Consider the following axiom system: Undefined terms: point, line, incident with Defined terms: Two lines are...

 

Transcribed Image Text:

Consider the following axiom system: Undefined terms: point, line, incident with Defined terms: Two lines are said to be parallel if and only if no point incident with both of them. AXIOMS: Axiom 1. Any two distinct points determine a unique line Axiom 2. Any line is incident with at least two points. Axiom3. There is exactly 4 distinct points. Axiom 4. No line incident with all points. Is the following statement independent from this Axiom System? Explain why? i. S: For any line é and any point P not on e, there is at least two lines incident with P and parallel to e Is this axiom system categorical? Explain. Is it a complete axiom system? Explain. ii. iii. iv. Give an example of a non-consistent axiom system, other than the one given in the lecture. Consider the following axiom system: Undefined terms: point, line, incident with Defined terms: Two lines are said to be parallel if and only if no point incident with both of them. AXIOMS: Axiom 1. Any two distinct points determine a unique line Axiom 2. Any line is incident with at least two points. Axiom3. There is exactly 4 distinct points. Axiom 4. No line incident with all points. Is the following statement independent from this Axiom System? Explain why? i. S: For any line é and any point P not on e, there is at least two lines incident with P and parallel to e Is this axiom system categorical? Explain. Is it a complete axiom system? Explain. ii. iii. iv. Give an example of a non-consistent axiom system, other than the one given in the lecture.