Download Section 2.2: Axiomatic Systems

MAT 333 Fall 2008

As we discovered with the Pythagorean
Theorem examples, we need a system of
geometry to convince ourselves why
theorems are true

But what is a “system”?

The idea of systematizing mathematics was
unheard of when Euclid created his Elements
in 300 BCE.

Euclid’s work consisted of
 definitions
 postulates (what we would call “axioms”)
 propositions (what we would call “theorems”)

Axioms are statements that we assume to be
true without proof

Why are axioms necessary?

Shouldn’t we always prove things and not
assume they are true without proof?

We want axioms to be
 as few in number as possible
 as simple or “obvious” as possible

Let’s look at Euclid’s axioms and see how he
measures up to these standards

Euclid’s Axioms are divided into 5 “common
notions” and 5 “postulates”

The common notions are algebraic in nature,
while the postulates refer to basic properties
of geometry
Things which equal the same thing are equal to
one another.
2. If equals are added to equals, then the sums are
equal.
3. If equals are subtracted from equals, then the
remainders are equal.
4. Things which coincide with one another are
equal to one another.
5. The whole is greater than the part.
1.
1.
2.
3.
4.
5.
A straight line segment can be drawn by joining any two
points.
A straight line segment can be extended indefinitely in a
straight line.
Given a straight line segment, a circle can be drawn using
the segment as radius and one endpoint as center.
All right angles are equal.
If two lines are drawn which intersect a third in such a
way that the sum of the inner angles on one side is less
than two right angles, then the two lines inevitably must
intersect each other on that side if extended far enough.

This can be restated in modern terms like this:

Given a line and a point not on that line, there
is exactly one other line passing through the
point and parallel to the line.

Is this “simple” or “obvious”?

Euclid’s “propositions” are statements that
logically follow from his axioms – we would
call these “theorems”

Euclid (and many mathematicians after him)
attempted to prove the 5th Postulate as a
theorem so that it did not have to be
assumed without proof


Euclid includes definitions for 23 terms at the
beginning of Elements, some of which are
listed here. How many can you define?
Point
Obtuse Angle
Semicircle
Line Segment
Acute Angle
Equilateral Triangle
Endpoints
Circle
Square
Line
Center of a Circle
Rhombus
Perpendicular
Diameter
Parallel Lines
Euclid did not define the term “distance.” Can
you?

Euclid’s definition of point is “that which has
no part.”

Euclid’s definition of line segment is “a
breadthless length”

What do you think of these definitions?

We have seen that any mathematical system
must rely on undefined terms and axioms

Without these, we wouldn’t have anything to
talk about or anything to base our proofs on

An axiomatic system is a list of undefined
terms together with a list of statements
(called “axioms”) that are assumed to be true
without proof.

Our goal will be to create an axiomatic system
for geometry, but first we will need to
understand how these systems work in
general.

Undefined terms: committee, member

Axiom 1: Each committee is a set of three
members

Axiom 2: Each member is on exactly two
committees

Axiom 3: No two members may be together
on more than one committee

Axiom 4: There is at least one committee

A model for an axiomatic system is a way to
define the undefined terms so that the
axioms are all true

Here is a model for the committees system.
Check that all the axioms are true.

Members: Alan, Beth, Chris, Dave, Elena, Fred
Committees: {A,B,C}, {A,D,E}, {B,D,F}, {C,E,F}

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
Undefined terms: element, product
Axiom 1: Given two elements x and y, the
product of x and y, denoted x * y, is a unique
defined element
Axiom 2: Given elements x, y, and z, the
equation (x * y) * z = x * (y * z) is always true
Axiom 3: There is an element e, called the
identity, such that e * x = x = x * e for all
elements x.

What models of the monoid system can you
think of?

elements = integers, product = *
elements = real numbers, product = *
elements = integers, product = +
elements = 2x2 matrices, product = matrix
multiplication




Here is a theorem for the committees system

Theorem: There cannot be exactly four
members.

The proof involves assuming that there can be
four members and reaching a contradiction.

An axiom is independent from the other axioms
in a system if it cannot be proven from the other
axioms.

Euclid wanted to prove that his 5th postulate was
dependent on the other axioms, but could not
find a proof

If you can find a model where the axiom is false,
but all the other axioms are true, then the axiom
is independent

This model shows that Axiom 1 of the
Committees system is independent of the
others

Members: Alan, Beth, Chris, Dave
Committees: {A,B}, {B,C,D}, {A,C}, {D}


If we could find a model where Euclid’s
axioms (without the 5th postulate) are all true,
and the 5th postulate is false, we will have
proved that the 5th postulate is independent

The only way to convince ourselves that such
a model cannot exist is to prove it!

An axiomatic system is consistent if there are
no internal contradictions among the axioms

If some of the axioms contradict each other,
then they can’t all be true all at the same time

So finding a model of an axiomatic system is
enough to prove the axioms are consistent

An axiomatic system is complete if all
statements that are true in the system can be
proved from the axioms

There is a famous fact called Gödel’s
Incompleteness Theorem that tells us there is
no “sufficiently complex” axiomatic system
that is both consistent and complete

We will be using this kind of framework to
develop our system of geometry

We will start with some undefined terms and
a short list of axioms

We will expand the list of axioms only when
necessary