Lecture: Axiom Systems and Models

A fundamental aspect of this course is to study different sets of axiom systems. For some of your Group Projects, each group will develop its own set of undefined terms, defined terms, axioms, and theorems. It is important for you to have a basic understanding of an axiomatic approach since it is used in the development of all of modern mathematics.

An axiomatic system has the following four components:
  1. Undefined terms: an axiomatic system contains a set of technical terms deliberately chosen as undefined terms and are subject to the interpretation of the reader

  2. Definitions: All other technical terms of the system are ultimately defined by means of the undefined terms. These terms are the definitions of the system.

  3. Axioms: An axiomatic system contains a set of statements, dealing with the relationships between the undefined terms and definitions, that are chosen to remain unproved. These are the axioms (or postulates) of the system and are accepted as true.

  4. Theorems: All other statements of the system must be logical consequences of the axioms. These derived statements are called the theorems of the axiomatic system.
NOTE: Historically, in what is called the classical axiomatic system, points, (no length, breadth or width); lines (one dimension length, infinite), and planes (2 dimensions, infinite) were defined. In modern axiomatic systems points, lines, and planes are left as undefined and the axioms specify relationships between the points, lines, and planes.

Before introducing some geometric axiom systems, we pause to develop an elementary system. (This material was adopted from Wallace, E.C., and West, S.F. (1998). Roads to Geometry 2nd Ed., Upper Saddle River, NJ: Prentice Hall.)

Undefined terms: Fe's, Fo's, and the relation ``belongs to.''

Axiom 1: There exist exactly three distinct Fe's in this system.

Axiom 2: Two distinct Fe's belong to exactly one Fo.

Axiom 3: Not all Fe's belong to the same Fo.

Axiom 4: Any two distinct Fo's contain at least one Fe that belongs to both.

Some theorems that could be proved from this axiom system are:

Theorem 1: Two distinct Fo's contain exactly one Fe

Theorem 2: There are exactly three Fo's.

Theorem 3: Each Fo has exactly two Fe's that belong to it.
At this point, this system seems is quite abstract and it is difficult to determine whether or not the theorems `make sense' or not. By giving a meaning to each of the undefined terms in a system, we can create an interpretation - or model - of the system. If, for a given interpretation of a system, all the axioms are ``correct'' statements, then we can use the interpretation to gain an intuitive understanding of the axiom system. The danger, as we will see later, comes in confusing statements about the model with statements about the axiom system.

Now suppose we define Fe's as people; and Fo's as committees, then the axioms become:

  1. There are exactly three distinct people.

  2. Two distinct people belong to exactly one committee.

  3. Not all people belong to the same committee.

  4. Any two distinct committees contain one person who belongs to both.
Now the relationships are more familiar and more easily interpreted. We can now test whether a particular arrangement of people on committees meet all of the axioms.


Arrangement #1

Test this arrangement against each axiom. You should find that this arrangement meets all four axioms. Thus, this arrangement is a model for the Fe-Fo axiom system.

Arrangement #2

Test this arrangement against each axiom. You should find that this arrangement fails to meet axiom #2. Thus, this arrangement is NOT a model for the Fe-Fo system.

Consider another way of defining the Fe's and Fo's. Designate the Fe's as books and the Fo's as horizontal shelves and the relation is ``is on''.

The axioms become:

  1. There are exactly three distinct books.

  2. Two books are on exactly one shelf.

  3. Not all books are on the same shelf.

  4. Any two distinct shelves contain one book that is on both.

Since Axioms 2 and 4 are not correct statements, this interpretation is NOT a model for the Fe-Fo system.

Now we turn to a geometric example

.

Incidence Axioms for points, lines and planes:


Undefined terms:


point, line, plane, space



Definitions:


Universal set - set of all points in space - ( S)

collinear - two or more points that lie on the same line

concurrent - two ore more lines or planes which meet at the same point

coplanar --two or more points or lines that lie in the same plane


Axioms


1. Given any two distinct points, there is exactly one line containing them.

2. Given any three distinct noncollinear points, there is exactly one plane containing them.

3. If two points lie in a plane, then any line containing those two points lies in that plane.

4. If two planes intersect, then their intersection is a line.

5. Space consists of at least four noncoplanar points, and contains at least three non-collinear points. Each plane contains at least three noncollinear points, and each line contains a least two distinct points.


Theorems:


1. If two lines intersect, then they have one point in common.

2. If two planes intersect, then they intersect in a line.

3. Given a line and a point not on the line, there is exactly one plane containing the line and the point.

4. If two lines intersect, then their union lies in exactly one plane.

5. There are at least six lines and at least four planes.



Note that the first two axioms assert two things that an object exists and what is more, there is only one such object. For instance, given distinct points P and Q we are given 1) that there is a line m that contains P and Q and 2) if n is a line containing P and Q, then m=n. Similarly, if P,Q and R are three noncollinear points, there is 1) a plane E that contains the points P,Q, and R and 2) if F is a plane containing P,Q and R, then F=E.

The proofs of Theorems 1 and 2 are both indirect proofs (i.e., one demonstrates the contrapositive or that the negation of the statement leads to a contradiction). We give the proof of theorem 2 as an example.

Theorem 2: If a line intersects a plane not containing it, then the intersection is a single point.

Proof. Let L be a line intersecting a plane E, but not lying in E. Note that LÇ E contains at least one point P. For the sake of contradiction, suppose there exists another point Q contained in LÇ E

Since both P and Q are contained in L, axiom 1 yields that L=PQ. (If PQ and L were distinct lines they would have to intersect in exactly 1 point by Thm.1; since they intersect in 2 points, they are the same line.) Now, P and Q also lie in plane E since they are in the intersection of L and E. Hence axiom 3 implies that L lies in plane E by Since this contradicts the assumption that L is not contained in E, the theorem is established.

It is possible to have direct proofs, where one moves from the hypothesis to the conclusion in a sequence of logical steps. For example:

Theorem: If the geometry contains five points, no three of which are collinear, then the geometry contains at least ten lines.

Proof. Suppose that S={P,Q,R,S,T} is a set of five points, no three of which are collinear. Then each pair of points determines a distinct line. (Suppose not: Then S contains points P1,P2,P3 and P4 such that at P1, P2,P3 are distinct and P1P2=P3P4. Then P1,P2 and P3 are collinear. But this contradicts the assumption that no three points of S are collinear). Hence each of the lines PQ,PR,PS,PT,QR,QS,QT,RS,RT and ST are distinct and consequently the geometry contains at least ten lines.



Models of Axiom Systems


Given an axiom system, one would typically like to know two things about it: are the axioms consistent and are the axioms independent. An axiom system is consistent if the statements do not imply a contradiction. For instance an axiom system that required lines to contain a finite number of points and also to contain an infinite set of points would not be consistent. A statement is said to be independent of a set of axioms provided the axioms cannot be used to either prove or disprove the statement.

One way to test for an axiom systems consistency is to construct a model of the system; this is, in effect, what we did when gave meaning to Fe and Fo in our first example. One constructs a model of an axiomatic system by finding an interpretation of the undefined terms using a collection of objects that one has an understanding of and that stand in relation to one another in the manner specified by the axioms. For instance, a model for a geometry is a collection of objects that have been defined to be points, lines, et cetera that satisfy the axioms of the geometry. This is best understood through some examples. In order to keep our examples simple, we shall restrict ourselves to some models of the following axiom system.


Incidence Axioms for the plane

1. Any two points are contained in exactly one line.

2. Any line contains at least two points

3. There are at least three noncollinear points.


To build a model for this system we have to find some objects to play the role of points and lines. Note that if we can find such objects, then the axioms are consistent.


A three point geometry.

Let the points be the numbers 1,2,3 and the lines be the sets {1,2},{1,3} and {2,3}. Clearly these points and lines satisfy the incidence axioms for the plane.


A four point geometry.

Let the points be the numbers 1,2,3,4 and the lines be the sets {1,2}, {1,3}, {1,4}, {2,3}, {2,4}, {3,4}. Note that these points and lines also satisfy the incidence axioms for the plane.


A nine point geometry.

Let the points be the numbers 1,2,3,4,5,6,7,8,9 and the lines be the sets {1,2,3}, {4,5,6}, {7,8,9},{1,4,7}, {2,5,8}, {3,6,9},{1,6,8}, {2,4,9}, {3,5,7}, {1,5,9}, {2,6,7}, {4,8,3}.


A seven point geometry.

Let the points be 1,2,3,4,5,6,7 and the lines be {1,2,3}, {1,7,4}, {1,6,5}, {3,4,5}, {2,7,5}, {3,7,6}, {2,6,4}.


In order to test the independence of a statement in relation to a given axiom system, one procedure is to construct two models: one which satisfies the axioms and the statement and another that satisfies the axioms and the negation of the statement. For instance, if one wanted to show the statement


``given a line and a point not on it, there is a line containing the given point that does not intersect the given line''


is independent from the incidence axioms for the plane one could observe that the statement is not true in the seven point geometry and true in the nine-point geometry.

 

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