Talk:Hilbert's axioms
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Isn't the link to French wikipedia broken? 4C 13:08, 8 Jun 2005 (UTC)
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[edit] Axiom I.4
Is axiom I.4: Given any three points not contained in one line, there exists a plane containing all three points. Every plane contains at least one point. translated correctly?
I was wondering about the sentence "Every plane contains at least one point.", so I checked the German version of this page and noticed that this sentence/statement is not present in that axiom.
It just says I.4: Given any three points not contained in one line, there exists a plane containing all three points.
Dfwiki 07:43, 12 April 2007 (UTC)
- I do not know about the translation, but the extra sentence indeed seems to be independent on the other axioms: take any model of the axioms, and include an additional plane which does not contain any points or lines. Then all the axioms remain valid. -- EJ 10:51, 12 April 2007 (UTC)
- Actually, there are more differences in the German version. Axiom I.3 reads "Auf einer Geraden gibt es stets wenigstens zwei Punkte, in einer Ebene gibt es stets wenigstens drei nicht auf einer Geraden gelegene Punkte." If I understand it correctly, the second part does not mean "given any line, there exists at least one point not on it", but "a plane contains at least three noncollinear points". Go figure. -- EJ 14:54, 12 April 2007 (UTC)
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- I just did a translation from the German version into English, but upon double checking I found the following file Übungsblatt Universität Konstanz which led me to believe that it is the German version is actually can't be trusted. Maybe there should be placed a comment on the German page (I don't feel like doing that). -- Dfwiki 00:15, 13 April 2007 (UTC)
[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)
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- 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] Discussion
Removing Axioms I.4-8 means there is no assurance of the existence of points.
[edit] Area Axioms
How come there are no area axioms in Euclidean Geometry? It is interesting though that there is a proof of Pythagoras theorem from only the concept of similar triangles (and arithmetic). However, Euclid's proofs contain the concept of area. What gives?
[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.
[edit] Rays
The concept of "ray" seems not to be defined, even in terms of the undefined primitives. What does it mean? Hairy Dude 16:01, 20 August 2007 (UTC)
[edit] Axioms as published in Townsend translation vs these paraphrases?
From the earliest version of this page, it has used what so far as I could tell were personalized paraphrases of the axioms, not the standard published axioms. I assumed this was due to requirement to honor copyright and followed the same principle myself in editing some erroneous paraphrases. However, I now see that the Project Gutenberg e-book provides license to use, duplicate, or disseminate the Townsend translation with no restrictions whatsoever. Accordingly I see no reason for these idiosyncratic paraphrases to continue, which invariably seem worse-written to me than Hilbert's own axioms as translated.
Unless there are objections I don't see, any reason I should not revert the axioms to the standard renditions as per the translation? —Preceding unsigned comment added by 68.226.15.19 (talk) 23:32, 14 October 2007 (UTC)
[edit] What does I.3 add that I.4 does not include?
While I.3 and I.4 (and all other axioms) are exactly as in The Foundations of Geometry, to me I.3 seems implicit from I.4. How am I wrong, which I must be: what is the difference requiring two axioms where one would, to me, seem to do?
[edit] Pasch Pastiche
While rooting through this page I was horrified to find myself forced to make these edits ([1], [2]). It seems that some previous editor substituted Pasch's theorem for Pasch's axiom.
I'm too tired and too busy to pontificate at my usual length on the idiocy thus offered to the reading public, to lament the death of rationality, or to condemn the foolishness of allowing the unwashed masses to edit unsupervised. I'll merely say that whoever thought he was watching this page, wasn't. — Xiong熊talk* 23:59, 29 January 2008 (UTC)
- Now I'm the idiot. The original wording of II.4 was correct; it was merely mislabeled. Fixed. There remains the difficulty that what is given in the article as Hilbert's discarded 21st axiom is also Pasch's Theorem. Fixed. I refer to the Townsend translation of Hilbert's paper.
