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Transcript of Axiomatic system2
Introduction
Components
Axiomatic Systems
Example
Finite Projective Planes
Properties
EnrichmentJennifer C. Bunquin
Axiomatic
System
The study of any mathematicsrequires an understanding of the nature ofdeductive reasoning; frequently, geometryhas been singled out for introducing thismethodology to secondary school students.This topic introduces the terminologyessential for a discussion of deductivereasoning so that the extraordinaryinfluence of the history of geometry on themodern understanding of deductivereasoning will become evident.
Components
Axiomatic Systems
Example
Finite Projective
Planes
Properties
Enrichment
Introduction
Words differently arranged have a different meanings and meanings differently arranged have a different effect.
- Blaise Pascal (1623-1662)
Deductive reasoning takes place inthe context of an organized logical structurecalled an axiomatic ( or deductive) system.
One of the pitfalls of working with adeductive system is too great a familiaritywith the subject matter of the system. Weneed to be careful with what we areassuming to be true and with sayingsomething is obvious while writing a proof.
Components
Axiomatic Systems
Example
Finite Projective
Planes
Properties
Enrichment
Introduction
Remember:
it is crucially important in a proof to useonly the axioms and the theorems whichhave been derived from them and notdepend on any preconceived idea orpicture. diagrams should be used as an aid, sincethey are useful in developing conceptualunderstanding.
Components
Axiomatic Systems
Example
Finite Projective
Planes
Properties
Enrichment
Introduction
Components of an Axiomatic System
Undefined Terms
Certain terms are left undefined toprevent circular definitions. Examples ofundefined terms (primitive terms) ingeometry are point, line , plane, on andbetween.
Undefined terms are of two types:terms that imply objects, called elements,and terms that imply relationships betweenobjects, called relations.
Introduction
Axiomatic Systems
Example
Finite Projective
Planes
Properties
Enrichment
Components
Components of an Axiomatic System
Defined Terms
They are not necessary, but in nearlyevery axiomatic system certain phrasesinvolving undefined terms are usedrepeatedly. Thus, it is more efficient tosubstitute a new term, that is, a definedterm, for each of these phrases wheneverthey occur.Example: lines that do not intersect = parallel lines
Introduction
Axiomatic Systems
Example
Finite Projective
Planes
Properties
Enrichment
Components
Components of an Axiomatic System
Axioms
Statements that are accepted withoutproof.
Early GREEKS:Axiom = an assumption common to all
sciencesPostulate = an assumption peculiar to
the particular sciencebeing studied
Introduction
Axiomatic Systems
Example
Finite Projective
Planes
Properties
Enrichment
Components
Components of an Axiomatic System
A system of logic
From the axioms, otherstatements can be deduced or provedusing the rules of inference of asystem of logic (usually Aristotelian).
Introduction
Axiomatic Systems
Example
Finite Projective
Planes
Properties
Enrichment
Components
Components of an Axiomatic System
Theorems
New statements which arededuced or proved using the axioms,system of logic and previoustheorems.
Introduction
Axiomatic Systems
Example
Finite Projective
Planes
Properties
Enrichment
Components
Axiomatic Systems
Four Point Geometry
AXIOMS 4P:1. There exist exactly four points.2. Each two distinct points are on
exactly one line.3. Each line is on exactly two
points.
Axiomatic Systems
Introduction
Components
Example
Finite Projective
Planes
Properties
Enrichment
Axiomatic Systems
Consistency
Axiom 4P.1 explicitly guarantees theexistence of exactly four points. However,even though lines are mentioned inAxioms 4P.2 and 4P.3,new cannotascertain whether or not lines exist untiltheorems verifying this are proved, sincethere is no axiom that explicitly insurestheir existence.
Four Point Geometry
Introduction
Components
Example
Finite Projective
Planes
Properties
Enrichment
Axiomatic Systems
Consistency
Axiom 4P.2 and 4P.3. like manymathematical statements, are disguised“if… then” statements. Axiom 4P.2 shouldbe interpreted as follows: If two distinctpoints exist, then these two points areon exactly one line. Axiom 4p.3 should be:If there is a line, then it is on exactly twopoints.
Four Point Geometry
Introduction
Components
Example
Finite Projective
Planes
Properties
Enrichment
Axiomatic Systems
Consistency
MODEL A
Four Point Geometry
Introduction
Components
Example
Finite Projective
Planes
Properties
Enrichment
Axiomatic Systems
1
3
4
2
Undefined Term
Interpretation
Points
Lines
On
Dots denoted 1, 2, 3, 4Segments illustrated by the figure.A dot is an endpoint of a segment or vice-versa
Consistency
MODEL B
Four Point Geometry
Introduction
Components
Example
Finite Projective
Planes
Properties
Enrichment
Axiomatic SystemsUndefined Term Interpretation
Points LinesOn
Letters A, B, C, DColumns of letters belowContains, or is contained in
LINES A A A B B CB C D C D D
Independence
The independence of this axiomaticsystem is demonstrated by thefollowing three model, all of whichinterpret points as letters of thealphabet and lines as the columns ofletters indicated.
Four Point Geometry
Introduction
Components
Example
Finite Projective
Planes
Properties
Enrichment
Axiomatic Systems
Models for Independence
Four Point Geometry
Introduction
Components
Example
Finite Projective
Planes
Properties
Enrichment
Axiomatic Systems
Model 4P. 1Axiom 4P.1: There exist exactly four
points.Axiom 4P.1’: There do not exist four
points.
Points Lines
A, BAB
Models for Independence
Four Point Geometry
Introduction
Components
Example
Finite Projective
Planes
Properties
Enrichment
Axiomatic Systems
Model 4P. 2Axiom 4P.2: Each two distinct points are on
exactly one line.Axiom 4P.2’: There are two distinct points
not on one line..
Points Lines
A, B, C, DA C B D
Models for Independence
Four Point Geometry
Introduction
Components
Example
Finite Projective
Planes
Properties
Enrichment
Axiomatic Systems
Model 4P. 3Axiom 4P.3: Each line is on exactly two
points.Axiom 4P.3’: There are lines not on exactly
two points.
Points Lines
A, B, C, DA A B CB D D DC
Completeness
Four Point Geometry
Introduction
Components
Example
Finite Projective
Planes
Properties
Enrichment
Axiomatic SystemsIf all models of a system are pairwise
isomorphic, it is clear that each model hasthe same number of points and lines.Hence, if all models of the system arenecessarily isomorphic, it follows that thesystem is complete.
Completeness
Four Point Geometry
Introduction
Components
Example
Finite Projective
Planes
Properties
Enrichment
Axiomatic SystemsIn Four Point geometry, it is
clear that Model A and Model B areisomorphic.
Four Point Geometry
THEOREMS:
1. The four point geometry hasexactly six (6) lines.
2. Each point of the geometry hasexactly three (3) lines
Axiomatic Systems
Introduction
Components
Example
Finite Projective
Planes
Properties
Enrichment
Axiomatic Systems
Examples
Three Point Geometry
AXIOMS:1. There exist exactly three points.2. Each two distinct points are on
exactly one line.3. Not all the points are on the same
line.4. Each two distinct line are on at least
one point.
Introduction
Components
Axiomatic System
Finite Projective
Planes
Properties
Enrichment
Examples
Three Point Geometry
MODEL:
Note:Nothing is said about the type of
line whether it is straight or curved.
Examples
Introduction
Components
Axiomatic System
Finite Projective
Planes
Properties
Enrichment
Examples
Three Point Geometry
THEOREMS
1. Each two distinct lines are onexactly one point.
2. There exist exactly three lines.
Examples
Introduction
Components
Axiomatic System
Finite Projective
Planes
Properties
Enrichment
Examples
Five Point Geometry
Examples
Introduction
Components
Axiomatic System
Finite Projective
Planes
Properties
Enrichment
Examples
AXIOMS:
1. There exist exactly five points.2. Each two distinct points have exactly
one line on both of them.3. Each line has exactly two points.
Five Point Geometry
Examples
Introduction
Components
Axiomatic System
Finite Projective
Planes
Properties
Enrichment
Examples
MODEL
3
5
1
2
4
Note: No point exists where there are lines that intersects.
Five Point Geometry
Examples
Introduction
Components
Axiomatic System
Finite Projective
Planes
Properties
Enrichment
Examples
THEOREMS:
1. The five point geometry has exactlyten (10) lines.
2. Each point of the geometry hasexactly four (4) lines
What is a MODEL?
A model of an axiomatic system isobtained if we can assign meaning to theundefined terms of the axiomatic systemwhich converts the axioms in to truestatements about the assigned concepts.
It is also used to establishconsistency.
Models are in two types.
Introduction
Components
Example
Finite Projective
Planes
Properties
Enrichment
Axiomatic Systems
What is a MODEL?
TWO TYPES : Concrete ModelsAbstract Models
A model is concrete if the meaningsassigned to the undefined terms are objectsand relation adapted from the real world. Amodel is abstract if the meanings assignedto the undefined terms are objects andrelations adapted from another axiomaticdevelopment.
Introduction
Components
Example
Finite Projective
Planes
Properties
Enrichment
Axiomatic Systems
Properties
Definition 1
An axiomatic system is said to
be consistent if there do not exist inthe system any two axiom, any axiomand theorem, or any two theoremsthat contradict each other. This canbe proved using a model.
Introduction
Components
Example
Finite Projective
Planes
Axiomatic Systems
Enrichment
Properties
Properties
Consistency
If the model is obtained by usinginterpretations that are objects andrelation adapted from the real world, wehave established absolute consistency. If amodel is obtained using the interpretationsfrom another axiomatic system, we havedemonstrated relative consistency.
Introduction
Components
Example
Finite Projective
Planes
Axiomatic Systems
Enrichment
Properties
Properties
Definition 2
An axiom in an axiomaticsystem is independent if it cannot beproved from the other axioms. Ifeach axiom of a system isindependent, the system is said tobe independent.
Introduction
Components
Example
Finite Projective
Planes
Axiomatic Systems
Enrichment
Properties
Properties
Independence
Independence is not necessaryrequirement for an axiomatic system;whereas, consistency is necessary. Theverification that an axiomatic system isindependent is also done via models. Theindependence of Axiom A in an axiomaticsystem S is established by finding a model ofS’ where S’ is obtained by replacing Axiom Awith a negation of A.
Introduction
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Example
Finite Projective
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Axiomatic Systems
Enrichment
Properties
Properties
Independence
Thus, to demonstrate that a systemconsisting of n axioms is independent, nmodels must be exhibited – one for eachaxiom.
Introduction
Components
Example
Finite Projective
Planes
Axiomatic Systems
Enrichment
Properties
Properties
Definition 3
An axiomatic system is completeif every statement containing undefinedterms of the system can be proved validor invalid, or in other words, if it ispossible to add new independent axiomto the system.
Introduction
Components
Example
Finite Projective
Planes
Axiomatic Systems
Enrichment
Properties
Properties
Completeness
It is impossible to demonstratedirectly that a system is complete.However, if a system is complete, therecannot exist two essentially differentmodels. This means all models of thesystem must be pair wise isomorphic andthe axiomatic system is categorical.
Introduction
Components
Example
Finite Projective
Planes
Axiomatic Systems
Enrichment
Properties
Properties
Definition 4
Two models of an axiomatic systemare said to be isomorphic if there exists aone-to-one correspondence Φ from the setof points and lines α onto the set of pointsand lines of β that preserves all relations.
Introduction
Components
Example
Finite Projective
Planes
Axiomatic Systems
Enrichment
Properties
Properties
Definition 4
In particular if the undefined terms ofthe system consist of the terms “point”,“line”, and “on”, then Φ must satisfy the ff.conditions:
1. For each point P and line l in α, Φ(P)and Φ(l) are a point and line in β.
2. If P is on l, then Φ(P) is on Φ(l)
Introduction
Components
Example
Finite Projective
Planes
Axiomatic Systems
Enrichment
Properties
Properties
Definition 5
An axiomatic system is categorical ifevery two models of the system areisomorphic.
thus for a categorical axiom systemone my speak of the model; the one andonly interpretation in which its theoremsare all true.
Introduction
Components
Example
Finite Projective
Planes
Axiomatic Systems
Enrichment
Properties
Properties
Definition 6
In a geometry with two undefinedterms, the dual of an axiom or theorem isthe statement with the two terms beinginterchanged.EXAMPLE:
The dual of “ A line contains at leasttwo points”, is “A point contains at least twolines”.
Introduction
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Example
Finite Projective
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Axiomatic Systems
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Properties
Definition 6
The dual of a statement isobtained by replacing eachoccurrence of the word “point” by theword “line” and vice-versa,(consequently, the words“concurrent” and “collinear” mustalso be interchanged.
Introduction
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Example
Finite Projective
Planes
Axiomatic Systems
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Properties
Definition 7
An axiomatic system in whichthe dual of any axiom or theorem isalso an axiom or theorem is said tosatisfy the principle of duality.
Introduction
Components
Example
Finite Projective
Planes
Axiomatic Systems
Enrichment
Properties
Finite Projective Planes
Introduction
Components
Example
Axiomatic System
Properties
Enrichment
Finite Projective
Planes
An axiomatic system for animportant collection of finitegeometries known as finite projectiveplanes. In a finite projective plane,each pair of lines intersect; that is,there is no parallel lines. This pairwiseintersection of lines leads to severalother differences between projectiveplanes and Eucledian planes.
Finite Projective Planes
Introduction
Components
Example
Axiomatic System
Properties
Enrichment
Finite Projective
Planes
AXIOMATIC SYSTEM
Undefined Terms: Point, Line, IncidentDefined Terms: Points incident with
the same line are said tobe collinear. Linesincident with the samepoint are said to beconcurrent.
Finite Projective Planes
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Axiomatic System
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Finite Projective
Planes
AXIOMATIC SYSTEM
Axiom FPP.1: There exist at least four distinctpoints, no three of which are collinear.
Axiom FPP.2: There exists at least one linewith exactly n + 1 (n > 1) distinct pointsincident with it.
Axiom FPP.3: Given two distinct points,there is exactly one line incident with both ofthem.
Axiom FPP.4: Given two distinct lines,there is at least one point incident with both ofthem.
Finite Projective Planes
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Finite Projective
Planes
Any set of points and linessatisfying these axioms is called aprojective plane of order n. Notethat the word “incident” has beenused in place of the undefined term“on” in this axiom system.
Finite Projective Planes
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Finite Projective
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CONSISTENCY
MODEL A
Points Lines
A, B, C, D, E,F, G
A A B A B C CB D D F E D EC E F G G G F
MODEL B
Points Lines
Dots denoted1, 2, 3, 4, 5, 6, 7
Segments illustrated in the figure.
Finite Projective Planes
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Example
Axiomatic System
Properties
Enrichment
Finite Projective
Planes
CONSISTENCY
4
65 7
21 3
Finite Projective Planes
Introduction
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Example
Axiomatic System
Properties
Enrichment
Finite Projective
Planes
Models A and B have threepoints on each line, three lines oneach point, and a total of sevenpoints and seven line. To determineif finite projective planes exist withmore points and lines, it is clearlyimpractical to employ trial-and-errorprocedures.
Finite Projective Planes
Introduction
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Example
Axiomatic System
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Finite Projective
Planes
Geometers developed a seriesof theorems that lead to a generalresult regarding the number ofpoints and lines in a finite projectiveplane of order n.
Finite Projective Planes
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Finite Projective
Planes
The proofs of these theoremsare simplified by noting that thisaxiom system satisfies the principleof duality, which Coxeter hasdescribed as “ one of the mostelegant properties of projectivegeometry. ( Coxeter, 1969, p. 231)
Finite Projective Planes
Introduction
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Axiomatic System
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Finite Projective
Planes
In an axiomatic system that satisfiesthe principle of duality, the proof of anytheorem can be “turned into” a proof of adual theorem merely by dualizing theoriginal proof. To show that an axiomsystem has the property of duality, it isnecessary to prove that the duals eachaxiom are the theorems of the system.
Finite Projective Planes
Introduction
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Example
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Finite Projective
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Theorem FPP.1 (Dual of Axiom FPP.1)
There exist at least four distinct lines, no three of which are concurrent.
Theorem FPP.2 ( Dual of Axiom FPP.3)
Given two distinct lines, there is exactly one point incident with both of them.
Finite Projective Planes
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Finite Projective
Planes
Theorem FPP.3 (Dual of Axiom FPP.4)
Given two distinct points, there is at least one line incident with both of them.
Theorem FPP.4 ( Dual of Axiom FPP.2)
There exists at least one point with exactly n+1 (n>1) distinct lines incident with it.
Finite Projective Planes
Introduction
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Finite Projective
Planes
Theorem FPP.5There is exactly
n +1 lines through a point P.
Theorem FPP.6 Each of these
lines contains exactly n + 1 points, that is, npoints addition to P.
Pn+1P2P1
P
l1 l2 ln+1 ln+2
Finite Projective Planes
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Finite Projective
Planes
Theorem FPP.7
A projective plane of order n contains exactly points and
lines.
Proof:The total number of points is (n+1)
n+1 = . A dual argument verifies that the total number is also .
12
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