Section 2.2: Axiomatic SystemsMAT 333 Fall 2008

Goals

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”?

Euclid’s 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”)

What are axioms?

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?

Axioms

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

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

The Common Notions

1. 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.

The Postulates

1. A straight line segment can be drawn by joining any two points.

2. A straight line segment can be extended indefinitely in a straight line.

3. Given a straight line segment, a circle can be drawn using the segment as radius and one endpoint as center.

4. All right angles are equal.5. 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.

The Fifth Postulate

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”?

Theorems

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

Definitions

Euclid includes definitions for 23 terms at the beginning of Elements, some of which are listed here. How many can you define?

Euclid did not define the term “distance.” Can you?

Point Obtuse Angle Semicircle

Line Segment Acute Angle Equilateral Triangle

Endpoints Circle Square

Line Center of a Circle Rhombus

Perpendicular Diameter Parallel Lines

Euclid’s Definitions

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?

Undefined Terms and Unproved Truths

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

The Modern View

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.

An Example: Committees

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

Models

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}

Another Example: Monoid

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.

Models of Monoids

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

An Example Theorem

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.

Independence

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

An Example

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}

Independence of the 5th Postulate

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!

Consistency

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

Completeness

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

Onward to Geometry

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