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"
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.