Re: Two results of set geometry

On Aug 28, 4:10 am, WM <mueck...@xxxxxxxxxxxxxxxxx> wrote:
Two results of set geometry

Set geometry, a new branch of mathematics, is devised to investigate
relations of finite and infinite sets by means of geometrical
representations. Up to now there have been two important results.

The first result is that the ordinal number of the set of natural
numbers is not unique. It has been shown that if {1, 2, 3, ...} =
omega, then {1, 2, 3, ...} = omega + 1 too.

I believe the confusion here is that by themselves, _sets_ do not
have "ordinal numbers" at all. One has to define an _order_ on
the set, and if the order happens to be a wellorder, then it is
isomorphic to an ordinal. For the set N, if the order is <, then
it is isomorphic to omega, and if the order happens to be <<,
defined as:

m << n iff (1 < m < n or 1 = n < m)

then it is order-isomorphic to omega + 1. Indeed, in ZFC or any
standard set theory, N is isomorphic to any countable ordinal,
depending on the wellorder.

The second result shows that the set of real numbers is countable.

WM provided a link to the old threads about the binary tree. I
already pointed out that WM desires to have the properties of
finite trees transfer to infinite trees.

Both Robinson's hyperreals and Conway's surreals have been
mentioned several times in these Cantor threads. In this case
I believe that the surreals are more relevant, because they
are often represented by a binary tree:

http://www.valdostamuseum.org/hamsmith/surreal.html

As you can see in the link above, the binary tree of surreals
has every path, even the infinite ones, ending in a leaf node,
with such surreals as omega, pi, e, sqrt(2), 2/3, iota, -iota,
and -omega being generated after infinitely many steps -- what
Conway calls "birthday omega." Thus every path, whether finite
or infinite, has a leaf node, and thus there cannot be more
paths than nodes in the binary tree of surreals.

Of course, the surreal binary tree still doesn't have every
property that WM wants. In this case, there are at least as
many nodes as standard reals, and so both the set of nodes
and the set of paths are uncountable.

Also, ironically, once one allows surreals or hyperreals, WM's
"first result" no longer holds. When one considers the set of
positive omnific integers (or hyperintegers) rather than N,
several things happen. First of all, < is no longer a well
ordering, because omega - 1, omega - 2, omega - 3, etc. (or
the hyperreal equivalent) is an infinitely descending set of
surreals (respectively hyperreals). But if we consider the
set of positive hyperintegers less than a given infinite
integer W, then < and << (defined above) are order isomorphic
because in {1, 2, 3, ..., W - 3, W - 2, W - 1} maps to
{2, 3, 4, ..., W - 2, W - 1, 1} by mapping each hypernatural
less than W - 1 to its successor and W - 1 to 1. The surreals
are more problematic because they form a proper class and not
a set, but something similar happens here.

.

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