- #1

lugita15

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- 15

So Euclid's fourth Postulate is not redundant for the reason that beginning students might think. But my question is, is it nevertheless a redundant postulate, although for far less trivial reasons? David Hilbert, in his Foundations of Geometry (Grundlagen der Geometrie in German), claims to prove Euclid's fourth Postulate in theorem 15 (on page 19 of the PDF or page 13 according to the book's internal page numbering), prefacing the proof by saying "it is possible to deduce the following simple theorem, which Euclid held - although it seems to me wrongly - to be an axiom."

Now it's fair to say that Hilbert was working in a different (and more rigorous) system of axioms than Euclid was, but I think Hilbert's proof should be seriously considered for two reasons. First of all, why would he dub Euclid's decision to call "all right angles are equal" a Postulate as "wrong" if it merely reflected a stylistic difference concerning what you choose as starting assumptions and what you consider theorems? But more importantly, by tracing back all the assumptions used in the proof of theorem 15, it seems to me that only four of Hilbert's axioms are ultimately used: IV 3, IV 4, IV 5, and IV 6. And I don't think Euclid would have objected to any of these statements:

1. IV 3 follows directly from Euclid's Common Notion 2.

2. IV 4 is partly stated in Euclid's Book I Proposition 23, which doesn't depend on the fourth postulate, and the part of IV 4 which (I think) is not stated is easily provable in Euclid's system.

3. IV 5 follows from Euclid's Common Notion 1.

4. IV 6 is just part of Euclid's Book I Proposition 4, which doesn't depend on the fourth postulate at all.

So could Euclid have proven his fourth Postulate as a theorem instead of just assuming it?

Any help would be greatly appreciated.

Thank You in Advance.