\documentclass[12pt]{article}
\usepackage{graphicx}
\usepackage{url}
\usepackage[left=1.0 in, right=1.5 in, top=1.8 in, bottom=1.2 in]{geometry}
\setlength{\parindent}{0em}
\setlength{\marginparwidth}{1.2in}
\newcommand\margin[1]{\-\marginpar[\raggedleft\footnotesize #1]%
{\raggedright\footnotesize #1}}

\def\tri{\bigtriangleup}
\def\ang{\angle}
\def\so {.\cdot .}
\setlength{\parskip}{2ex}
\title{Axiomatic Systems for Geometry\footnote{From 
textit{Post-Euclidean Geometry: Class Notes and Workbook}, UpClose Printing 
\& Copies, Champaign, IL 1995, 2004} 
}
\author{George Francis\footnote{Prof. George K. Francis, Mathematics 
Department, University of 
Illinois, 1409 W. Green St., Urbana, IL, 61801. (C) 2010 Board of Trustees.}  
}
\date{composed 6jan10, adapted 27jan15} 

\begin{document}

\maketitle

\section{Basic Concepts}
\margin{Adaptation to the current course is in the margins.}
An axiomatic system contains a set of {\it primitives} and {\it axioms}. The 
primitives are object names, but the objects they name are left 
undefined. This is why the primitives are also called {\it undefined 
terms}. The axioms are sentences that make assertions about the 
primitives. Such assertions are not provided with any justification, 
they are neither true nor false. However, every subsequent assertion 
about the primitives, called a {\it theorem}, must be a rigorously logical 
consequence of the axioms and previously proved theorems. There are 
also formal {\it definitions} in an axiomatic system, but these serve only 
to simplify things. They establish new object names for complex 
combinations of primitives and previously defined terms. This kind of 
definition does not impart any `meaning', not yet, anyway.
\margin{See Lesson A6 for an elaboration.}

If, however, a definite meaning is assigned to the primitives of the 
axiomatic system, called an {\it interpretation}, then the theorems 
become meaningful assertions which might be true or false. If for a 
given interpretation all the axioms are true, then everything asserted
by the theorems also becomes true. Such an interpretation is called 
a {\it model} for the axiomatic system. 

In common speech, `model' is often used to mean an example of a class 
of things. In geometry, a model of an axiomatic system is an 
interpretation of its primitives for which its axioms are true. Since 
a contradiction can never be true, an axiom system in which a 
contradiction can be logically deduced from the axioms has no model. 
Such an axiom system is called {\it inconsistent}. On the other hand, 
if an abstract axiom system does have a model, then it must be 
consistent because each axiom is true, each theorem is a logical 
consequence of the axioms, and hence it is true, and a contradiction 
cannot be true. 

\margin{Hvidsten, in his textbook ``Geometry with Geometry Explorer", 
McGrawHill, 2004, devotes a large part of Chapter 1 giving examples for
these concepts using finite geometries. Earlier editions of 
MA402 treats this subject as well. In this edition of the course, we 
regrettably dispense with it in the interest of having more time on
other topics.} 
Finally, an axiom system might have more than one model. If two models
of the same axiom system can be shown to be structurally equivalent, 
they are said to be {\it isomorphic}. If all models of an axiom system
are isomorphic then the axiom system is said to be {\it categorical}. 
Thus for a categorical axiom system one may speak of {\it the} model; 
the one and only interpretation in which its theorems are all true. 

All of these qualities: truth, logical necessity, consistency, 
uniqueness were tacitly believed to be the hallmark of classical 
Euclidean geometry. At the start of the 19th century, a scant 200 
years ago, philosophers and theologians, physicists and mathematicians
were all persuaded that Euclidean geometry was absolutely the one and 
only way to think about space, and therefore it was the job of 
geometers to develop their science in such a way as to demonstrate 
this necessary truth. By the end of the century, this belief had 
been thoroughly discredited and abandoned by all mathematicians.\footnote{ 
Curiously, it persists even today among some irresponsible, but 
influential amateurs. In her column, ``Ask Marilyn" in
Parade Magazine, November 21, 1993, the world's most intelligent
woman [sic] rejects Hyperbolic Geometry on nonsensical grounds, see
\url{http://en.wikipedia.org/wiki/Marilyn_vos_Savant}.}

The main theme of our course concerns the evolution of this 
idea, and its replacement by the much richer, far more illuminating, 
post-Euclidean geometry of today. It is about a method of thought,
called the {\it axiomatic method.} Although at one time this method 
may have developed merely from a practical need to verify the rules 
obtained from the careful observation of physical experiments, this
changed with the Greek philosophers. The axiomatic method has formed
the basis of geometry, and later all 
of mathematics, for nearly twenty-five hundred years. It survived a 
crisis with the birth of non-Euclidean geometry, and remains today one
of the most distinguished achievements of the human mind.

As we noted earlier, the transition of geometry from \textit{inductive 
inference} to \textit{deductive reasoning} resulted in the development of 
axiomatic systems.  Next, we look at four axiom systems for 
Euclidean geometry, and close by constructing a model for one of them.

\section{Euclid's Postulates:} 

Earlier, we referred to the basic assumptions as `axioms'. 
Euclid divided these assumptions into two categories --- 
postulates and axioms. The assumptions that were directly related 
to geometry, he called {\it postulates}. Those more related to 
common sense and logic he called \textit{axioms}. Although modern geometry no 
longer makes this distinction, we shall continue the ancient custom and 
refer to axioms for geometry also as postulates. 

Here is a paraphrase\footnote{Thomas L. Heath, ``The Thirteen Books of
Euclid's Elements'', Cambridge, 1908.} 
of the way Euclid expressed himself. 

Let the following be postulated:
\margin{MA402 lessons A2 and E1 ff} 
\begin{description}
\item[Postulate 1:] To draw a straight line from any point to any point.
\item[Postulate 2:] To produce a finite straight line continuously in a straight line.
\item[Postulate 3:] To describe a circle with any center and distance.
\item[Postulate 4:] That all right angles are equal to one another.
\item[Postulate 5:] That, 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.  
\end{description}

Note that the wording suggests construction. 
Euclid assumed things that he felt were too obvious to justify further. 
This caused his axiomatic system to be logically incomplete. 
Consequently, other axiomatic systems were devised in an attempt 
to fill in the gaps. We shall consider three of these, due to David 
Hilbert, (1899),
George Birkhoff, (1932) and the School Mathematics Study Group (SMSG), 
a committee that began the reform of high school geometry in the 1960s. 
\newpage
\section{Hilbert's Postulates:}      
\margin{The following sections may be skimmed with profit now, but 
will not be easy to follow until additional lessons on Euclid's 
Elements and the his Parallel Postulate have been studies.}
In the late 19th century began the critical examinations into the 

foundations of geometry. It was around this time that 
David Hilbert(1862 - 1943) introduced his axiomatic system. 
The primitives in Hilbert's system are the sets of {\it points},
{\it lines}, and {\it planes} and relations, such as 

\begin{tabular}{ll}
incidence: & as in `a point $A$ is on line $\ell$' \\
 order:& as in `$C$ lies between points $A$ and $B$'\\
congruence:& as in ` line segments $AB \cong A'B'$' \\
\end{tabular}

An example of a formal definition would be that of a {\it line segment}
$AB$ as the set of points $C$ between $A$ and $B$.  
He partitioned his axioms into five groups; axioms of connection,order, 
parallels, congruence and continuity.\footnote{cf. Wallace and West, 
``Roads to Geometry", Pearson 2003, Chapter 2 for a more detailed 
discussion of Hilbert's axioms.} 
Hilbert's axiom system is important for the following two 
reasons. It is generally recognized as a flawless version of what 
Euclid had in mind to begin with. It is purely geometrical, in that 
nothing is postulated concerning numbers and arithmetic. Indeed,
it is possible to model formal arithmetic inside Hilbert's axiomatic
system. 

We wish to show how Euclidean geometry can be modelled inside 
arithmetic.\footnote{The historical significance of these two exercises
in building models of formal systems is the irrefutable demonstration 
that geometry and arithmetic are equi-consistent. That means, if you 
believe the one to be without contradiction, then you are obliged to 
accept the
other also, and vice-versa. Hilbert's program for a proof that one, and hence
both of them are consistent came to naught with G\"odel's Theorem. According
to this theorem, any formal system sufficiently rich to include arithmetic,
for example Euclidean geometry based on Hilbert's axioms, contains true but
unprovable theorems.} For this purpose,  we want the shortest possible list
of primitives and postulates, for then, we have less to check. Birkhoff 
meets this requirement.  

\section{Birkhoff's postulates}
\margin{These concepts are treated in greate detail in the  module on 
Cartesian Geometry, Lesson C1. And also in Hvidsten \textit{op. cit.} 
Chapter 3}
The primitives here are the set of {\it points}, a system of subsets of
points called {\it lines}, and two real-valued functions, `distance' and 
`angle'. That is, for any pair of points, the distance $d(A,B)$ is a 
positive real number. For any ordered triple of points $A,Q,B$, 
the real number $m\ang AQB$ is well defined\footnote{To 
distinguish the figure $\ang AQB$, which we call an `angle', the number
$m\ang AQB$ is called the {\it angular measure} of the angle. Moreover,
two real numbers that differ by a multiple of $2\pi$ measure the same 
angle.} modulo $2\pi$.

\begin{description}
\item[Euclid's Postulate:] 
A pair of points is contained by one and only one line.
\item[Ruler Postulate:]
For each line there is a 1:1 correspondence between its points and the 
real numbers, in such a way that if $A$ corresponds to the real number
$t_A$ and $B$ corresponds to $t_B$ then \[ d(A,B) = |t_B - t_A| \].
\item[Protractor Postulate:]
For each point $Q$, there is a 1:1 correspondence of 
its rays\footnote{Note that once we can apply 
a ruler to a line, we can identify one of the two half-lines, or rays, at
a point $Q$ as those points $P$ on the line for which $t_P > t_Q$.}
and the real numbers\footnote{We might call these the {\it circular
numbers} because they lie on the {\it
number circle}, just as one speaks of the real numbers lying on 
{\it number line}.} modulo $2\pi$, in such 
a way that if ray $r$ corresponds to the circular number $\omega_r$ and ray
$s$ to $\omega_s$ then \[m\ang RQS = \omega_s - \omega_r (mod 2\pi) \] 
where $R$ is a second point on $r$ and $S$ on $s$ . 

\item[simSAS Postulate:]
If  $m\ang PQR = m\ang P'Q'R'$ and $d(PQ):d(P'Q')=d(QR):d(Q'R')=k$ then
the other four angles are pairwise equal, and the remaining side pairs have 
the same ratio. 
\end{description}

One says that such triangles are {\it similar}, $\tri PQR \sim \tri P'Q'R'$
with {\it scaling factor} $k$. Of course, for $k=1$, 
$\tri PQR \cong \tri P'Q'R'$ .

\section{The SMSG Postulates}

There are 22 of these,\footnote{Cf. Appendix of Wallace and West,
{\it op.cit.} } and they combine the flavor of Hilbert and Birkhoff.
With Birkhoff, rulers and protractors are postulated, under the valid 
impression that children already know how to deal with real numbers by
the time they study geometry. There are many postulates so that proofs 
of interesting theorems can be constructed without the tedium of proving
hundreds of lemmas first. Of course, unlike Birkhoff's foursome, the 
SMSG postulates are redundant, in that some postulates can be logically
derived from others. The pedagogical wisdom and usefulness of the SMSG 
axiom system is a matter of some debate among educators.

\section{A Cartesian Model of Euclidean Geometry}

\margin{Hvidsten 3.6. Class lesson C2 }
We next give an example of an axiomatic system and a model for it. 
For this purpose we choose a very familiar area of mathematics in 
which to interpret the primitives and to test the truth of the axioms.
We all know analytic plane geometry from high school, also known as
{\it Cartesian geometry}.  Birkhoff's 
four postulates for Euclidean geometry appear compact enough for us 
not to lose our way. 

We interpret the {\it points} $A,B,C ...$ as ordered pairs, $(x,y)$, of real 
numbers. {\it Lines} shall be solution sets to 
linear equations of the form 
$ax+by+c=0$. A point $(p,q)$ is {\it incident} to the line 
$ax+by+c=0$ if it 
satisfies the equation, i.e. if $ap+bq+c=0$ is true. Remember that 
the distance between two points and the angle measure are also 
primitives and need an interpretation. We shall do that later.

With just this much we can already attempt to verify the first 
postulate which asserts the existence and uniqueness of a line through
two given points. You could do this yourself by deriving the formula for
the line through two points $(x_0,y_0), (x_1,y_1)$ in any of the many 
ways you learned to do this in high school. 
Here we do this by solving 
this system of two linear equations for the as yet unknown 
parameters $a,b,c$ :
\[
\begin{array}{clc}
ax_0 + by_0 + c &=& 0 \\
ax_1 + by_1 + c& =& 0 \\
a(x_1 - x_0) + b(y_1 - y_0)& =& 0 \\
\end{array}
\]
The third equation
eliminates $c$ for the moment; we can recover it as soon as we know $a,b$,
for example thus:
\[
c = -ax_0 - by_0. 
\]
One plausible choice for a,b would be
\[
a = -(y_1-y_0),\,  b = (x_1-x_0)
\]
because it fits the third equation and yields
\[
c = x_0y_1 - x_1y_0 = \left| \begin{array}{cc}
                              x_0 & y_0 \\
                              x_1 & y_1   
                              \end{array}
                       \right|
\]

While this shows that both points lie on some line, it does not 
demonstrate the uniqueness of this line. Indeed, our interpretation is
incomplete. If we really want the first postulate to hold we must 
agree that the same line may have more than one equation, provided the
same set of points is the solution set for each. We therefore amend 
our interpretation of a line by stressing that 
\[ \begin{array}{c}
a_0x + b_0y + c_0 = 0 \\
a_1x + b_1y + c_1 = 0
\end{array}\]

define the same line provided the parameters are proportional:
\[
a_0:a_1 = b_0:b_1 = c_0:c_1.
\]

The distance function $d(A,B)$ Birkhoff had in mind is, of course, the 
Euclidean distance as derived from the Pythagorean theorem:

For $A = (x_0,y_0), B = (x_1, y_1),$ 
\[
d(A,B) = \sqrt{(x_1-x_0)^2 + (y_1-y_0)^2}
\]


We are now ready to verify Birkhoff's ruler postulate in a 
particularly useful fashion. First we give our two arbitrary points 
more mnemonic names:
\(
Q = (x_0,y_0), I = (x_1,y_1). 
\)
Now there is a canonical way of labelling
all other points on the line $QI$ with real numbers $t$ in several useful 
ways as follows:
\[
\left[ \begin{array}{c} x_t \\ y_t \end{array} \right] 
\left[ \begin{array}{c} x_0 \\ y_0  \end{array}  \right]                      
(1-t) + t 
\left[ \begin{array}{c} x_1 \\ y_1  \end{array} \right]
\]

In vector notation this might be written as
\[
P_t = Q(1-t) + tI = Q + t(I-Q).
\]
Notice that $P_0=Q$ and $P_1=I$, 
and that the points on the segment $QI$ 
are given by the set 
\[ \{P_t | 0 < t < 1 \}.\] 

There is still something to prove here, namely that the Euclidean 
distance is in fact measured by our ruler. Once again we were too 
hasty in ruling lines. For the Euclidean distance
\[
d(Q,I) = \sqrt{(x_1-x_0)^2+(y_1-y_0)^2}
\]
which need not equal the parametric distance, which is 1. We may, 
however, rescale our ruler by a unit 
\( u=d(Q,I)\), to yield another parameter, \( s= tu\), for which 
the points $P_s$ on the line are given by \(P_s = Q + sU\), where
$U$ is the unit vector, \( U= (I-Q)/d(I,Q)\), in the direction
if $I$ from $Q$ . This way, $I$ is the correct distance, $d(I,Q)$,
away from $Q$ on this ruler for the line.  

For the remaining pair of Birkhoff's postulates we need a protractor, 
i.e. a device for measuring angles. The simplest way of doing this in 
our model is to recall the definition of the dot product of two 
vectors and interpret:
\[
m\ang AQB = \arccos(\frac{A-Q}{d(A,Q)} \cdot \frac{B-Q}{d(B,Q)}).
\]


Birkhoff's axiom system achieved its remarkable economy by postulating
what turns out to be the quintessential property of Euclidean 
geometry. What distinguishes it from non-Euclidean geometry are 
the properties of {\it geometric similarity}. Two shapes are similar if they
differ only in scale. Birkhoff postulates that two triangles with a 
similar corner, are wholly similar. By a corner we mean, of course, a 
vertex and its adjacent edges. If the proportionality factor is 1, 
then this postulate says that two triangles are congruent as soon as 
they have one congruent corner. 

We shall verify the simSAS postulate, which makes an assertion about two 
triangles, by carefully measuring one triangle. Just as today we 
exchange goods by means of their price, instead of bartering items for
each other, so modern geometry compares shapes by comparing their 
measurements. 

Given a triangle $\tri ABC$, vital statistics consists of six numbers, the 
three angles and sides, 
\[
\begin{array}c
\alpha = m\ang A\\ 
\beta  = m\ang B \\
\gamma = m\ang C \\
a = d(B,C) \\
b = d(C,A) \\
c = d(A,B).
\end{array}
\]

The law of cosines, which generalizes the Pythagorean theorem to 
arbitrary triangles by resolving the square of a side in terms of the
opposite corner:
\[
c^2 = a^2 + b^2 - 2ab \cos \gamma. 
\]
allows us to measure $c$ in terms of the measures of two sides and
the included angle.

\section{Problems}

These problems are assigned for submission elsewhere. As you study this
lesson and as you solve these problems, enter their solutions into your Journal for
future reference.

{\bf Problem 1:} 

Above we have given the parametric equations for a line.
Recall how to eliminate the paramater $t$ to obtain the
non-parametric equation of a line you may be more 
familiar with.

{\bf Problem 2:}  

To demonstrate that you understood our verification of Birkhoff's 
Ruler Axiom, see if you can do likewise for his Protractor Axiom.
In other words, you first define the angle measure of every angle
as some circular number between $0$ and $2\pi$. Recall that 
was done in terms of a unit circle centered at the vertex of 
the angle. The angle measure is the length of the arc between
the two points the legs of the angle cross the circle.

Next, show that parametrization of this unit circle by the cosine
and sine of a real number  establishes a bijection to the 
circular reals  of the angle from the from the positive x-axis.
There remains to prove that any angle situated at that origin
is measured by the difference of the parameters. 

As you complete this non-trivial exercise, you will discover
that you need to use a well known trig identity.

{\bf Problem 3:} 

To finish the proof that Birkhoff's simSAS axiom holds in Cartesian
geometry, you should first
use vector algebra and the definition of the dot product to verify the
law of cosines. Hint: Multiply out
\[
C^2 = (B-A)^2 = ((B-C)-(A-C))^2.
\]

Thus, knowing $a,b$ and $\gamma$, we calculate $c$. If $a$ and $b$ are 
stretched, or shrunk by the same factor, so is $c$, provided $\gamma$ 
remains the same. 

{\bf Problem 4:}

Continuing.
Thus, knowing $a,b$ and $\gamma$, we calculate $c$. If $a$ and $b$ are 
stretched, or shrunk by the same factor, so is $c$, provided $\gamma$ 
remains the same. 

Apply the law of cosines to the other two sides to calculate 
$\alpha$ and $\beta$ as functions of $a,b$ and $\gamma$.   

{\bf Problem 5:}

A generalization of Euclid's proof of the Pythagorean theorem
leads to another proof of the law of cosines. Label an 
arbitrary acute triangle in the standard way. Construct squares on
two of its sides, say $b$ and $c$. Extending the altitudes
from $C$ and $B$ partitions the squares into rectangles
\[ \begin{array}{c} 
b^2 = b b_1 + b b_2 \\ 
c^2 = c c_1 + c c_2 \\ 
\end{array}\]
Euclid's argument (do it!) proves that $c c_1 = b b_2 $

Now drop the third altitude from $A$. Of course (can you
prove this?) it passes through the same point where the 
first two altitudes intersected,\footnote{This point is
called the {\it orthocenter} of the triangle.} and 
partitions the third square into two rectangles. 


\[ \begin{array}{rl}
a a_2 =& b b_2 = a b \cos C \\
c^2   =& c c_1 + c c_2 \\ 
      =& b b_2 + a a_1 \\
      =&( b^2 - b b_1) + (a^2 - a a_2)  \\
      =& b^2 + a^2 - 2 a b \cos C  \\
\end{array}\]

So, we can measure the rectangle, summarize our 
inferences and come up with the law of cosines.

Work your way through this argument, and supply details
which you had use to understand it, or had to look up
elsewhere. 

{\bf Problem 6:}

Generalize the above argument to work also for an 
obtuse triangle. Hint: Sometimes you need to 
add instead of subtract and vice-versa.




\end{document}





 

 

 




