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Historical Overview of Axiomatic System and Finite
Geometry
Historically, when axiomatic systems were first being studied, distinctions
were made between axioms and postulates. The word postulate was used by
Euclid and other early Greek mathematicians to stand for an assumed truth
peculiar to one particular science, while an axiom was used as an assumption
common to all sciences. This difference was commonly made by early Greek
mathematicians. An example of the way these terms were used is shown in
Euclid’s book the Elements where Euclid made distinctions between postulates
and common notions (axioms). Today, however, both terms are used
interchangeably to refer to an assumed statement.
Undefined terms are included in axiomatic systems from historical
problems that early geometers ran into. Geometers, such as Proclus, Euclid, and
Plato, tried to define most terms in their work. Common examples of these terms
that they attempted to define were point, line, and plane; however, they quickly
ran into problems. While attempting to define these terms, geometers would
frequently run into the problem of needing to define a term with another
term. This process would lead to circular definitions or an infinite chain of terms
and their definitions. An example of Euclid’s use of circular definitions can be
seen in Euclid's definitions of point and line; Euclid defined a point as ‘that which
has no part’ and a line as ‘length without breadth.’ The questions then arise:
What is 'no part'? What is 'length'? What is 'breadth'? From this common fault
that mathematicians ran into, the mathematician, David Hilbert, was quoted as
saying, “we may as well be talking about chairs, coffee tables and beer mugs.
David Hilbert (1862–1943), a German number
theorist and mathematician, is one of the best known
mathematicians. Hilbert began his career in
algebra and number theory, and then moved on to
study geometry. His work in geometry was based
on Euclid's work in geometry from about 2000
years earlier. Euclid’s proofs, although
revolutionary for his time, did contain gaps
where he had made tacit assumptions, or
assumptions that were not mathematically
warranted. One such example is seen in “the
proof of Proposition I 16, where Euclid
unconsciously assumed the infinitude of straight
lines.” Hilbert’s early work with Euclid’s proofs and
axiomatic systems “has been more helpful in enabling mathematicians to pursue
the foundations of geometry.” His work transformed the flawed method used in
Euclid’s proofs to the format of axiomatic systems used today.
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An exciting time for Hilbert came in 1900 when he was asked to speak at
the International Congress of Mathematicians in Paris. During his speech, he
proposed 23 problems he challenged fellow mathematicians to solve in the
upcoming century. His professional life was remarkable; however, during the end
of his life, his career was “disappearing under the ideological onslaught from the
Nazi government” during World War II. This political government was the reason
that this distinguished mathematician’s funeral was attended by less than 12
people.
In 1904, Hilbert went to work on investigating mathematic logic and
proving the consistency of mathematics. His ideas triggered tension between
himself and other mathematicians, the most notable being Kurt Gödel.
Kurt Gödel (1906–1978), an Austrian
mathematician,
is
well-known
for
his
Incompleteness Theorem, which exposed
Hilbert’s ideas. The Incompleteness Theorem
was
published
in
1931
in
MonatsheftfürMathematik und Physik, a
German mathematical journal, with the title
“On Formally Undecidable Propositions of
Principia Mathematica and Related Systems.”
This theorem demonstrated that even in
elementary parts of arithmetic there exist
propositions which cannot be proved or disproved
within the system.
Gödel’s mathematical career and life was also affected by the Nazi
government, however, not to the same extent that Hilbert faced. Gödel decided
to leave Europe and the threat of the Nazi government in 1939 after receiving a
letter accusing him of interacting with Jews. After arriving in the United States,
Gödel remained active with mathematics and became close friends with Albert
Einstein.
Finite geometry followed the axiomatic systems in the late 1800’s. Finite
geometry was developed while attempting to prove the properties of consistency,
independence, and completeness of an axiomatic system. Geometers wanted
models that fulfilled specific axioms. Often the models found had finitely many
points which contributed to the name of this branch of geometry.
Gino
Fano
(1871–1952),
an
Italian
mathematician, is credited with being the first person
to work with finite geometry (in 1892). His first work
in this new geometry included 15 points, 35 lines,
and 15 planes, all in a three-dimensional plane.
Fano, like Hilbert and Gödel and many Europeans
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during the 1930’s and 40’s, was affected by the Nazi government. Fano also was
forced to leave Europe but did continue his mathematics career and began to
teach in the United States.
Even with Fano’s early work, it wasn’t until the early 1900’s that finite
geometry obtained a well-known role in mathematics. Considering the relatively
short history of finite geometries, there are still unsolved problems actively being
researched by leading mathematicians today.
Finite Geometry
Gino Fano (1871–1952) is credited with being the first person to explore
finite geometries beginning in 1892. He worked primarily in projective and
algebraic geometry. He was born in 1871 in Mantua, Italy. He initially studied in
Turin. Later, he moved to Göttingen and worked with Felix Klein. Fano served as
a professor of mathematics in Turin until he was forced to leave during World
War II. He also taught in Switzerland and the United States. He died in Verona,
Italy in 1952.
Fano initially considered a finite three-dimensional geometry consisting of
15 points, 35 lines, and 15 planes. Here, we obtain a finite geometry by
restricting the system to one of the planes.
In order for a configuration of points and lines to be considered a finite
geometry, several properties should be satisfied. These include finiteness,
uniformity, uniqueness, and existence. The desirable properties are:
1.
2.
3.
4.
5.
6.
7.
8.
The number of points is finite.
The number of lines is finite.
Each line is on the same number s of points, where s≥2.
Each point is on the same number t of lines, where t≥2.
Each pair of distinct points is on at most one line.
Each pair of distinct lines is on at most one point.
Not all points are on the same line.
There exists at least one line.
Axiomatic System

It is a list of undefined terms together with a list of statements
(called “axioms”) that are presupposed to be “true.” A theorem is
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


any statement that can be proven using logical deduction from the
axioms.
It is any set of specified axioms from which some or all of those
axioms can be used, in conjunction along with derivation rules or
procedures, to logically derive theorems.
It is a set of axioms used to derive theorems. What this means is
that for every theorem in math, there exists an axiomatic system
that contains all the axioms needed to prove that theorem.
It is a logic structure in which we prove statements from a set of
assumptions. It consist of four main parts: undefined terms, defined
terms, axioms/postulates (accepted or unproven statements), and
proved statements (theorems).
An axiom is a statement that is considered true and does not require
proof. It is considered as the starting point of reasoning. Axioms are used to
prove other statements. They are basic truths. For example, the statement that
all right angles are equal to each other is an axiom and does not require a proof.
While theorem is any statement that can be proven using logical deduction from
the axioms.
Most early Greeks made a distinction between axioms and postulates.
Evidence exists that Euclid made the distinction that an axiom (common
notion) is an assumption common to all sciences and that a postulate is an
assumption peculiar to the particular science being studied. Now in modern times
no distinction is made between the two; an axiom or postulate is an assumed
statement.
Axiomatic System has:
1. Undefined terms/primitive terms – terms that are self-defined
2. Defined terms – terms that have a definition
3. Axioms or postulates – statements about terms that are true without
evidence
4. Theorems – statements that build using terms and postulate and logic.
Two types of undefined terms:
1. Elements – terms that imply objects
2. Relations – terms that imply relationships between objects
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Examples of undefined terms (primitive terms) in geometry are point, line,
plane, on, and between. For these undefined terms, on and between would
indicate some undefined relationship between undefined objects such as point
and line. An example would be: A point is on a line. Early geometers tried to
define these terms:
point
Pythagoreans, “a monad having position"
Plato, “the beginning of a line"
Euclid, “that which has no part"
line
Proclus, “magnitude in one dimension", “flux of
a
point"
Euclid, “breadthless length"
Basic Ingredients of an Axiomatic System
1. Set theory - it is needed to understand and interpret the primitive terms.
2. Logic - it provides the rules of syllogisms/arguments. This will be the
standard, the benchmark use in deciding whether one statement follows
directly equivalent, or inconsistent with another statement/s.
3. Language – the medium use for conveying the axioms, definitions and the
theorem. It must not be ambiguous.
Example:
Consider the following statements:
i.
P and Q are points.
ii.
P and Q are two points.
iii.
P and Q are two distinct points.
iv.
P and Q are distinct points.
 On the assumption that (i), (iii) and (iv) are already clearly understood, and
it can be concluded that (iii) and (iv.) are equivalent if one can count up to
two.
 On the other hand, (i) could mean either P or Q are distinct points or P
and Q are the same point.
 However, for (ii) different mathematicians will have different
interpretations. But in this example, it can be adopted that (ii) has the
same meaning as (iii).
Remember!
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The above example only shows that it is important for the language to be
clear and easy to understand.
The Axiomatic Method
The Axiomatic Method is a procedure by which we prove that results
discovered by experimentation, observation, trial and error or even intuition are
indeed correct.
The proof of a result is a sequence of statements, each of which follows
logically from the ones before and leads from a statement that is known to be
true to the statement which is to be proven.
The axiomatic method consists of:
1. A set of technical terms that are chosen as undefined and are subject to
the interpretation of the reader.
2. All other technical terms are defined by means of the undefined terms.
These are the definitions of the system.
3. A set of statements dealing with undefined terms and definitions that are
chosen to remain unproven. These are the axioms of the system.
4. All other statements of the system must be logical consequences of the
axioms. These are the theorems of the system.
By giving each undefined term in the system a particular meaning, we
create an interpretation of the system. If for a given interpretation of a system, all
of the axioms are “correct” statements we call the interpretation a model.
The Axiomatic Method Development
The axiomatic method is often discussed as if it were a unitary approach,
or uniform procedure. With the example of Euclid to appeal to, it was indeed
treated that way for many centuries. Up until the beginning of the nineteenth
century it was generally assumed in European mathematics and philosophy that
the heritage of Greek mathematics represented the highest standard of
intellectual finish (development more geometrico, in the style of the geometers).
That approach, in which axioms were supposed to be self-evident and
thus indisputable, was swept away during the course of the nineteenth century.
One important episode in this was the development of Non-Euclidean geometry,
based on denial of Euclid's parallel postulate (or axiom). It was found that
consistent geometries can be constructed by denying that postulate, taking as an
axiom that more than one parallel to a given line can be drawn through a point
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outside that line, or a different axiom that no parallel can be drawn—both of
those result in different and consistent geometric systems that may or may not be
applicable to an experienced world.
Other challenges to the supposed self-evidence of axioms came from the
foundations of real analysis, from Georg Cantor's set theory, and from the failure
of Frege's work on foundations. Russell was able to derive a paradox—a kind of
contradiction—from Frege's axioms for set theory, thus showing that Frege's
axiomatic system was not consistent, and this showed that the supposed selfevidence of Frege's axioms was mistaken.
Another challenge came from David Hilbert's 'new' use of axiomatic
method as a research tool. For example, group theory was first put on an
axiomatic basis towards the end of that century. Once the axioms were clarified
(that inverse elements should be required, for example), the subject could
proceed autonomously without reference to the transformation group's origins of
those studies.
Therefore, there are at least three 'modes' of axiomatic method current in
mathematics, and in the fields it influences. In caricature, possible attitudes are:
1. Accept my axioms and you must accept their consequences;
2. I reject one of your axioms and accept extra models;
3. My set of axioms defines a research program.
The first case is the classic deductive method. The second goes by the
slogan be wise, generalise; it may go along with the assumption that concepts
can or should be expressed at some intrinsic 'natural level of generality'. The
third was very prominent in the mathematics of the twentieth century, in particular
in subjects based around homological algebra.
Models
-
It is a well-defined set, which assigns meaning for the undefined terms
presented in the system, in a manner that is correct with the relations
defined in the system.
Two Types of Model:
1. Concrete Models
Have interpretations of the undefined terms adapted from the real world.
The existence of a concrete model proves the consistency of a system.
When a concrete model has been exhibited, an absolute consistency of
the axiomatic system has been established.
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2. Abstract Models
-
Have interpretations of the undefined terms taken from some other
axiomatic system like the real number system. If an abstract model has
been exhibited where the axioms of the first systems are theorems of
the second system, then the first axiomatic system is relatively
consistent.
Models of an axiomatic system are isomorphicif there is a one-toone correspondence between their elements that preserves all relations. That is,
the models are abstractly the same; only the notation is different. An axiomatic
system is categoricalif every two models of the system are isomorphic.
Principle of Duality
In a geometry with two undefined primitive terms, the dual of an axiom or
theorem is the statement with the two terms interchanged. For example, the dual
of "A line contains at least two points," is "A point contains at least two lines." An
axiom system in which the dual of any axiom or theorem is also an axiom or
theorem is said to satisfy the principle of duality.
Example:
Axiom 1. Every ant has at least two paths.
Axiom 2. Every path has at least two ants.
Axiom 3. There exists at least one ant.
a. Write the dual of this system.
The dual of this axiomatic system is formed by interchanging ant and path
in each axiom.
Dual of Axiom 1. Every path has at least two ants.
Dual of Axiom 2. Every ant has at least two paths.
Dual of Axiom 3. There exists at least one path.
b. How do the system and its dual compare?
Axiom 1 and Axiom 2 are duals of each other. Axiom 3 is the dual
of Theorem 1. Hence the system and its dual are equivalent. Therefore,
this axiomatic system satisfies the principle of duality.
Properties of an Axiomatic System
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Consistency
A set of axioms is consistent if it is impossible to deduce from the axioms
a theorem that contradicts any axiom or previously proven theorem. It is said to
be consistent if it lacks contradiction. Equivalently, an axiomatic system is
inconsistent if it implies a contradiction, that is, if it is possible to prove in it that
some statement is both true and false
Since contradictory axioms are usually not desired in an axiomatic system,
consistency is consider to be a necessary condition for an axiomatic system. An
axiomatic system that does not have the property of consistency has no
mathematical value and is generally not of interest.
Example:
The following axiomatic system is inconsistent: (Any one of the axioms
can be proven false using the other axioms.)
A1. There are exactly 2 boys.
A2. There are exactly 3 girls.
A3. Each boy likes exactly 2 girls.
A4. No two boys like the same girl.
To establish the absolute consistency of a set of axioms you need to
produce a concrete model.
Example:
The following axiomatic system is consistent:
A1. There are at least two houses.
A2. For any two distinct houses, there is exactly one street connecting them.
A3. No street connects all houses.
A4. Given any street S and any house H not on S, there is exactly one street on
which H lies, but none of the houses on S lie on this street.
To establish the absolute consistency of a set of axioms you need to
produce a concrete model. Producing a concrete model to establish absolute
consistency is not always possible. Alternatively, we can establish relative
consistency by producing an abstract model.
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Independence
An axiom is independent if it cannot be logically deduced from the other
axioms in the system. The entire set of axioms is independence if each axiom is
independent. Lack of independence means that the system has redundancy in its
axioms, meaning that one or more of its axioms is not needed. Independence is
not a necessary requirement for an axiomatic system; whereas, consistency is
necessary.
To show that an axiom is independent you produce a model in which that
axiom is incorrect and the rest of the axioms are correct. Remember from logic
that only correct statements may be logically deduced from correct statements.
Example:
Axiom 1. Every ant has at least two paths.
Axiom 2. Every path has at least two ants.
Axiom 3. There exists at least one ant.
a. Show the axioms are independent.
We need to produce a model that does not satisfy the axiom we are
showing to be independent but does satisfy the other two axioms. This
demonstrates that the axiom cannot be proved using the other two axioms,
i.e., the axiom cannot be a theorem.
First, we show Axiom 1 is independent. In the following model, Axiom 2
and Axiom 3 are true, but Axiom 1 is not true. Axiom 1 is not true since ant A
has only one path AB. Axiom 2 is true, since path AB has two ants A and B.
exists an ant A.
Axiom 3 is true, since there
Ant
A, B
Path
AB
b.
Next, we show Axiom 2 is independent. The following model has
Axiom 1 and Axiom 3 true, but Axiom 2 is not true. Axiom 1 is true, since the
ant is the dot with two paths represented by the segments. Axiom 2 is not
true, since each path (segment) has only one ant (dot). Axiom 3 is true, since
there is one ant represented by the dot.
c.
The dot is an ant and segments are paths.
Finally, we show Axiom 3 is independent. A model where Axiom 1 and
Axiom 2 are true, but Axiom 3 is not true. Consider a model with no ants and no
paths. The model satisfies both Axiom 1 and Axiom 2 vacuously. But, since there
are no ants, Axiom 3 is not true.
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Completeness
An axiomatic system is complete if it is impossible to add an additional
consistent and independent axiom without adding additional undefined terms. It
is complete if for every statement containing the undefined and defined terms of
the system can be proved valid or invalid.
It is often quite difficult to prove that a set of axiom is complete. An easier
approach is to prove that the set of axioms is categorical which means that each
of its models is isomorphic to every other model. It is a proven result, that if a
system of axioms is categorical, then it is complete. Recall that, two models are
isomorphic if there is a 1-1 correspondence between the elements of the
models which preserves all relations existing in either model.
Example of non-categorical system:
A1. There exist five points.
A2. Each line is a subset of those five points.
A3. There exist two lines.
A4. Each line contain at least two points.
Example of a categorical system:
A1. There exist exactly three distinct points.
A2. Any two distinct points have exactly one line on them.
A3. Not all points are on the same line.
A4. Any two distinct lines have at least one point in common.
Exercise: consider a set of undefined elements, S, and the undefined relation R
which satisfies the following axioms:
A1. If a ∈ S then aR a.
A2. If a, b ∈ S and aRb then bRa.
A3. If a, b, c ∈ S, aRb, and bRc, then aRc.
1. Devise a concrete model for the system.
Let S be the set of students on the UEP campus. Let R be the relation
“has the same hair color” or “has the same weight”.
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2. Devise an abstract model for the system.
Let S be the set of triangles. Let R be the relation “is congruent to” or “is
similar to.”
3. Are the axioms independent?
To show that the axioms are independent, consider the following:
A1 is independent:
Let S = {x, y, z}and R ={(x, x),(y, y),(x, y),(y, x)}. Clearly the relation R on S
satisfies axioms A2 and A3 but not A1. A second example could be: Let S = {Jim,
Amy, Mary} and R = {(Jim, Jim), (Amy, Amy), (Amy, Jim), (Jim, Amy)} where (x,
y) means: “x likes y”.
A2 is independent:
Let S be the set of real numbers, and let R be the relation “is less than or
equal to”.
A3 is independent: Let S be the set of students on the UEP campus and
let R be the relation “is an acquaintance of”.
Finite Geometry Samples
Finite geometry describes any geometric system that has only a finite
number of points.
In Figure 1. It have a space with five points
labeled 1, 2, 3, 4, and 5. There are six lines.
Consider the line connecting points 1 and 2; we
label that line {1, 2}. Similarly, the other lines
containing point 1 are labeled {1, 3}, {1, 4}, and {1,
5}. Then there is the line connecting points 2 and
5, giving us {2, 5}. Lastly there is the line
connecting points 3, 4, and 5, which we label {3, 4,
5}. So we have our space S = {P, L} with P = {1, 2,
3, 4, 5} and L = {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 5}, {3, 4, 5}}. Since L is a set of
lines, and each line is a set of points, L must be written as a composition of sets.
It is very important to note here that {3, 4, 5} is one line containing three
points. If we were dealing with two different lines containing only two points
each, we would label them {3, 4} and {4, 5} and draw them so that they didn’t
look like one line. The diagrams contained henceforth are carefully drawn to
distinguish between these two cases.
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In the same way plane geometry is based on postulates, finite geometries
are based on certain given facts called axioms. Although we could list some of
the axioms that the space represented by Figure 1 satisfies, the list would
continue forever. We could include axioms like: each line must contain no more
than three points; there exist five points; and there exists no line containing
twenty seven points. It is much more practical and interesting to start with a set
of axioms and see which spaces satisfy them. Consider the following axiomatic
system:
1. There are five points and two lines.
2. Each line contains at least two points.
3. Each line contains at most three points.
It might seem that there would be many
possible spaces that follow the axioms, but we
can systematically consider the possibilities.
We know that each line must contain at least
two points. So we first consider all spaces in
which both lines contain two points each,
discovering the spaces represented by Figures
2A and 2B. Looking at spaces in which one line
contains two points and the other line contains
three points gives us Figures 2C and 2D. And
looking at spaces in which both lines contain
three points gives us Figure 2E. Since each line contains at most three points,
we know we have considered all possible spaces.
Mathematicians working with finite geometries have a few standard ways
to represent spaces. This is more technical than mathematically significant, but it
helps to keep things in order. For example, when we looked for all spaces that
satisfied the previous axiomatic system, we looked at them by the number of
points on each line, in increasing order. This is one way to ensure uniqueness of
each space, and to guarantee that we have found each space.
Another practice we have already seen comes into play when
representing a space in set notation. When we described Figure 1 as S, we
wrote out the points in numerical order: P = {1, 2, 3, 4, 5}. But when we labeled
lines, we listed the points within each line in numerical order: {1, 2}, {1, 3}, {1, 4},
{1, 5}, {2, 5}, and {3, 4, 5}. We also listed the lines themselves in a numerical
order, starting with all lines that contain point 1, then all other lines that contain
point 2, and so on. These are just some standard ways for working with finite
geometries.
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Additionally, we must touch on consistency and independence.
Consistency is whether or not a set of axioms produces a space. A consistent
system, like the one we just saw, produces at least one space. An inconsistent
system does not. One example of an inconsistent system is the following:
1. There exist four points and six lines.
2. There is at least one line with three points.
When looking at all spaces with four points, we see that the only space
that contains four points and six lines is the space S = {P, L} where P = {1, 2, 3,
4} and L = {{1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 4}}. Because no lines have three
points, no such system satisfies this axiomatic system, and this system is
inconsistent. For our purposes, we will only deal with systems that are
consistent.
Independent systems are systems in which no postulate can be derived
from the others. Every system we have dealt with so far has been independent.
A dependent system is one in which at least one of the axioms can be derived
from the other axioms. For example, the following axiomatic system is
dependent:
1. There are five points and three lines.
2. Each line has at most five points.
This system is dependent because the second axiom follows directly from
the first axiom. As long as a space has only five points, no line can contain more
than five points. In this case, it is trivial to see that the system is dependent. In
other cases, a good way to test dependency is to consider the systems formed
using all but one of the axioms. If a subset of the axioms produces the same
spaces that the entire set of axioms produces, then the set is dependent.
Consistency and dependence are not too important when studying basic
finite geometries because it only makes sense to work with consistent,
independent systems. So although it is important to be familiar with those terms,
intricately testing the independence of an axiomatic set is not of utter importance.
FanoPlane
Fano's work in the area of finite geometry included the discussion of a 3dimensional finite geometry which consisted of 15 points, 35 lines, and 15 planes
where each line had 3 points on it and each plane had 7 points. Fano had a long
career as a teacher at the University of Turin, where he had also been a student.
Due to his Jewish background he was forced out of his position in Turin and went
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to Switzerland for the war years. By the time the war was over Fano was quite
elderly, but this did not prevent him from traveling to the United States and
lecturing in Italy. Fano's sons Ugo and Robert had distinguished
careers. Ugo earned a doctorate in mathematics but pursued a career in physics,
and Robert taught engineering at MIT, where he did research on the
mathematical problem of efficient data compression. The finite geometry shown
in the diagram below is now called the Fano Plane in honor of Gino Fano.
(Figure 1: One way of drawing a diagram representing the Fano Plane.)
The Fano Plane has 7 points, and in the diagram above they are
represented by the dark dots P1, ..., P7, which is a finite set of points. What are
the lines of the Fano Plane? A typical line, say L7, consists of a set of three
points. L7, as can be read from the diagram, consists of the points P1, P7, and
P3. Looking at the way the diagram is drawn it may be tempting to think that the
point P7 is "between" the points P1 and P3. However, this is not the case. Line
L7 is a set of points but for this geometry we will not be able to give meaning to a
concept of "betweenness." Also, the way that the line L7 has been drawn may
make it appear as if there are lots of points between P7and P1 but, in fact, this
line has exactly three points on it. Many geometers like to have a diagram such
as this figure to illustrate the Fano Plane but for those who find the diagram
"confusing," one can revert to thinking of the Fano Plane as a set of 7 points and
seven lines, where the exact points that make up each line have been specified.
Some people find another aspect of Figure 1 confusing. The line L6 also
has three points on it: P7, P6, and P2. Yet this line is drawn as if it is a circle!
Again, the diagram is an aid to insight and if one finds the fact that one of the
lines is drawn in a "non-straight" fashion confusing, one can go back to just the
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set of points and the list of lines and points that make them up without using any
diagram whatsoever. You may ask, why draw the line L6 as if it is circular? Is it
possible to locate 7 points in the Euclidean plane in such a way that the points
are in positions where all 7 lines of the Fano Plane lie along "straight" lines in the
Euclidean plane? The answer is "no!" In fact, the diagram we have chosen is at
least somewhat appealing because 6 of the 7 lines are represented as being on
"straight"
lines
in
the
Euclidean
plane.
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REFERENCES
http://study.com/academy/lesson/the-axiomatic-system.html
http://wespace.ship.edu/jehamb/F07/333/axsystems.pdf
http://www.math.upd.edu.ph/faculty/jbasilla/classes/201011AM1/
http://web.mnstate.edu/peil/geometry/C1AxiomSystems.htm
http://web.mnstate.edu/peil/geometry/C1AxiomSystem/AxSysWorksheet.htm
http://web.mnstate.edu/peil/geometry/C1AxiomSystem/history.htm
http://www.newworldencyclopedia.org/entry/AxiomaticSystems/20100901handou
t12.pdf
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