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### Hyperbolic Geometry

Date: 03/24/2003 at 20:31:21
From: Lis
Subject: Hyperbolic Geometry

Can you explain to me this assumption:

Assuming that Euclidean geometry is consistent, had any of the failed
attempts to prove Euclid's 5th Postulate from the other axioms
succeeded, they would have actually completely destroyed Euclidean
geometry as a consistent body of thought.

Date: 03/25/2003 at 04:20:59
From: Doctor Jacques
Subject: Re: Hyperbolic Geometry

Hi Lis,

There is some special logic involved here.

Hyperbolic geometry states that you can draw more than one line
parallel to a given line through an external point. This implies that
the "5th postulate" is false.

You can read more on Non-Euclidean geometry in the Dr. Math archives:

Euclid's Parallel Postulate
http://mathforum.org/library/drmath/view/54750.html

Hyperbolic Geometry and the Euclidean Parallel Postulate
http://mathforum.org/library/drmath/view/52832.html

or other pages that you can find by using the Dr. Math searcher:

http://mathforum.org/library/drmath/mathgrepform.html

It has been proven that, IF Euclidean geometry is consistent, then
there is a geometry that negates the 5th postulate and is also
consistent.

The idea of the proof is to build a model of non-Euclidean geometry
within Euclidean geometry itself, and to prove that that model
satisifies all the other axioms of Euclidean geometry, and therefore
all conclusions that can be drawn from them.

There are a few different such models. In two dimensions, one of the
most widely known is due to Poincare. In that model the "plane" is
the interior of a disk in the Euclidean plane, and the "straight
lines" are circular arcs orthogonal to the perimeter of the disk. You

World of Mathematics - Eric Weisstein
http://mathworld.wolfram.com/PoincareHyperbolicDisk.html

The important thing is that the model is built within the Euclidean
plane itself, using properties of Euclidean geometry. The fact that
the "plane" and "straight lines" (mind the quotes) satisfy the
postulates of Euclidean geometry (except the 5th) can be proven
*using Euclidean geometry itself*.

The conclusion is that, IF Euclidean geometry is valid with the 5th
postulate, it is also valid with its negation.

Now, if we were able to prove the 5th postulate from the other axioms,
then, as a consequence, we could also prove that it is false, and this
would be a contradiction. In such a case, the conclusion would be that
Euclidean geometry implies a contradiction, i.e. it is inconsistent.
That is, in my opinion, the meaning of the sentence you refer to.

Euclidean geometry - as exposed is Euclid's Elements - is far from a
rigorous discipline. Euclid assumes implicitly many more things than
his explicit postulates. However, this can be remedied with a more
rigorous approach, and the conclusion remains the same.

The problem encountered here is rather common in mathematics, for
example in set theory. It is usually not possible to prove that a
system of axioms is consistent, but, in some cases, it can be proven
that, IF you assume that a system of axioms is consistent, then it
remains consistent if you add another axiom to it.

some more, or if you have any other questions.

- Doctor Jacques, The Math Forum
http://mathforum.org/dr.math/
Associated Topics:
College Euclidean Geometry
College Logic
College Non-Euclidean Geometry

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