From: Fred Galvin <galvin@math.ukans.edu>
Subject: Re: Combinatorial concept of "set".
Date: Tue, 12 Sep 2000 20:32:42 -0500
Newsgroups: sci.logic,sci.math
Summary: [missing]

On Wed, 13 Sep 2000, it was written:

> > This notion appears quite clearly in many AC-using proofs, e.g. the proof
> > of the existence of a partition of R^3 into non-parallel lines.
> 
> Wow! Could you describe the proof? Or a reference?

There's not much too it if you're familiar with transfinite induction.
Let c be the least ordinal such that {n: n < c} has the same
cardinality as R, which is also the cardinality of R^3. List all the
points of R^3 in a transfinite sequence (P_n: n < c}. Define a
transfinite sequence of lines (L_n: n < c) as follows. Suppose that
L_i has already been defined for all i < n. If P_n is not in the union
of the family {L_i: i < n}, let P = P_n; otherwise, let P be any point
of R^3 not in that union. Choose a line L through P which is distinct
from  all the lines L_i, i < n. Each of those lines L_i lies on at most
one of the (continuum many) planes containing L, so we can choose a
plane H containing L which contains none of the lines L_i with i < n.
For each i < n, the plane determined by P and L_i intersects the plane
H in a line M_i. Let L_n be some line through P which lies in H and is
distinct from all of the lines M_i (i < n). In this way, we get a
family of noncoplanar lines L_n (n < c) covering every point of R^3.

-- 
"Any clod can have the facts, but having opinions is an art."--McCabe