Talk:Hilbert's axioms

From Wikipedia, the free encyclopedia
Jump to: navigation, search
WikiProject Mathematics (Rated C-class, Mid-priority)
WikiProject Mathematics
This article is within the scope of WikiProject Mathematics, a collaborative effort to improve the coverage of Mathematics on Wikipedia. If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks.
Mathematics rating:
C Class
Mid Priority
 Field: Geometry

Isn't the link to French wikipedia broken? 4C 13:08, 8 Jun 2005 (UTC)

Axiom I.4[edit]

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)
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)

21st axiom[edit]

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. 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. 00:52, 17 August 2006 (UTC)


Is the word "distinct" missing in a couple of the postulates? Amcfreely 09:00, 18 February 2006 (UTC)

There is a statement just before the list of axioms: "All points, straight lines, and planes in the following axioms are distinct unless otherwise stated". (talk) 12:11, 14 September 2009 (UTC)

A fair bit could be added to this entry[edit]

  • 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? 18:42, 17 March 2006 (UTC)

II.4: Axiom of Pasch[edit]

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)

What is the reference that this axioms was proved to be redundant? I can't find any source that says so. --flyingfeline 23 July 2008 —Preceding unsigned comment added by (talk) 13:00, 23 July 2008 (UTC)

I've looked this up and I can prove that Pasch's axiom is not redundant at all. I'm really curious what the (wrong) reference for this statement could possibly be... —Preceding unsigned comment added by Flyingfeline (talkcontribs) 20:47, 23 July 2008 (UTC)


"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)


"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+1CD, 1≤j<n, with B between A1 and An.") --Henrygb 23:06, 2 April 2006 (UTC)


"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.


Removing Axioms I.4-8 means there is no assurance of the existence of points.

Area Axioms[edit]

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?


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.


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)

Axioms as published in Townsend translation vs these paraphrases?[edit]

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 (talk) 23:32, 14 October 2007 (UTC)

What does I.3 add that I.4 does not include?[edit]

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?

Pasch Pastiche[edit]

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. — Xiongtalk* 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.
So tell me why this amateur has to go editing the reference work he desires to study? Where did the editors go? — Xiongtalk* 00:31, 30 January 2008 (UTC)

Well, im not really understand what is the real pasch's theorem, because this link: [3] and also this one: [4] sais, that pasch's theorem makes axiom II. 5 (related to triangles) redundant. So i guess pasch's prooved, that II. 5 is redundant. II. 4. (ABCD ordering) is really an axiom, according to the links. --Gabor8888 (talk) 22:03, 13 February 2010 (UTC)

Neither of your sources mentions anything related to Hilbert's axioms. I'm afraid you are confused.—Emil J. 11:29, 15 February 2010 (UTC)


I've made a couple of changes which concern the best translation of Hilbert's German:
(1) Hilbert's primitives are "Punkte" (points), "Gerade" (straight lines) and "Ebenen" (planes). "Gerade" had been translated as "lines" when introduced in the definition but then as "straight lines" when used in the axioms - which meant that the term "straight line" had no legitimacy - so I have made the translation consistently "straight line". I could of course instead have changed all occurrences of "straight line" to "line" and this would indeed accord with Unger's translation (Open Court, 1971), however pace Unger, "Gerade" does mean "straight" rather than "line" (it is a noun derived from the adjective "gerade" meaning "straight").
Problems with this choice of translation:Hilbert's method is for his "primitive concepts" to derive their meaning not from any prior understanding we may have of what is meant by "point", "straight line" or "plane" but implicitly from the axioms. He could indeed have invented completely new terms for them. This means that it is preferable to have them as single words as they are in German, to which the axioms can then attach meaning. So "straight line" is sub-optimal as it suggests the need for prior definitions of "straight" and "line" when it is in fact a primitive term. This is probably why Unger preferred the translation "line" alone. Unger's choice is good because it forces one to come to see that what "line" stands for has properties that a straight line has only after grasping the axioms. However, as I said, the fact remains that "Gerade" means "straight line", or "a straight" and not "line", so the only accurate one-word options would be either to translate it as "straight" giving the primitives as "points, straights, planes", or to hyphenate it as "straight-line"
(2) I have changed "Axioms of Incidence" into "Axioms of Combination". Unger's translation of "Axiome der Verknüpfung" as "Axioms of Incidence", substitutes a mathematical term with a specific meaning ("incidence" means a "lying on" or "passing through") for Hilbert's much more general term. "Verknüpfung" is a common German word meaning connection or combination: Hilbert describes these axioms as "producing a combination (or connection) between the concepts defined above(points, straight lines, planes)"[Die Axiome dieser Gruppe stellen zwischenden oben erklärten Begriffen Punkte, Geraden und Ebenen eine Verknüpfung her]. The principles behind my change are (1) it more accurately translates the original (2)it is true to Hilbert's method for meaning as much as possible to be determined through the axioms rather than depend on a prior mathematical understanding.Nickomidgley (talk) 17:17, 20 April 2009 (UTC)

While you are most likely correct in your translations (my German is not good enough to verify your statements) doing this translation in Wikipedia is considered Original Research and is verboten! I have changed everything back to what it is in the Unger translation since that is a published source. Bill Cherowitzo (talk) 03:55, 1 October 2013 (UTC)

Poorly defined[edit]

I think this article is very poorly written. For example:

Axiom I.1: Two distinct points A and B always completely determine a straight line a. We write AB = a or BA = a. Instead of “determine,” we may also employ other forms of expression; for example, we may say “A lies upon a”...

This is a highly ambiguous way of defining the axiom. All that can be inferred from this statement is that there exists some injective function mapping pairs of points onto lines. It also appears to introduce a new ternary operator '=' not listed in the definitions at the beginning of the section. According to another source, linked to below, what is meant by this axiom is simply "For every two points A and B, there exists a unique line that contains both of them.", a statement easily translated into symbolic logic and thus much more precise and formal.

Conversely, the statement which we currently have does not even mention the containment operator. — Preceding unsigned comment added by (talk) 04:31, 12 August 2011 (UTC)

So I suggest the section be re-written, using the above source as a style reference. — Preceding unsigned comment added by (talk) 04:30, 12 August 2011 (UTC)

Moore and the genius[edit]

I have read in the David Hilbert article that R. L. Moore proposed a set of axioms similar to Hilbert, but independently. He was 19 then, so it must have been in 1901 or so. The wikis in other languages translated this, too.

After that, I have read in the R. L. Moore article that RLMoore proved that one of Hilbert's axioms was redundant.

After that, I have read in this Hilbert's axioms article that E. H. Moore proved that one of Hilbert's axioms was redundant.

After that, I have read in the E. H. Moore article that EHMoore proved that one of Hilbert's axioms was redundant, but simultaneously, and independently, a 20yrs old RLMoore gave a more elegant proof of the same topic (ie that one of Hilbert's axioms was redundant).

So in the end I don't know which version is right, but a lexicon shouldn't include paradoxons (IMO). (talk) 22:44, 12 February 2012 (UTC) (DJS from Hungary)

I switched the R.L. in this article to E.H. based on a footnote in Foundation of Geometry (Unger translation) which gave a reference to E.H.'s article of 1902. R.L.'s work was not mentioned. MacTutor's article on R.L. does mention this work in geometry, but doesn't give a reference for it - it is possible that it wasn't published. There are certainly written biographies of R.L. that one can try to get for a reference to this detail, but until someone does that, I'd stick with E.H. Bill Cherowitzo (talk) 06:56, 14 February 2012 (UTC)
Upon further investigation I have found that both E.H. and R.L. have proofs that a Hilbert axiom (II.4 in the earlier editions, in particular in the Townsend translation) is redundant. E.H. published his result, R.L. did not directly publish his result, but his argument was published by his teacher G.B. Halsted. A few years later E.H. revised his argument and wrote a paper which is based on the slicker approach of R.L. I will document all of this and add it to the article. Bill Cherowitzo (talk) 04:11, 1 October 2013 (UTC)

20 or 21 axioms?[edit]

It is stated in the article that the axioms are 20 (or 21 with the discarded axiom). I probably can't count but it seems to me that they are 8+4+6+1+2=21 (and must have been 22 with the discarded one). I don't believe I am the first one that has counted them in the last 110 years, so there must be some explanation. Does somebody know it? — Preceding unsigned comment added by (talk) 22:30, 24 January 2014 (UTC)

This is a bit of a complicated story as the number has changed several times over the years. The original 1899 edition had 20 axioms, the French translation of that edition had 21 (this is usually considered the 2nd edition because Hilbert made some changes). The first English edition (translated by Townsend) has the same 21. The distribution was 7-5-1-6-1-1. Several changes were made in the editions up to the 7th, at which time the distribution was 8-4-1-5-2. An axiom of order was discarded (mentioned in the article) and also an axiom of congruence (not mentioned in the article), but an axiom of incidence was added. This was the distribution of axioms in the 10th edition which was the basis for the second English translation (by Unger). Our article is not faithful to any one edition. Most of it follows Unger, but the axioms of congruence follow Townsend. I hope this helps. Bill Cherowitzo (talk) 04:36, 25 January 2014 (UTC)

Congruence of triangles, Axiom III.5, and other problems[edit]

In a couple of other languages' versions of this page (French and Hebrew), the congruence relation is said to apply to line segments, angles, and triangles, and axiom III.6 concludes that the two triangles are congruent. Quite confusing, but I actually think that expressing it only in terms of angles is cleaner.

Also, in these languages' pages, as well as a few other sites on the internet, Axiom III.5 (the one about the transitivity of angle congruence), is missing. Is it perhaps derivable from the others? In some other languages' versions of the page though, it is still included. Actually I'm pretty sure it's needed unless congruence is defined elsewhere to be an equivalence relation or something.

And yeah, anyway, this is supposed to be rigorous and axiomatic, and yet there are so many things which have not been defined:

  • What is the definition of a segment? (an unordered pair of the endpoints? an ordered pair of the endpoints? the set of all the points on the segment?)
  • What is the rule for when a point is on a segment, or a segment is on a line?
  • What does it mean to talk about a side of a point on a line (as in III.1)
  • What is the definition of an angle? (an unordered pair of intersecting lines? intersecting rays? an ordered pair of intersecting lines? intersecting rays? an ordered triplet of points?)
  • "Let an angle ∠ (h,k) be given in the plane α". What does this mean? It hasn't even said what h and k are.
  • "Suppose also that, in the plane α′, a definite side of the straight line a′ be assigned". I get what it's saying, but it hasn't defined this rigorously.
  • What is the definition of a ray? (an ordered pair of the endpoint and another point on it? the set of all the points on the ray?)
  • What is the rule for when a point is on a ray, or a ray is on a line?
  • "∠ (h, k), or ∠ (k, h)". Soooo, the order doesn't matter?
  • "There exist at least three points that do not lie on a line." This sentence is ambiguous. For every line there exist at least 3 points that do not lie on it? Or there exist at least 3 points such that there does not exist a line they all lie on? Same with "There exist at least four points not lying in a plane".
  • And as someone said before, the line "For every plane there exists a point which lies on it" does not occur on the German page.
  • "n segments CD constructed contiguously from A, along the ray from A through B". Once again, "constructed continguously", not defined.

Oh, and one site I found has II.2 as "For two points A and C, there always exists at least one point B on the line AC such that C lies between A and B". Important difference. (By the way, not really relevant to the wikitalk itself, but I can't seem to prove from the axioms that of three points A,B,C on a line, one has to be between the other two. Nor can I seem to prove that given a line and a point not on it, some parallel runs through the point).

In any case, yeah, I see that "These axioms and their numbering are taken from the Unger translation (into English) of the 10th edition of Grundlagen der Geometrie", but in my opinion there's so much left to be desired. And why does every source seem to differ on the details of the axioms? :( AndreRD (talk) 17:58, 3 July 2015 (UTC)

Two comments. First of all, most of the inconsistencies that you've noted come from the fact that Hilbert kept changing the wording of the axioms in the different editions of the book, so what a web page reports depends on which edition was being used as a source. The two English translations do not agree with each other in several respects, so choices had to be made in terms of what is written on this page. Secondly, most of your issues have to do with definitions, which are not part of what this article is about. Wikipedia is not a textbook, it is not meant to be a complete treatment of the subject. If you want to study what Hilbert said, look at his book ... it is available on-line. Bill Cherowitzo (talk) 19:10, 3 July 2015 (UTC)