The Axiomatic Method

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The Axiomatic Method
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Axiomatic System

• Axiomatic Systems have
• four essential components
• Undefined terms
• Defined Terms
• Axioms
• Theorems

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Definition

• A definition is the statement of a single,
  unambiguous idea that the word, phrase or symbol
  represents.
• Given any object, one should be able to determine
  whether or not that the object satisfies the
  definition.
• The word, phrase or symbol being defined becomes
  a label and is not used to label anything else.

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Undefined Term

• Undefined terms are necessary in order to avoid
  circular definitions.
• One cannot have an infinite regression of terms,
  each defined in turn by more basic terms. There
  must be a starting point.

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Axiom (Postulate)

• Just as one cannot expect to define every term,
  so one cannot expect to prove every mathematical
  statement.
• To have some statements to start one has to have
  some statements that are assumed without proof.
• These statements are called axioms.

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Theorem

• The statements that are derived from the axioms,
  undefined terms, defined terms, and previously
  derived theorems by strict logical proof are
  called theorems.

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Types of Axiomatic Systems

• A material axiomatic system considers its
  undefined terms to have meanings derived from
  reality.
• A formal axiomatic system considers its undefined
  terms to have no meaning at all and to behave
  much like algebraic symbols.

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Vocabulary of Axiomatic Systems

• An interpretation of an axiomatic system is any
  assignment of specific meanings to the undefined
  terms of that system.
• If an axiom becomes a true statement when its
  undefined terms are interpreted in a specific
  way, then we say that the interpretation
  satisfies the axiom.
• A model for an axiomatic system is an
  interpretation that satisfies all the axioms of
  the system.

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Vocabulary of Axiomatic Systems

• Mathematics constructed
• in a formal axiomatic system is called pure
  mathematics.
• Mathematics constructed
• in a material axiomatic system is called applied
  mathematics.

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Vocabulary of Axiomatic Systems

• The negation of a statement is another statement
  that is true when the original statement is false
  and false when the original statement is true.

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Vocabulary of Axiomatic Systems

• An axiomatic system is inconsistent if it is
  possible to prove both a statement and its
  negation from the axioms.
• A system is consistent if it is not inconsistent.

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Vocabulary of Axiomatic Systems

• An axiom A in a consistent axiomatic system is
  said to be independent if the axiomatic system
  formed by replacing A with its negation is also
  consistent.
• An axiom is dependent if it is not independent.
• An axiomatic system is independent if each of its
  axioms is independent.

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Vocabulary of Axiomatic Systems

• An axiomatic system will be called complete if
  for every statement, either itself or its
  negation is derivable.
• An axiomatic system is incomplete if even a
  single statement and its negation is derivable.

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Example of a Formal Axiomatic System

• Undefined terms hinkle, pinkle, babble
• Axioms
• 1. There are exactly three hinkles.
• 2. Every hinkle babbles at least two pinkles.
• 3. No pinkle is babbled by more than two hinkles.
  

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Example of a Material Axiomatic System

• Undefined terms squirrel, tree, climb
• Axioms
• 1. There are exactly three squirrels.
• 2. Every squirrel climbs at least two trees.
• 3. No tree is climbed by more than two squirrels.

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Example of a Formal Axiomatic System

• Undefined terms X,Y related to
• Axioms
• 1. There exist at least one X and one Y.
• 2. If a and b are distinct Xs then exactly one Y
  is related to both of them.
• 3. If c and d are distinct Ys then exactly one X
  is related to both of them.
• At least three Xs are related to any Y.
• Not all Xs are related to the same Y.

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Example of a Material Axiomatic System

• Undefined terms brass ring, wire related to
• Axioms
• 1. There exist at least one brass ring and one
  wire.
• 2. If a and b are distinct brass rings then
  exactly one wire is related to both of them.
• 3. If c and d are distinct wires then exactly one
  brass ring is related to both of them.
• At least three brass rings are related to any
  wire.
• Not all brass rings are related to the same wire.

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Example of a Material Axiomatic System

• Undefined terms hinkle, pinkle, babble
• Axioms
• 1. There are exactly three hinkles.
• 2. Every hinkle babbles at least two pinkles.
• 3. No pinkle is babbled by more than two hinkles.

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Example Using a triangle to prove the
squirrel-and-tree axiom system is consistent.

• Undefined terms vertex, side, is on
• Axioms
• 1. There are exactly three vertices.
• 2. Every vertex is on at least two sides.
• 3. No side is on more than two vertices.
• If we interpret squirrels as vertices,
  trees as sides, and climb as on, then it
  is clear that all axioms are satisfied.

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Examples of non-mathematical use of the axiomatic
method

• Ethics by Spinoza in 1677 is in five parts, each
  consisting of a list of definitions, a set of
  axioms, and a number of propositions derived from
  them.
• Declaration of Independence, the U.S.
  Constitution (We hold these truths to be
  self-evident . . .)

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Examples of use of the axiomatic method

• Principia by Newton in 1677 which contained
  development of the laws of motion and universal
  graviation.
• Grundbegriffe der Wahrscheinlichkeitsrechnung by
  Andrei Kolmogorov set up the axiomatic basis for
  modern probability theory in 1933.

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Euclids Axiomatics

• Undefined terms point, line, plane
• Common Notions
• Things equal to the same thing are equal.
• If equals are added to equals, the results are
  equal.
• If equals are subtracted from equals, the results
  are equal.
• Things that coincide with one another are equal
  to one another.
• The whole is greater than the part.

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Euclids Axiomatics

• Undefined terms point, line, plane
• Postulates
• Exactly one straight can be drawn from any point
  to any other point.
• A finite straight line can be extended
  continuously in a straight line.
• A circle can be formed with any center and
  distance (radius).
• All right angles are equal to one another.
• If a straight line falling on two straight lines
  makes the sum of the interior angles on the same
  side less than two right angles, then the two
  straight lines, if extended indefinitely, meet on
  that side on which the angle sum is less than two
  right angles.

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Euclids Parallel Postulate

• If a straight line falling on two straight lines
  makes the sum of the interior angles on the same
  side less than two right angles, then the two
  straight lines, if extended indefinitely, meet on
  that side on which the angle sum is less than two
  right angles.

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Flaws in Euclid's Postulates

• Euclid takes existence of points for granted,
  never stating such existence as a postulate. In
  Hilbert's system these assumptions are stated in
  axiom I-2 and I-3.
• Euclid takes betweenness and line separation for
  granted, never stating the properties he uses in
  any axioms or postulates. In Hilbert's system
  these properties are stated in axioms II-1, II-2,
  II-3, and II-4.

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Flaws in Euclid's Postulates

• Euclid has a faulty proof of SAS where he assumes
  that certain motions are possible without stating
  in postulates or axioms that such motions are
  possible. Some modern treatments of geometry do
  assume motions in their axioms. However,
  Hilbert's does not, so in Hilbert's system SAS
  has to be taken as an axiom (axiom III-5).
• Euclid takes continuity properties for granted.
  For example, he assumes (without stating it as an
  axiom) that if two circles are sufficiently close
  together then they intersect in two points.

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David Hilbert (1862 1943)
Grundlagen der Geometrie 1899 The
Problems of Mathematics 1900 Grundlagen der
Mathematik 1934 Hilbert's work in
geometry had the greatest influence in that area
after Euclid. A systematic study of the axioms of
Euclidean geometry led Hilbert to propose 21 such
axioms and he analyzed their significance. He
made contributions in many areas of mathematics
and physics.
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Hilbert's Axioms Undefined Terms

• Points
• Lines
• Planes
• Lie on, contains
• Between
• Congruent

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Hilbert's Axioms Incidence Axioms

• The following axioms set out the basic incidence
  relations between lines, points and planes. They
  also characterize the concept of dimension that
  we associate with these notions.
• Postulate I-1.
• For every two points A, B there exists a line a
  that contains each of the points A, B.
• Postulate I-2.
• For every two points A, B there exists no more
  than one line that contains each of the points A,
  B.

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Hilbert's Axioms Incidence Axioms

• Postulate I-3.
• There exists at least two points on a line. There
  exist at least three points that do not lie on a
  line.
• Postulate I-4.
• For any three points A, B, C that do not lie on
  the same line there exists a plane a that
  contains each of the points A, B, C. For every
  plane there exists a point which it contains.
• Postulate I-5.
• For any three points A, B, C that do not lie on
  one and the same line there exists no more than
  one plane that contains each of the three points
  A, B, C.

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Hilbert's Axioms Incidence Axioms

• Postulate I-6.
• If two points A, B of a line a lie in a plane a
  then every point of a lies in the plane a.
• Postulate I-7.
• If two planes a, ß have a point A in common then
  they have at least one more point B in common.
• Postulate I-8.
• There exist at least four points which do not lie
  in a plane.

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Hilbert's Axioms Order (Betweenness) Axioms

• These axioms were almost ignored by Euclid except
  the second one below. Their importance was
  noticed by M. Pasch who saw how they were
  implicitly being used in many proofs. This is one
  problem with self-evident truths we often
  forget to state some of the axioms and then the
  geometry is incomplete without them. Euclid
  regarded it as intuitive that, if three distinct
  points lie on a line, then exactly one lies
  between the other two. Hilbert made this an
  axiom, thus defining the concept of betweenness.
  Some other axioms are needed to complete the
  characterization. The following axioms make clear
  the notion of a point lying between two other
  points.

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Hilbert's Axioms Order (Betweenness) Axioms

• Postulate II.1.
• If a point B lies between a point A and a point C
  then the points A, B, C are three distinct points
  of a line, and B then also lies between C and A.
• Postulate II.2.
• For two points A and C, there always exists at
  least one point B on the line AC such that C lies
  between A and B.

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Hilbert's Axioms Order (Betweenness) Axioms

• Postulate II.3.
• Of any three points on a line there exists no
  more than one that lies between the other two.
• Postulate II.4.
• Let A, B, C be three points that do not lie on a
  line and let a be a line in the plane ABC which
  does not meet any of the points A, B, C. If the
  line a passes through a point of the segment AB,
  it also passes through a point of the segment AC,
  or through a point of the segment BC.

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Hilbert's Axioms Congruence Axioms

• Hilbert's resolution was to introduce axioms to
  define congruence. Instead of moving figures, he
  postulated the ability to construct an exact copy
  of a figure at any place on the plane, rotated
  through any angle, and oriented either way. The
  SAS condition is one of the axioms.

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Hilbert's Axioms Congruence Axioms

• Postulate III.1.
• If A, B are two points on a line a, and A' is a
  point on the same or on another line a' then it
  is always possible to find a point B' on a given
  side of the line a' such that AB and A'B' are
  congruent.
• Postulate III.2.
• If a segment A'B' and a segment A"B" are
  congruent to the same segment AB, then segments
  A'B' and A"B" are congruent to each other.

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Hilbert's Axioms Congruence Axioms

• Postulate III.3.
• On a line a, let AB and BC be two segments which,
  except for B, have no points in common.
  Furthermore, on the same or another line a', let
  A'B' and B'C' be two segments which, except for
  B', have no points in common. In that case if
  ABA'B' and BCB'C', then ACA'C'.

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Hilbert's Axioms Congruence Axioms

• Postulate III.4.
• If ABC is an angle and if B'C' is a ray, then
  there is exactly one ray B'A' on each "side" of
  line B'C' such that A'B'C'ABC. Furthermore, every
  angle is congruent to itself.
• Postulate III.5. (SAS)
• If for two triangles ?ABC and ?A'B'C' the
  congruences ABA'B', ACA'C' and BAC B'A'C'
  are valid, then the congruence ?ABC ?A'B'C' is
  also satisfied.

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Parallel Axiom

• The axiom in this section caused the most
  controversy and confusion of all. The axioms of
  parallels (which is also an incidence axiom) is
• Postulate IV.1.
• Let a be any line and A a point not on it. Then
  there is at most one line in the
• plane that contains a and A that passes through A
  and does not intersect a.

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Hilbert's Axioms Continuity Axioms

• These axioms are the axioms which give us our
  correspondence between the real line and a
  Euclidean line. These are necessary to guarantee
  that our Euclidean plane is complete.

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Hilbert's Axioms Continuity Axioms

• Postulate V.1. (Archimedes Axiom)
• If AB and CD are any segments, then there exists
  a number n such that n copies of CD constructed
  contiguously from A along the ray AB will pass
  beyond the point B.

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Hilbert's Axioms Continuity Axioms

• Postulate V.2. (Line Completeness)
• An extension of a set of points on a line with
  its order and congruence relations that would
  preserve the relations existing among the
  original elements as well as the fundamental
  properties of line order and congruence (Axioms
  I-III and V-1) is impossible.

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