Date: Jan 12, 2013 3:07 AM
Author: mueckenh@rz.fh-augsburg.de
Subject: Matheology § 190
Matheology § 190

The Binary Tree can be constructed by aleph_0 finite paths.

0

1, 2

3, 4, 5, 6

7, ...

But wait! The Binary Tree has aleph_0 levels. At each level the number

of nodes doubles. We start with the (empty) finite path at level 0 and

get 2^(n+1) - 1 finite paths within the first n levels. The number of

all levels of the Binary Tree is called aleph_0. That results in

2^(aleph_0 + 1) - 1 = 2^aleph_0 finite paths.

The bijection of paths that end at the same node proves 2^aleph_0 =

aleph_0.

This is the same procedure with the terminating binary representations

of the rational numbers of the unit interval. Each terminating binary

representation q = 0,abc...z is an element out of 2^(aleph_0 + 1) - 1

= 2^aleph_0.

Or remember the proof of divergence of the harmonic series by Nicole

d'Oresme. He constructed aleph_0 sums (1/2) + (1/3 + 1/4) + (1/5 + ...

+ 1/8) + ... requiring 2^(aleph_0 +1) - 1 = 2^aleph_0 natural numbers.

If there were less than 2^aleph_0 natural numbers (or if 2^aleph_0 was

larger than aleph_0) the harmonic series could not diverge and

mathematics would deliver wrong results.

Beware of the set-theoretic interpretation which tries to contradict

these simple facts by erroneously asserting aleph_0 =/= 2^aleph_0.

Regards, WM