fe's, Fo's, and the relation "belongs to."
• Axiom 1 says there exist exactly three distinct Fe's in this system.
• Axiom 2 says any two distinct Fe's belong to exactly one Fo.
• Axiom 3 says not all Fe's belong to the same Fo.
• Axiom 4 says any two distinct Fo's contain at least one Fe that belongs to both.
• Fe-Fo Theorem 1: Two distinct Fo's contain exactly one Fe.
• Fe-Fo Theorem 2: There are exactly three Fo's.
• Fe-Fo Theorem 3: Each Fo has exactly two Fe's that belong to it.

Models

• Each axiomatic system contains undefined terms with no inherent meaning, open to the reader's interpretation.
• An interpretation of the system occurs when each undefined term is assigned a meaning.
• A model is created if all the axioms are "correct" statements for a given system interpretation.

Example 1.2.2

• Fe's are designated as people, and the Fo's are designated as committees; the axioms become:
• Axiom 1: There are exactly three people.
• Axiom 2: Two distinct people belong to exactly one committee.
• Axiom 3: Not all people belong to the same committee.
• Axiom 4: Any two distinct committees contain one person who belongs to both.
• With Bob, Ted, and Carol as the people, and Entertainment (Bob and Ted), Finance (Ted and Carol), and Refreshments (Bob and Carol) as the committees, the axioms become "correct" statements.
• This interpretation becomes an example of a model.

Example 1.2.3

• Fe's are designated as books, Fo's as horizontal shelves, and the relation "belongs to" as "is on".
• Axiom 1: There are exactly three books.
• Axiom 2: Any two books are on exactly one shelf.
• Axiom 3: Not all books are on the same shelf.
• Axiom 4: Any two distinct shelves contain one book that is on both.
• This interpretation is not a model for the system, because Axioms 2 and 4 are untrue statements.

Four-Point Geometry

• The four-point geometry derives its name from its first axiom.
• The undefined terms are point, line, and on.
• The following set of three axioms will be assumed:
• Axiom 1: There exist exactly four points.
• Axiom 2: Any two distinct points have exactly one line on both of them.
• Axiom 3: Each line is on exactly two points.
• Models offer insight into an axiomatic system.
• If points are interpreted as dots on the paper and lines as pencil lines, a model of the four-point geometry can be represented.

Definitions and Theorems

• Two lines on the same point intersect and are called intersecting lines.
• Two lines that do not intersect are called parallel lines.
• Four-Point Theorem 1: If two distinct lines intersect in the four-point geometry, then they have exactly one point in common.
• Four-Point Theorem 2: The four-point geometry has exactly six lines.
• Four-Point Theorem 3: Each point of the four-point geometry has exactly three lines on it.
• Four-Point Theorem 4: Each distinct line in the four-point geometry has exactly one line parallel to it.

Fano's Geometry

• Axiom 1: There exists at least one line.
• Axiom 2: There are exactly three points on every line.
• Axiom 3: Not all points are on the same line.
• Axiom 4: There is exactly one line on any two distinct points.
• Axiom 5: There is at least one point on any two distinct lines.
• Fano's Theorem 1: Two distinct lines in Fano's geometry have exactly one point in common.
• Fano's Theorem 2: Fano's geometry contains exactly seven points and seven lines.

Young's Geometry

• Axiom 1: There exists at least one line.
• Axiom 2: There are exactly three points on every line.
• Axiom 3: Not all points are on the same line.
• Axiom 4: There is exactly one line on any two distinct points.
• Axiom 5: For each line l and each point P not on l, there exists exactly one line on P that does not contain any points on l.
• Young's Theorem 1: Every point in Young's geometry is on at least four lines.

Axioms for Incidence Geometry

• Undefined terms consist of point, line, and on.
• Incidence Axiom 1: For each two distinct points, there exists a unique line on both of them.
• Incidence Axiom 2: For every line, there exist at least two distinct points on it.
• Incidence Axiom 3: There exist at least three distinct points.
• Incidence Axiom 4: Not all points lie on the same line.
• Incidence geometry satisfies all four incidence axioms.

Incidence Theorems

• Incidence Theorem 1: If two distinct lines intersect, then the intersection is exactly one point.
• Incidence Theorem 2: For each point, there exist at least two lines containing it.
• Incidence Theorem 3: There exist three lines that do not share a common point.
• If we consider any line l and any point P where P is not on l, then three possibilities exist for a parallel axiom:
  • There exist no lines on P that are parallel to l.
  • There exists exactly one line on P that is parallel to l.
  • There exists more than one line on P parallel to l.
• An incidence geometry is Euclidean or has the Euclidean parallel property if it assumes alternative 2 or has axioms implying an equivalent statement.
• If the incidence geometry assumes alternative 1 or 3 or its axioms implying an equivalent statement, then it is non-Euclidean.
• The four-point geometry has the Euclidean parallel property since its axioms imply alternative 2.
• Young's geometry has the Euclidean property since its Axiom 5 is equivalent to alternative 2.
• Fano's geometry has the non-Euclidean property since its Axiom 5 is equivalent to alternative 1.