Incompleteness of Euclidean Geometry: Proving the Parallel Postulate

[junglebeast]

"For example, Euclidean geometry without the parallel postulate is incomplete; it is not possible to prove or disprove the parallel postulate from the remaining axioms."

http://en.wikipedia.org/wiki/Gödel's_incompleteness_theorems

The parallel postulate says that, if a line segment intersects 2 lines that both have angles less than 90 degrees, then those two lines must intersect.

http://en.wikipedia.org/wiki/Parallel_postulate

Why is it be impossible to prove this postulate? This seems intuitively obvious and seems like it would be very easy to prove on the basis of simply calculating the intersection point.

 

[sylas]

"For example, Euclidean geometry without the parallel postulate is incomplete; it is not possible to prove or disprove the parallel postulate from the remaining axioms."

http://en.wikipedia.org/wiki/Gödel's_incompleteness_theorems

The parallel postulate says that, if a line segment intersects 2 lines that both have angles less than 90 degrees, then those two lines must intersect.

http://en.wikipedia.org/wiki/Parallel_postulate

Why is it be impossible to prove this postulate? This seems intuitively obvious and seems like it would be very easy to prove on the basis of simply calculating the intersection point.

What axioms are you going to use to prove it? What assumptions do you make in your calculation?

There are quite a number of axioms which are equivalent to the parallel postulate, in the sense that they can be proved from the postulate, and which allow you to prove the postulate.

There are also geometries which satisfy all the other axioms of Euclid, and in which the parallel postulate is not true. For example... a hyperboloid geometry. This shows it must be impossible to prove the postulate from the other axioms.

Cheers -- sylas

 

[CRGreathouse]

Why is it be impossible to prove this postulate?

It's not.

It's impossible to prove it from Euclid's postulates. Stronger postulates could allow it to be proven. Examples:
* Trivially: Euclid's axioms + the parallel postulate
* Less trivial: Euclid's axioms + "a rectangle exists"

One nice way to think of it is that hyperbolic geometry is a model of Euclid's axioms where the parallel postulate fails.