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Math 10 > Module 3: Axiomatic Systems > Flashcards

Module 3: Axiomatic Systems Flashcards

1
Q

The language in mathematics refers to the _______________________.

A

Undefined and Defined Terms

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2
Q

The rules in Mathematics refer to the ______________.

A

Axioms and Theorems

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3
Q

Axiomatic Method

A

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

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4
Q

The axiomatic method originated from the __________.

A

Greeks, 600 B.C.

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5
Q

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

A

Euclid, 300 B.C.

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6
Q

Definition

A

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

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7
Q

Characteristic Definition

A

Provides a single unambiguous complete idea

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8
Q

Circular Definition

A

Uses terms that are themselves defined by the term being defined

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9
Q

Undefined Terms

A

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

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10
Q

2 types of undefined terms

A

elements and relations

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11
Q

Axioms or Postulates

A

statements that are accepted as true without proof

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12
Q

Axioma was coined by _________.

A

Aristotle

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13
Q

A=A

A

Reflexivity Axiom

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14
Q

If a=b, then b=a

A

Symmetry Axiom

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15
Q

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

A

Transivity Axiom

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16
Q

1st Postulate of Euclid

A

A straight line segment can be drawn joining any two points

17
Q

2nd Postulate of Euclid

A

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

18
Q

3rd Postulate of Euclid

A

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

19
Q

4th Postulate of Euclid

A

All right angles are congruent.

20
Q

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

A

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.

21
Q

Theorem

A

Statements proven from axioms and previously proven theorems.

22
Q

Proof

A

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

23
Q

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

A

TRUE

24
Q

Considerations on Proofs

A

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

25
Q

Q.E.D

A

quod erat demonstandum (which was to be demonstrated)

26
Q

Tombstone (a small rectangle with its shorter side horizontal)

A

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

27
Q

Halmos (filled in square)

A

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

28
Q

Conjecture

A

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

29
Q

Consistent Axiomatic System

A

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)

30
Q

Interpretation

A

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

31
Q

Model of the Axiomatic System

A

an interpretation that satisfies all the axioms of the system

32
Q

An interpetation satisfies the axiom if ____________________.

A

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

33
Q

Complete Axiomatic System

A

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

34
Q

Axiom P in a consistent system is independent if ______________.

A

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

35
Q

Dependent Axiom

A

not independent

36
Q

Independent Axiomatic System

A

each of its axioms is independent

37
Q

An axiom in a consistent system is independent if

A

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

38
Q

David Hilbert’s 2nd Problem

A

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

39
Q

Kurt Godel’s Incompleteness Theorem

A

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

Math 10 (4 decks)

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