From: Ken.Pledger@vuw.ac.nz (Ken Pledger)
Newsgroups: sci.math
Subject: Re: Pasch Axiom
Date: Mon, 22 Jun 1998 09:57:52 +1200

In article <1bRi1.772$dT6.415111@news.tpnet.pl>, "Kaziu"
<machuram@friko3.onet.pl> wrote:

> Every models of Euklides geometry are isomorphic. ( i read it in polish book
> Borsuk, Szmielew "Podstawy Geometrii"), but I heared many about Pasch Axiom.
> In aritmetic model Pasch axiom holds. My question is : is Pasch Axiom is
> realy very important? Can we proof Pasch Axiom ???
> 
> Kaziu

     You can't prove it from the other usual axioms (i.e.  Pasch's axiom
is independent of them); but I'm sorry I have no reference to a proof of
that meta-theorem.   Some years ago Dr L.W. Szczerba was working on it in
Poland.

     This may help you in thinking about it.   (I shall refer to the 1960
English translation of Borsuk & Szmielew.)   The other axioms of order,
O1-O8 (in Section 6), are purely 1-dimensional, so they lead to the theory
of order on a line.   But when you start looking at a plane, they give no
way to relate the order on one line to the order on another line.   That
is roughly why you also need a 2-dimensional order axiom such as O9
(Section 18) or the equivalent Pasch axiom (Section 26).

          Ken Pledger.