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Topic: Matheology § 190
Replies: 15   Last Post: Jan 14, 2013 3:35 PM

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 mueckenh@rz.fh-augsburg.de Posts: 18,076 Registered: 1/29/05
Matheology § 190
Posted: Jan 12, 2013 3:07 AM

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