Module 3: Axiomatic Systems Flashcards
The language in mathematics refers to the _______________________.
Undefined and Defined Terms
The rules in Mathematics refer to the ______________.
Axioms and Theorems
Axiomatic Method
No mathematical claim is accepted unless it can be proven from basic axioms.
The axiomatic method originated from the __________.
Greeks, 600 B.C.
He organized known mathematics in The Elements and formalized the axiomatic method.
Euclid, 300 B.C.
Definition
The statement of a single, unambiguous idea that the word, phrase, or symbol defined represents
Characteristic Definition
Provides a single unambiguous complete idea
Circular Definition
Uses terms that are themselves defined by the term being defined
Undefined Terms
Terms that cannot be defined, but its meaning can be derived from context
2 types of undefined terms
elements and relations
Axioms or Postulates
statements that are accepted as true without proof
Axioma was coined by _________.
Aristotle
A=A
Reflexivity Axiom
If a=b, then b=a
Symmetry Axiom
If a=b and b=c, then a=c
Transivity Axiom
1st Postulate of Euclid
A straight line segment can be drawn joining any two points
2nd Postulate of Euclid
Any straight line segment can be extended indefinitely in a straight line.
3rd Postulate of Euclid
Given any straight line segment, a circle can be drawn having the segment as radius an one endpoint as center.
4th Postulate of Euclid
All right angles are congruent.
5th Postulate of Euclid (not really a postulate?)
If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough.
Theorem
Statements proven from axioms and previously proven theorems.
Proof
A logically sound argument that progresses from accepted ideas to the statement being assessed.
If the negation of the theorem is false, then the theorem is ____.
TRUE
Considerations on Proofs
Experiments, visualizations, and popular opinions are not considered as proof; proving is a matter of rigor; theorems must be proven through systematic reasoning.
Q.E.D
quod erat demonstandum (which was to be demonstrated)
Tombstone (a small rectangle with its shorter side horizontal)
the death of suspicion of the validity of the statement that was to be proven
Halmos (filled in square)
a symbol representing the end of proof; introduced by Paul Halmos
Conjecture
A conclusion or proposition based on incomplete information for which no rigorous proof has been found
Consistent Axiomatic System
If it has no statement such that the statement and its negation are axioms or theorems of the system (i.e the axioms do not contradict one another)
Interpretation
any assignment of specific meanings to the undefined terms of that system
Model of the Axiomatic System
an interpretation that satisfies all the axioms of the system
An interpetation satisfies the axiom if ____________________.
an axiom becomes a true statement when its undefined terms are interpreted in a specific way
Complete Axiomatic System
it is possible to prove or disprove any statement about the objects in the system from the axioms alone; can be interpreted by a unique model
Axiom P in a consistent system is independent if ______________.
the axiomatic system formed by replacing P with its negation is also consistent
Dependent Axiom
not independent
Independent Axiomatic System
each of its axioms is independent
An axiom in a consistent system is independent if
a model can be provided for which the axiom is false while all the other axioms are true
David Hilbert’s 2nd Problem
Proving that mathematics itself could be reduced to a consistent set of axioms that is complete or finding axioms from which all mathematical truths could be proven
Kurt Godel’s Incompleteness Theorem
In any “sufficienlty complex” consistent axiomatic system, there must exist true statements that cannot be proven
