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Mathematics and Axioms

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- 1. Math 8 – Mathematics as an Axiomatic System Ms. Andi Fullido © Quipper
- 2. Objectives • At the end of this lesson, you should be able to: • define an axiomatic system; and • enumerate the parts of an axiomatic system.
- 3. An axiomatic system is a system composed of the following: • Undefined terms • Definitions or Defined terms • Axioms or Postulates • Theorems
- 4. Undefined Terms • Undefined terms are terms that are left undefined in the system. Instead of providing a definition for them, we resort to a description, illustration, or demonstration.
- 5. Examples: • Point - represented by a dot, it has no length, width, or thickness. • Line - represented by a straight line with two arrowheads, it has no thickness but its length extends and goes on forever in both directions.
- 6. Definitions • Definitions or defined terms on the other hand are terms defined from the undefined terms in the system.
- 7. Examples • Angle - A figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle. • Parallel lines - Lines in a plane which do not meet.
- 8. Postulates • Postulates are statements that are considered true without proof or validation.
- 9. Examples • Postulate: Through any two points there is exactly one line. • Postulate: If two lines intersect, then they intersect at exactly one point. • These statements need not to be proven.
- 10. Theorems • Theorems are statements proved to be true using postulates, definitions, other established theorems, and logic.
- 11. Vertical Angles Theorem: Vertical angles are equal in measure. • Proof:
- 12. Consistency • An axiomatic system is said to be consistent if there are no axiom or theorem that contradict each other. So if the following statement is an axiom or a theorem: • “There exist two lines that are parallel.” • Then its negation should not be an axiom or a theorem: • “No two lines are parallel.” • An axiomatic system should be consistent for it to be logically valid. Otherwise, the axiomatic system and its statements are all flawed.
- 13. Independency • A postulate is said to be independent if it cannot be proven true using other axioms in the system. An axiomatic system is said to be independent if all of its axioms are independent.
- 14. Try it! • Which of the following is not an undefined term? • {point, ray, line, plane}
- 15. Sol’n • The ray is the only defined term in the list.
- 16. “Axiom” • The word "axiom" can be used interchangeably with "postulate." If a statement is an axiom, it is regarded as true whether it makes sense or not.
- 17. Summary An axiomatic system is a system composed of the following:[ • Undefined terms: Terms that have no definition, but instead are either described or illustrated. • Definitions or Defined terms: Terms that are defined using undefined terms in the system. • Axioms or Postulates: Statements that are considered true without proof or validation. • Theorems: Statements proved to be true using postulates, definitions, other established theorems, and logic.

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