Re: Infinities

Anton Sherwood (
Mon, 17 Nov 1997 18:41:26 -0800 (PST)

John K Clark explains
: > Or you can prove that all lines are composed of the same number of points
: > regardless of length. Draw 2 parallel lines, a short one and a long one
: > below it, pick a point midway along the short line but above it.

: > /\
: > / \
: > /________\
: > / \
: > /________________\

: > Draw a line from that point to any place on the short line, then continue
: > it until you hit the long line. You've made a one to one correspondence
: > between all the points in the short line and all the points in the long
: > line, so they must have an equal number of points.

Wolfkin asks:
: How did you get this? It seems obvious that because we admitted that
: the longer line *was* longer, after you have used all of the
: (infinity of) points on the shorter line, there will be points on the
: longer line in between the lines you drew across.

Do it the other way. Draw a line through the apex and any point on the
longer segment. If the shorter segment contains only half as many points
as the longer segment, then half of the time your new line will intersect
*no* point on the shorter segment. I think Euclid would disagree.

Anton Sherwood *\\* +1 415 267 0685 *\\*