AXIOMATIC SYSTEM Flashcards
What are the four essential components of an axiomatic system?
- Defined terms
- Undefined terms
- Axioms
- Theorems
What is an axiom?
A statement accepted as true without proof.
What is a theorem?
A new result that evolves from undefined terms, defined terms, and axioms.
What is the purpose of axioms in mathematics?
To provide the basic rules upon which theorems can be established.
What is a definition in the context of an axiomatic system?
A statement of a single, unambiguous idea that the term represents.
What characterizes a good definition?
It must be unambiguous and not circular.
True or False: Circular definitions are acceptable in mathematics.
False
What are undefined (primitive) terms?
Terms that form a fundamental vocabulary for defining other terms.
Give an example of an undefined term.
Point, Line, Plane, Set
What constitutes a sound argument in mathematics?
An argument that is valid and whose premises are all true.
How is a proof defined?
A logically sound argument that progresses from accepted ideas to the statement in question.
What is the difference between a concrete model and an abstract model in axiomatic systems?
Concrete models use real-world objects and relations, while abstract models use terms from another axiomatic development.
What does it mean for an axiomatic system to be consistent?
It means there is no statement such that both it and its negation are axioms or theorems.
What is the role of an interpretation in an axiomatic system?
To assign specific meanings to the undefined terms of the system.
Fill in the blank: A __________ is an interpretation that satisfies all the axioms of an axiomatic system.
model
What is an example of a circular definition?
Define number as quantity. Define quantity as amount. Define amount as number.
What is the significance of the axiomatic method in mathematics?
It provides a structured approach to derive theorems from basic definitions and axioms.
What kind of terms do undefined terms imply?
They imply objects and relationships between objects.
What is a characteristic property in a definition?
A condition that allows us to determine whether an object satisfies the definition.
True or False: Axioms can be proven.
False
What is the first step in proving conjectures in mathematics?
Start with some assumed information, often provided by axioms.
What does it mean for a mathematical claim to be accepted?
It must be provable from basic axioms.
Give an example of a theorem derived from axioms.
Pythagorean Theorem
How does the axiomatic method distinguish mathematics from other disciplines?
It relies on rigorous proofs derived from accepted axioms and definitions.
What is an exercise to determine the accuracy of a definition?
Identify whether the statement is circular or not characteristic and provide an example.
What is an axiomatic system?
A collection of statements about undefined terms
What makes an axiomatic system consistent?
It has a model where the axioms are satisfied
What is an inconsistent axiomatic system?
A system containing contradictory axioms, rendering it of no practical value
Give an example of a consistent axiomatic system.
Giraffe System, Cat-Mouse-Catch System, Ant-and-Path System
What is a non-example of an axiomatic system?
A system with contradictory axioms, such as A matches B and A does not match B
What can be concluded if two axioms contradict each other?
The system is inconsistent and has no models
Define a theorem in the context of an axiomatic system.
A statement derived from the axioms by strict logical proof
What are the undefined terms in the ant-and-path axiomatic system?
- Ant
- Path
- Has
Prove that there exists at least one path in the ant-and-path system.
There exists at least one ant (Axiom 3), every ant has at least two paths (Axiom 1), therefore, there exists at least one path.
What is the conclusion reached in the proof that there are exactly two paths?
There are exactly two paths, P1 and P2
What does the filled-in square at the end of a proof signify?
It indicates that the proof is complete
What are the undefined terms in the road-town-stop sign axiomatic system?
- Road
- Town
- Stop Sign
What does Axiom 2 state in the road-town-stop sign system?
Every stop sign is on exactly two roads
What is the conclusion of the proof that there is at least one stop sign in the town?
There must be at least one stop sign in the town
What abbreviation did mathematicians like Euclid use at the end of a proof?
Q.E.D. (quod erat demonstrandum)
What is the purpose of using a tombstone symbol in proofs?
To signify the end of doubt about the validity of the statement
Fill in the blank: Axioms 1 and 2 in the point-and-line system are contradictory, leading to an _______ system.
inconsistent
What is the relationship indicated by the term ‘has’ in the ant-and-path axiomatic system?
It indicates a relationship between ant and path
