Wouldn't a true "non-Euclidian Geometry" result from throwing out the 1st Postulate rather than the 5th?

Question

The 1st Postulate, namely that "a line may be drawn between any two points", leads directly to Zeno's Paradox, that motion would be impossible. It also allows the construction that results in the Pythagorean Theorem which leads to the need to create the irrational numbers.

Answer 1

These are interesting questions.

Modern versions of
Euclid's Postulates.

1. For every point P and every point Q not equal to P there exists a unique line that passes through P and Q.
2. For every segment AB and for every segment CD there exists a unique point E such that B is between A and E and segment CD is congruent to segment BE.
3. For every point O and every point A not equal to O there exists a circle with center O and radius OA.
4. All right angles are congruent to each other.
5. For every line l and for every point P that does not lie on l there exists a unique line m through P that is parallel to l.

Note that the 1st postulate justiifies the use of the straight edge to draw lines connectiong two points.

Note that the 2nd postulate is used to justify adding two line segments together onto the same line.

Note that the 3rd postulate is used to justify the use of the compass in constructing the circle.

Note that the 4th postulate is used to justify the notion that
all squares have identical shape.

The first 4 postulates are fundamental to the process of constructing geometric figures with straight edge and compass.

Without the first postulate, we would not be justified in declaring that any line we drew had consistent qualtities.

The first four postulates permit us to build a geometry.
The fifth postulate determines which geometry it is.

It's possible that you might replace the 1st postulate with some other postulate that would enable us to draw a staight line, but in such a way as to produce a different geometry. I have no idea as yet what such an alternative 1st postulate might be.

By the way, Zeno's Paradox is no longer considered a paradox.
The ancient greeks thought it paradoxical because they did not know that it was possible for a sum of an infinite number of numbers to be finite.

Today we are quite familiar with the notion that

1 + 1/2 + 1/4 + 1/8 + 1/16 + . . . for an infinite number of terms
is the finite number 2.

If in the first second I travel 1 foot.
if in the next half second I travel 1/2 foot more
if in the next fourth second I travel 1/4 foot more
if in the next eighth second I travel 1/8 foot more
if in the next 1/16 second I travel 1/16 foot more,
etc
for an infinite number of steps like this,

then in 2 seconds I will have traveled 2 feet.

Also,
It's not the fault of the Pythagorean theorem that we needed to create the irrational numbers.

The irrational numbers exist on the number line, and the Pythagorean theorem merely allowed us to discover it.

Answer 2

Without it, geometry itself is impossible.

The reason the fifth is so interesting is that, of Euclid's Postulates, it alone is *not* essential for a whole internally consistent geometry to be derived. This is the whole basis of Riemannian and Lobachevskian geometries that are pivotal in understanding a curved universe in modern cosmology.