Euclid and the Elements
Section 1.2
1.
Axiomatic System:
Generally, an axiomatic system consists of

Undefined terms

Defined terms

Axioms

Propositions, and

Rules of inference, or logic
2.
An axiomatic system is said to be consistent if neither the axioms nor the
propositions of the system contradict one another.
3.
An axiom is said to be independent of the other axioms if it cannot be derived
from the other axioms.
4.
An axiomatic system is complete if it possible for every properly posed statement
to be proved or disproved. Alternatively, it is not possible to add a new independent
axiom to the system.
Kurt Godel showed that axiomatic systems such as arithmetic and geometry are
incomplete.
The concepts of consistency, independence, and completeness of axiomatic systems a re
usually introduced using models of finite geometries.
Finite Geometries
Finite Geometries can be traced back to Gino Fano (1892) with some ideas going back to
von Staudt (1852).
A finite geometry consists of a finite set of points and an associated set of terms and
axioms.
Four-point geometry
Undefined terms: 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.
Axiom 1: There exists exactly four points
Axiom 2: Two distinct points are on exactly one line
Axiom 3: Each line is on exactly two points
Model:
Points: A, B, C, D
Lines: AB, AB, AD, BC, BD, CD
Note:
1.
The model demonstrates consistency – verify this!
2.
Independence – Verifying an axiomatic system is independent is also done
through models. To establish that an axiom A in axiomatic system is independent, we
first construct a new axiomatic system just by replacing the axiom A by its negation and
leaving the rest of the axioms as they are, and then we find a model for this new
axiomatic system. Thus if the original system has n axioms, we repeat the above process
n times, one for each axiom. If all the axioms of the original system can be shown to
independent, then it is an independent axiomatic system.
Axioms of the four-point geometry are independent
Negate Axiom 1
Axiom 1: There are exactly 2 points
Axiom 2: Two distinct points are exactly on one line
Axiom 3: Each line is on exactly two points
Points: A, B
Line: AB
Verification: Axiom 2, and 3 are true
Negate Axiom 2
Axiom 1: There are exactly 4 points
Axiom 2: Two distinct points are not exactly on one line
Axiom 3: Each line is on exactly two points
Points: A, B, C, D
Line: AB, CD
Verification: Axiom 1, and 3 are true
Negate Axiom 3
Axiom 1: There are exactly 4 points
Axiom 2: Two distinct points are exactly on one line
Axiom 3: Each line is not on exactly two points
Points: A, B, C, D
Line: ABC, AD, BD, CD
Verification: Axiom 2, and 3 are true
The Geometry of Euclid
Euclidean geometry is based on defined terms and 5 axioms. He defined the terms point
& line. Euclid’s 5 axioms are (restated in contemporary English):
1. A line may be drawn between any two points;
2. A segment of any length may be constructed in any line;
3. A circle with any radius any center may be drawn;
4. All right angles are equal;
5. If a straight line falling on two straight lines makes the interior angles on the same
side less than two right angles, the two straight lines, if produced indefinitely,
meet on that side on which the angles are less than two right angles.
Equivalently(logically), this axiom is also stated as:
Through a point not on a given line, exactly one line may be drawn parallel to the
given line – called Playfair’s Postulate.
Exercise:
Three Point Geometry is defined as follows:
Undefined Terms: Point, line, on.
Axiom 1: There exists exactly three points
Axiom 2: Every pair of points is on exactly one line
Axiom 3: Every pair of lines is on at least one point
Axiom 4: No three points are collinear
a) Is this system consistent? Justify.
b) Is this system independent? Justify
[This is an example of a complete system]
Some Theorems in 3 point geometry
Theorem 1.1 : Two distinct lines are on exactly one point.
Theorem 1.2 : The three point geometry has exactly three lines.
Theorem 1.1: Two distinct lines are on exactly one point.