Talk:Pasch's axiom

If the Pasch's axiom cannot be derived from Euclid's axioms, does it exist a geometry in which Pasch's axiom is false? --Roberto.zanasi 10:44, 16 June 2008 (UTC)

I don't think there are any (useful) ordered geometries for which Pasch's axiom is false, but there are unordered geometries (such as projective geometry) for which betweenness doesn't have any meaning. In this case, that means there is no distinction between intersecting an edge of a triangle and intersecting the line on which it lies. Eebster the Great (talk) 06:28, 17 February 2009 (UTC)

Pasch's axiom fails in three-dimensional geometry. In the context of the Euclidean geometry of the plane, "line" is often used with the meaning of "line segment," where Pasch's axiom as stated in this article fails if one end of the "line" so interpreted is inside the triangle, pointing up the need to be careful about the meaning of terms and the exact formulation of the axiom, one shouldn't take these things for granted. Pasch's axiom can also fail when lines are not Dedekind-complete, a possibility no one could visualize prior to 1872 when Dedekind plugged the holes a line might sneak through with Dedekind cuts. Pasch's axiom is weaker than the requirement of Dedekind completeness to the extent that geometries allowing such gaps might also lack any line in a position to witness the failure of Pasch's axiom. For example a sufficient model for Euclid's postulates is K^2 where K is the smallest ordered field closed under square root, which is countable whence every line segment has measure zero, i.e. is mostly empty, yet Pasch's axiom holds because non-parallel lines (of the infinite kind as opposed to line segments) that each contain two points of K^2 necessarily intersect at a point of K^2. --Vaughan Pratt (talk) 16:16, 22 May 2009 (UTC)

Removal of reference

I just removed the Hersh and Davis reference since they confused Pasch's axiom with Pasch's theorem (and they weren't even consistent ... once referring to it as the axiom and at another point referring to the same statement as the theorem.)Bill Cherowitzo (talk) 08:09, 23 December 2011 (UTC)

Statement of Axiom

I reverted the Hilbert version of the axiom because the version given by Beutelspacher and Rosenbaum is closer to the orginal statement of the axiom by Pasch. Pasch's concern was the introduction of order into Euclidean geometry in a precise way. Hilbert, because he introduced order through the primitive concept of betweeness, did not have to emphasize the order relation when he rephrased Pasch's axiom in his own work. This "sloppiness" on Hilbert's part (he was often criticized for this) has led many mathematicians to confuse Pasch's axiom with the Veblen axiom mentioned later in the article. The two versions of the axiom are the same, it is just that in the Hilbert version the reader has to infer that the word "or" in the statement needs to be rendered in the exclusive sense, which is not the way the word is used in modern mathematical writing. I suggest that for the sake of clarity, the Hilbert version of the statement of the axiom not be used in this article.Bill Cherowitzo (talk) 21:36, 3 February 2012 (UTC)

Let me be a bit more precise. The differences in the two versions are due to a difference in the meaning of a "side of a triangle". To Pasch, the side AB of triangle ABC meant the line determined by A and B. For a line m to meet the side AB internally, means that the point of intersection of these two lines is between A and B on line AB. For m to meet side AB externally, means that the point of intersection is not between A and B. For Hilbert, the side AB of triangle ABC is the line segment AB (at the point in the exposition that the axiom is introduced, triangles and their sides have not yet been defined). Thus, for Hilbert to say that line m meets "side" AB internally, he only has to say that m passes through a point of AB. Also, he would not make reference an external meeting, since there are no points outside of the segment. As a second issue, in the Hilbert version, the conclusion is that line a passes through a point of BC or a point of AC. In later editions of the book a comment is added that it can be proved that a does not pass through both of the segments, but Hilbert does not provide the proof (it was added in a supplement by P. Bernays in 1956). I find the Pasch version of the conclusion to be a bit more satisfactory in that the two possibilities are clearly incompatible and you don't need a theorem to tell you so. My objection to using the Hilbert version is that when taken out of context his statement can be misleading and the reader can miss the crucial point that this is an axiom about order and not just about intersections. Bill Cherowitzo (talk) 11:46, 4 February 2012 (UTC)

Having finally looked at the original statement of Pasch, I'm taking back what I said above. (It's not all wrong, but some essential parts are.) Bill Cherowitzo (talk) 06:22, 7 December 2014 (UTC)

Coxeter, H.S.M. (1969). Introduction to geometry (2nd ed.). John Wiley and Sons. p. 180. ISBN 0-471-50458-0. Zbl 0181.48101. "In his original formulation of Axiom 12.27, Pasch made the far stronger statement: If a line in the plane of a given triangle meets one side, it also meets another side (or passes through a vertex)" Axiom 12.27 is "If ABC is a triangle and [BCD] and [CEA], then there is on the line DE a point F for which [AFB]". Here [XYZ] is the fundamental relation Y is between X and Z. The form (12.27) is described as being "Peano's formulation". Deltahedron (talk) 19:44, 3 February 2013 (UTC)

Statement section = diagram

The color coding is confusing and should be fixed. (If you don't think so, please look at it as if you were seeing it for the first time, as if you knew nothing more than grade school geometry.) Also, the points ABC in the exposition should be labelled on the diagram as it is not self-evident (again, to a noob).

I'll concede your point. The diagram does not correspond to the original statement, but rather to the more modern version, so putting in the A, B, C labels would be confusing. The color coding was meant to help avoid a lengthy description of the internal and external portions of the sides of a triangle. I am open to suggestions (including ditching this diagram) and since I drew it, I can easily fix it. --Bill Cherowitzo (talk) 22:17, 23 September 2016 (UTC)

As a noob on the subject, I have no idea what the diagram is trying to show as related to the axiom. A full description of internal and external sides of the triangle would be extremely helpful. Trumblej1986 (talk) 20:20, 7 June 2019 (UTC)