Axiomatic System
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2011 Math 1
JB
Axiomatic System
Proofs
Julius Magalona Basilla
Institute of Mathematics
University of the Philippines-Diliman
[email protected]
The Plain Truth
The elements of an
axiomatic System
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Math 1
General Education Mathematics
Lecture
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The notion of truth
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Axioms of Set
Theory
Consistency Model
The plain truth
Axiomatic System
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J.M.Basilla
The Plain Truth
Axiomatic System
The elements of an
axiomatic System
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Proofs
What is the truth? What does it mean for a
sentence to be true or to be false?
Axioms of Set
Theory
Consistency Model
Two commonly used theories on truth
Axiomatic System
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Axiomatic System
Proofs
2. Coherence Theory
There is a collection of statement which are
presumed to be true. A statement is true if it
is consistent with respect to the initial
statements assumed to be true under some
rules of syllogism.
There is no such thing as an absolute truth.
The Plain Truth
The elements of an
axiomatic System
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1. Correspondence Theory
A statement is true if there exist an an
object which correspond to the statement.
The object is usually verifiable by the five
senses; sight, sound, smell, touch and
taste.
Axioms of Set
Theory
Consistency Model
The Blind Men and the Elephant
John Godfrey Saxe
Axiomatic System
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Proofs
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The Plain Truth
The elements of an
axiomatic System
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It was six men of Indostan
To learning much inclined,
Who went to see the Elephant
(Though all of them were blind),
That each by observation
Might satisfy his mind.
Axioms of Set
Theory
Consistency Model
The Blind Men and the Elephant
John Godfrey Saxe
Axiomatic System
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The Plain Truth
Axiomatic System
The elements of an
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The First approached the Elephant,
And happening to fall
Against his broad and sturdy side,
At once began to bawl:
"God bless me!-but the Elephant
Is very like a wall!"
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Proofs
Axioms of Set
Theory
Consistency Model
The Blind Men and the Elephant
John Godfrey Saxe
Axiomatic System
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J.M.Basilla
The Plain Truth
Axiomatic System
The elements of an
axiomatic System
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The Second, feeling of the tusk,
Cried: "Ho!-what have we here
So very round and smooth and sharp?
To me’t is mighty clear
This wonder of an Elephant
Is very like a spear!"
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Proofs
Axioms of Set
Theory
Consistency Model
The Blind Men and the Elephant
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The Third approached the animal,
And happening to take
The squirming trunk within his
hands,
Thus boldly up and spake:
"I see," quoth he, "the Elephant
Is very like a snake!"
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John Godfrey Saxe
Axiomatic System
J.M.Basilla
The Plain Truth
Axiomatic System
The elements of an
axiomatic System
Proofs
Axioms of Set
Theory
Consistency Model
The Blind Men and the Elephant
John Godfrey Saxe
Axiomatic System
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Axiomatic System
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Proofs
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The Plain Truth
The elements of an
axiomatic System
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The Fourth reached out his eager hand,
And felt about the knee.
"What most this wondrous beast is like
Is mighty plain," quoth he;
"’Tis clear enough the Elephant
Is very like a tree!"
Axioms of Set
Theory
Consistency Model
The Blind Men and the Elephant
John Godfrey Saxe
Axiomatic System
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The Plain Truth
The elements of an
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The Fifth, who chanced to touch the ear,
Said: "E’en the blindest man
Can tell what this resembles most;
Deny the fact who can,
This marvel of an Elephant
Is very like a fan!’
Axioms of Set
Theory
Consistency Model
The Blind Men and the Elephant
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The Sixth no sooner had begun
About the beast to grope,
Than, seizing on the swinging tail
That fell within his scope,
"I see," quoth he, "the Elephant
Is very like a rope!"
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John Godfrey Saxe
Axiomatic System
J.M.Basilla
The Plain Truth
Axiomatic System
The elements of an
axiomatic System
Proofs
Axioms of Set
Theory
Consistency Model
The Blind Men and the Elephant
John Godfrey Saxe
Axiomatic System
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Axiomatic System
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The Plain Truth
The elements of an
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And so these men of Indostan
Disputed loud and long,
Each in his own opinion
Exceeding stiff and strong,
Though each was partly in the right,
And all were in the wrong!
Axioms of Set
Theory
Consistency Model
Theory
Axiomatic System
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Axiomatic System
Proofs
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3. A theory is never true nor false. It can only remain
valid, plausible or debunked.
The Plain Truth
The elements of an
axiomatic System
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A guess or a conjecture which attempts to
give an explanation to some observable
characteristic/event. The validity of
statements are decided under the
assumption of a certain theory.
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2. Theory
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1. Every discipline has its own system of judging
whether a statement is true or false.
Axioms of Set
Theory
Consistency Model
Examples of theories
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1. Theories which attempts to explain the existence of
our solar system such as Nebular Theory, Cotme de
buffon Theory.
2. Theories which attempts to explain the origin of our
universe, Big bang Theory
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3. Theories which attempts some observable events
such as a falling object, Newton’s Universal Law of
Gravitation, Einstein’s Geometric Theory of Gravity.
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5. Theories explaining the origin of mankind:
Creationists versus evolutionists.
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4. Theories explaining light, particle theory vs wave
theory
Axiomatic System
J.M.Basilla
The Plain Truth
Axiomatic System
The elements of an
axiomatic System
Proofs
Axioms of Set
Theory
Consistency Model
Axiomatic System
Axiomatic System
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An axiomatic system or a postulate system consists
of some undefined terms and a list of statements,
called axioms or postulate, concerning the primitive
terms.
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A mathematical theory is obtained from an axiomatic
system by proving new statements called theorems
using only the axiom and/or previously proved
theorems.
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Definitions are made in the process in order to be
more concise and to facilitate discussions.
The Plain Truth
Axiomatic System
The elements of an
axiomatic System
Proofs
Axioms of Set
Theory
Consistency Model
The elements of an Axiomatic System
Primitive Terms
Axiomatic System
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Proofs
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The behavior and properties of primitive terms are
understood through the axioms of an axiomatic
system.
JB
Axiomatic System
The elements of an
axiomatic System
Primitive terms are undefined terms/objects.
Primitive terms form the very foundation of an
axiomatic system.
The Plain Truth
Axioms of Set
Theory
Consistency Model
The elements of an Axiomatic System
Axioms
Axiomatic System
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Any object that can be shown to satisfy the axioms
can be considered as an interpretation of the
primitive term.
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It is through these axioms that we come to
understood a primitive term.
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Axioms are rules/description of primitive terms which
are accepted as truth.
The Plain Truth
Axiomatic System
The elements of an
axiomatic System
Proofs
Axioms of Set
Theory
Consistency Model
Chess
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The Plain Truth
The elements of an
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Axiomatic System
There are exactly two persons playing in a game of
chess.
The board has sixty-four squares of alternating
colors.
Each player has a set of sixteen pieces with the
following composition
Pawns 8 Bishops 2 Knights 2
Rooks 2
Queen 1
King
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Primitive terms in the game ”Chess”
person(playing)
knight
chessboard
rook/castle/tower
pawn
queen
bishop
king
1. Some axioms
Axioms of Set
Theory
Consistency Model
Newtonian Physcis
Axiomatic System
J.M.Basilla
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some primitive terms in Newtonian Physics
matter energy mass
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Some axioms: Newtons law of motions
First law
A body persists its state of rest or of uniform
motion unless acted upon by an external
unbalanced force."
The Plain Truth
Axiomatic System
The elements of an
axiomatic System
Proofs
Axioms of Set
Theory
Consistency Model
Newtonian Physcis
Axiomatic System
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Axiomatic System
Proofs
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The Plain Truth
The elements of an
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Third law
To every action there is an equal and opposite
reaction.
J.M.Basilla
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Second law
Force equals mass times acceleration (F = ma)":
the net force on an object is equal to the mass of
the object multiplied by its acceleration.
Axioms of Set
Theory
Consistency Model
Set Theory
A mathematical example
Axiomatic System
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The Plain Truth
Axiomatic System
The elements of an
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Proofs
Primitive Terms : elements and sets
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Axioms: An element is a member of a set. If this is
the case, we say the element x is and element of the
set S, written x ∈ S.
Axioms of Set
Theory
Consistency Model
Definitions
Axiomatic System
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As an axiomatic system becomes more and more
complicated, definitions are introduced to facilitate
discussions and analyses.
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For example, in a game of chess, some of the
defined terms are
check
mate
stalemate
en passant castling
draw
In set theory, there is a definition, A set A is a subset
of B if and only if every element of A is also an
element of B.
The Plain Truth
Axiomatic System
The elements of an
axiomatic System
Proofs
Axioms of Set
Theory
Consistency Model
Theorems and Propositions
Axiomatic System
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3. The derivation of a theorem is the proof of truth of the
resulting statements once the hypothesis is satisfied.
4. Well known theorems are Four-color theorem;
Fermat(-Wiles) theorem, Pythagorean Theorem
The Plain Truth
Axiomatic System
The elements of an
axiomatic System
Proofs
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2. Often a theorem is stated with two parts; the
hypothesis and the conclusion. The hypothesis is a
condition that when satisfied results to the
occurrence/validity of the conclusion.
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1. A statement which has been proven on the basis of
previously proven and/or previously accepted true
statements suchs as other theorems and/or axioms.
Axioms of Set
Theory
Consistency Model
What is a proof?
Axiomatic System
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- A logically sound argument that progress from the
accepted ideas to the statement being proven.
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Proofs
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Axiomatic System
The elements of an
axiomatic System
A theorem is a statement that has been proven
logically from a set of axioms or previously proven
theorems.
- main kinds are the
method of counterexamples,
direct method,
indirect method
mathematical induction
The Plain Truth
Axioms of Set
Theory
Consistency Model
Axioms of Set Theory
Axiomatic System
Primitive terms : elements, sets, universal set.
Axioms
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elements are members of the universal set.
A collection of some(perhaps all) of the elements
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If every element of A is also an element of B, we call
A a subset of B. This idea is conveyed in symbol as
A ⊂ B.
Two sets A and B are said to be equal if and only if
A ⊂ B and B ⊂ A.
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If A ⊂ B and B ⊂ C then A ⊂ C.
The complement of A ∩ B is the union of the
complements of A and B.
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Some theorems
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’ denote the ”complement of”.
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(A ∩ B)0 = A0 ∪ B 0 ,
as
Axiomatic System
The elements of an
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Some Definitions
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The Plain Truth
Proofs
Axioms of Set
Theory
Consistency Model
Consistency Model
Axiomatic System
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J.M.Basilla
The Plain Truth
Axiomatic System
The elements of an
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Proofs
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Set Theory - Venn Diagrams
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An interpretation of the axioms demonstrating the
desired results
Axioms of Set
Theory
Consistency Model