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.
