11.5: How to prove that something cannot be proved
Many attempts were made to prove that any theorem in Euclidean geom- etry holds in neutral geometry. The latter is equivalent to the statement that Axiom V is a theorem in neutral geometry.
Some of these attempts were accepted as proof for long periods of time, until a mistake was found.
There is a number of statements in neutral geometry that are equivalent to the Axiom V. It means that if we exchange the Axiom V to any of these statements, then we will obtain an equivalent axiomatic system.
The following theorem provides a short list of such statements. We are not going to prove it in the book.
A neutral plane is Euclidean if and only if one of the following equivalent conditions holds:
(a) There is a line \(\ell\) and a point \(P \not\in \ell\) such that there is only one line passing thru \(P\) and parallel to \(\ell\).
(b) Every nondegenerate triangle can be circumscribed.
(c) There exists a pair of distinct lines that lie on a bounded distance from each other.
(d) There is a triangle with an arbitrarily large inradius.
(e) There is a nondegenerate triangle with zero defect.
(f) There exists a quadrangle in which all the angles are right.
It is hard to imagine a neutral plane that does not satisfy some of the properties above. That is partly the reason for the large number of false proofs; each used one of such statements by accident.
Let us formulate the negation of (a) above as a new axiom; we label it h-V as a hyperbolic version of Axiom V .
For any line \(\ell\) and any point \(P \not\in \ell\) there are at least two lines that pass thru \(P\) and parallel to \(\ell\).
By Theorem 7.1.1 , a neutral plane that satisfies Axiom h-V is not Euclidean. Moreover, according to the Theorem \(\PageIndex{1}\) (which we do not prove) in any non-Euclidean neutral plane, Axiom h-V holds.
It opens a way to look for a proof by contradiction. Simply exchange Axiom V to Axiom h-V and start to prove theorems in the obtained ax- iomatic system. In the case if we arrive at a contradiction, we prove the Axiom V in a neutral plane. This idea was growing since the 5th century; the most notable results were obtained by Saccheri in [16].
The system of axioms I–IV and h-V defines a new geometry which is now called hyperbolic or Lobachevskian geometry. The more this geome- try was developed, it became more and more believable that there is no contradiction; that is, the system of axioms I–IV, and h-V is consistent. In fact, the following theorem holds true:
The hyperbolic geometry is consistent if and only if so is the Euclidean geometry.
- Proof
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The claims that hyperbolic geometry has no contradiction can be found in private letters of Gauss, Schweikart and Taurinus.(The oldest surviving letters were the Gauss letter to Gerling in 1816 and yet more convincing letter dated 1818 of Schweikart sent to Gauss via Gerling.) They all seem to be afraid to state it in public. For instance, in 1818 Gauss writes to Gerling:
... I am happy that you have the courage to express yourself as if you recognized the possibility that our parallels theory along with our entire geometry could be false. But the wasps whose nest you disturb will fly around your head.
Lobachevsky came to the same conclusion independently. Unlike the others, he had the courage to state it in public and in print (see [ 13 ]). That cost him serious troubles. A couple of years later, also independently, Bolyai published his work (see [ 6 ]).
It seems that Lobachevsky was the first who had a proof of Theorem \(\PageIndex{2}\) altho its formulation required rigorous axiomatics which were not developed at his time. Later, Beltrami gave a cleaner proof of the "if" part of the theorem. It was done by modeling points, lines, distances, and angle measures of one geometry using some other objects in another geometry. The same idea was used earlier by Lobachevsky; in [ 14 , §34] he modeled the Euclidean plane in the hyperbolic space.
The proof of Beltrami is the subject of the next chapter.