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theorems of neutral geometry

In the first page, we said that any statement which can be proved from
the common axioms was provable in neutral geometry.
We then remarked that only euclidean and hyperbolic geometries actually
satisy these axioms. This suggests that any result which can be proved in
each of these geometries, perhaps with quite different proofs, should also
be classed as a theorem of neutral geometry.
This raises a point in Mathematical Logic. If a result holds in every system
which satisfies a set of axioms, then we say that the result is true in the
theory described by the axioms. In the present context, a result valid in
both euclidean and hyperbolic geometries is true in neutral geometry.
Obviously a result which is provable from a set of axioms is valid in any
system which satisfies those axioms. (We have a proof to hand!). But, in
general, there may exist results which are true (valid in every system),
but not provable from the axioms. This is a detail of the work of Godel.
For example, there is no set of axioms for number theory for which each
true statement is provable.
I believe that the situation here is simpler. Suppose that we can prove
a result in euclidean and in hyperbolic geometry. That is, in each case
we can obtain the result from the common axioms, together with either
the Parallel Axiom or the Alternative Parallel Axiom.

We know that, in neutral geometry, if for some point P, not on a line L,
there are two lines through P not meeting L then there are two lines for
any choice of P and L. We are then in a case where the Alternative
Parallel Axiom holds. All the axioms of hyperbolic geometry now hold
and we can use our hyperbolic proof. If there is no such point, then the
Parallel Axiom holds, and we can use our euclidean proof. Since such a
pair P,L either exists or it doesn't, we have a classic "proof by cases".
I am not sure that
I would want to put
much money on the
argument being
logically sound

We can now say that a result is a theorem of neutral geometry if either
(1) it can be proved from the common axioms, or
(2) can be proved in euclidean and hyperbolic geometries.
As an example of type (2), we have the medians theorem, which is true
in hyperbolic geometry and in euclidean geometry.
We do not propose to develop the theory of neutral geometry in general
To see how this is done, a good web source is David Royster's site.
We do give some theorems which are relevant to our study of hyperbolic
geometry.

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In the first page, we said that any statement which can be proved from the common axioms was provable in neutral geometry. We then remarked that only euclidean and hyperbolic geometries actually satisy these axioms. This suggests that any result which can be proved in each of these geometries, perhaps with quite different proofs, should also be classed as a theorem of neutral geometry. This raises a point in Mathematical Logic. If a result holds in every system which satisfies a set of axioms, then we say that the result is true in the theory described by the axioms. In the present context, a result valid in both euclidean and hyperbolic geometries is true in neutral geometry. Obviously a result which is provable from a set of axioms is valid in any system which satisfies those axioms. (We have a proof to hand!). But, in general, there may exist results which are true (valid in every system), but not provable from the axioms. This is a detail of the work of Godel. For example, there is no set of axioms for number theory for which each true statement is provable. I believe that the situation here is simpler. Suppose that we can prove a result in euclidean and in hyperbolic geometry. That is, in each case we can obtain the result from the common axioms, together with either the Parallel Axiom or the Alternative Parallel Axiom.
We know that, in neutral geometry, if for some point P, not on a line L, there are two lines through P not meeting L then there are two lines for any choice of P and L. We are then in a case where the Alternative Parallel Axiom holds. All the axioms of hyperbolic geometry now hold and we can use our hyperbolic proof. If there is no such point, then the Parallel Axiom holds, and we can use our euclidean proof. Since such a pair P,L either exists or it doesn't, we have a classic "proof by cases".	I am not sure that I would want to put much money on the argument being logically sound
We can now say that a result is a theorem of neutral geometry if either (1) it can be proved from the common axioms, or (2) can be proved in euclidean and hyperbolic geometries. As an example of type (2), we have the medians theorem, which is true in hyperbolic geometry and in euclidean geometry. We do not propose to develop the theory of neutral geometry in general To see how this is done, a good web source is David Royster's site. We do give some theorems which are relevant to our study of hyperbolic geometry.