Module 3: Axiomatic Systems

Question 1

The language in mathematics refers to the _______________________.

Answer 1

Undefined and Defined Terms

Question 2

The rules in Mathematics refer to the ______________.

Answer 2

Axioms and Theorems

Question 3

Axiomatic Method

Answer 3

No mathematical claim is accepted unless it can be proven from basic axioms.

Question 4

The axiomatic method originated from the __________.

Answer 4

Greeks, 600 B.C.

Question 5

He organized known mathematics in The Elements and formalized the axiomatic method.

Answer 5

Euclid, 300 B.C.

Question 6

Definition

Answer 6

The statement of a single, unambiguous idea that the word, phrase, or symbol defined represents

Question 7

Characteristic Definition

Answer 7

Provides a single unambiguous complete idea

Question 8

Circular Definition

Answer 8

Uses terms that are themselves defined by the term being defined

Question 9

Undefined Terms

Answer 9

Terms that cannot be defined, but its meaning can be derived from context

Question 10

2 types of undefined terms

Answer 10

elements and relations

Question 11

Axioms or Postulates

Answer 11

statements that are accepted as true without proof

Question 12

Axioma was coined by _________.

Answer 12

Aristotle

Question 13

A=A

Answer 13

Reflexivity Axiom

Question 14

If a=b, then b=a

Answer 14

Symmetry Axiom

Question 15

If a=b and b=c, then a=c

Answer 15

Transivity Axiom

Question 16

1st Postulate of Euclid

Answer 16

A straight line segment can be drawn joining any two points

Question 17

2nd Postulate of Euclid

Answer 17

Any straight line segment can be extended indefinitely in a straight line.

Question 18

3rd Postulate of Euclid

Answer 18

Given any straight line segment, a circle can be drawn having the segment as radius an one endpoint as center.

Question 19

4th Postulate of Euclid

Answer 19

All right angles are congruent.

Question 20

5th Postulate of Euclid (not really a postulate?)

Answer 20

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.

Question 21

Theorem

Answer 21

Statements proven from axioms and previously proven theorems.

Question 22

Proof

Answer 22

A logically sound argument that progresses from accepted ideas to the statement being assessed.

Question 23

If the negation of the theorem is false, then the theorem is ____.

Answer 23

TRUE

Question 24

Considerations on Proofs

Answer 24

Experiments, visualizations, and popular opinions are not considered as proof; proving is a matter of rigor; theorems must be proven through systematic reasoning.

Question 25

Q.E.D

Answer 25

quod erat demonstandum (which was to be demonstrated)

Question 26

Tombstone (a small rectangle with its shorter side horizontal)

Answer 26

the death of suspicion of the validity of the statement that was to be proven

Question 27

Halmos (filled in square)

Answer 27

a symbol representing the end of proof; introduced by Paul Halmos

Question 28

Conjecture

Answer 28

A conclusion or proposition based on incomplete information for which no rigorous proof has been found

Question 29

Consistent Axiomatic System

Answer 29

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)

Question 30

Interpretation

Answer 30

any assignment of specific meanings to the undefined terms of that system

Question 31

Model of the Axiomatic System

Answer 31

an interpretation that satisfies all the axioms of the system

Question 32

An interpetation satisfies the axiom if ____________________.

Answer 32

an axiom becomes a true statement when its undefined terms are interpreted in a specific way

Question 33

Complete Axiomatic System

Answer 33

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

Question 34

Axiom P in a consistent system is independent if ______________.

Answer 34

the axiomatic system formed by replacing P with its negation is also consistent

Question 35

Dependent Axiom

Answer 35

not independent

Question 36

Independent Axiomatic System

Answer 36

each of its axioms is independent

Question 37

An axiom in a consistent system is independent if

Answer 37

a model can be provided for which the axiom is false while all the other axioms are true

Question 38

David Hilbert's 2nd Problem

Answer 38

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

Question 39

Kurt Godel's Incompleteness Theorem

Answer 39

In any "sufficienlty complex" consistent axiomatic system, there must exist true statements that cannot be proven