Talk:Hilbert's axioms
From Wikipedia, the free encyclopedia
Isn't the link to French wikipedia broken? 4C 13:08, 8 Jun 2005 (UTC)
Contents |
[edit] 21st axiom
If there were originally 21 axioms, shouldn't the article give the 21st axiom (or the axioms that were different in the original system) and give the reason that it was changed?
- I've honestly never seen the 21st axiom, although it is surely included in Hilbert (1980) (anybody out there own a copy?) But I can tell you what killed axiom 21. In 1901, the 19 year old Robert Lee Moore, while enrolled in an undergraduate course at the University of Texas taught by George Halsted, derived axiom 21 from the other 20. This discovery earned him a Ph.D. fellowship at the University of Chicago. Moore went on to a world class career as a pioneering topologist.132.181.160.42 05:43, 22 March 2006 (UTC)
-
- In the article, it says E.H. Moore showed the dependence of the 21st axiom. Someone probably ought to work out which one is right. 128.135.96.222 00:52, 17 August 2006 (UTC)
[edit] Distinct
Is the word "distinct" missing in a couple of the postulates? Amcfreely 09:00, 18 February 2006 (UTC)
[edit] A fair bit could be added to this entry
- What are the primitive notions?
- Does removing the six axioms mentioning the word "plane" result in an axiomatization of Euclidian plane geometry? Howard Eves (1990) says so;
- Where can one find a detailed elaboration of geometry from these axioms?
- Has anyone ever written a high school text based on these axioms (or something near them)?
- Metamathematics: What is known about the independence of these axioms? Why is this axiomatization finite when Tarkski's is not?202.36.179.65 18:42, 17 March 2006 (UTC)
[edit] II.4: Axiom of Pasch
Why does it link to the Jewish festival of Passover? So a line which passes over one edge of a triange will pass over another edge. But that is not good enough. --Henrygb 22:29, 2 April 2006 (UTC)
- Thanks to User:Rain74 it is now a link to Moritz Pasch--Henrygb 22:15, 19 June 2006 (UTC)
[edit] III.1
"Given two points A,B, and a point A' on line m, there exist two and only two points C and D, such that A' is between C and D, and AB ≅ A'C and AB ≅ A'D."
Presumably A and B do not need to be on line m while C and D do need to be. But it does not say that --Henrygb 22:53, 2 April 2006 (UTC)
[edit] V.1
"Axiom of Archimedes. Given the line segment CD and the ray AB, there exist n points A1,...,An on AB, such that AjAj+1 ≅ CD, 1≤j<n. Moreover, B is between A1 and An."
I do not understand this. n is not qualified (I would expect something like "there is a natural number n such that there exist n points A1,...,An on AB, such that AjAj+1 ≅ CD, 1≤j<n, with B between A1 and An.") --Henrygb 23:06, 2 April 2006 (UTC)
[edit] V.2
"V.2: Line completeness. Adding points to a line results in an object that violates one or more of the following axioms: I, II, III.1-2, V.1."
While this is briefer than the earlier text, which was a little unwieldy, I don't think it's precise and it is somewhat confused and contradictory, or speaking of an impossibility as being an actuality, in the phrase "adding points to a line." Adding points would comprise specifying or including points additional to those existing in a line as already defined; any such points are in fact not points of the line and therefore really aren't -- can't be -- added to the line itself. Not only would an attempt result in an object that isn't a line, it just isn't correct to say that points are added to the line in such a process. They're being considered in combination with the line's complete set of points, but are not additions to the line. Any possible such points are off the line.
However, my only correction to the present rewrite would be going back to the previous text, which I admit was unwieldy. Perhaps someone can retain the advantage of the simplicity of the above, without the present problem.
[edit] References
Robin Hartshorne's book, "Geometry: Euclid and Beyond", (Springer, 2000) has a very nice chapter devoted to Hilbert's axioms and their relation to Euclid's axiom scheme. It would probably make a good reference to be attached to this article. Note that Hartshorne uses a subset of the Hilbert axioms, namely those necessary for plane geometry.
