Question

Answer the following questions. Please support your answers with appropriate reasoning. 1. Restate Euclid's fifth postulate in a more understandable form without changing the overall meaning of the statement. 2. Have any of the theorems proved by Euclid in the Elements been shown to be false in Euclidean geometry? If so, which ones? If not, why did Hilbert feel the need to revamp Euclid's work?

Answer 1

5th postulate :

if a straight line falling on two straight lines makes the interior angles on the same side of it taken together less than 180 , then the two straight lines will meet on that side on which the sum of angles is less than 180 if produces indefinitely.

None of the theorems proved by Euclid in the Elements been shown to be false in Euclidean geometry. But Hilbert feel the need to revamp Euclid’s work because Euclid’s work has few assumptions which are hidden. Due to hidden assumptions Euclid’s axiomatic system is not complete. Like, Euclid assumed the existence of points, none of his postulates implies that points exist. He didn’t mentioned parallel word in postulates 5, but the postulates is generally known as parallel postulates.

Definition for point by Euclid is “A point is that which has no part”, or a line as “A line is breathless length”. Euclid never defined breathless length. These descriptions are good for guide purpose but not sufficiently precise for justification for logical arguments.

Due to hidden assumptions Euclid’s axiomatic system is not complete. Like, Euclid assumed the existence of points, none of his postulates implies that points exist. He didn’t mentioned parallel word in postulates 5, but the postulates is generally known as parallel postulates. Euclid often tacitly assumed things that he felt were obvious.

So Hilbert felt that these hidden assumptions need to be revamped.

Hence none of Euclid’s theorems are valid.