Date: Jan 14, 2013 5:34 PM
Subject: Re: Matheology � 191
WM <email@example.com> wrote:
> On 14 Jan., 21:53, Virgil <vir...@ligriv.com> wrote:
> > > Here are all paths that I used:
> > > 0.
> > > 0 1
> > > 01 01
> > > ...
> > > and so on. Every node that you arrive at allows to continue left or
> > > right. And all that is countable.
> > The number of nodes involved is countable, but every countable set has
> > uncountably many subsets,
> but most subsets of the set of nodes do not describe paths.
It is easy enough to show that uncountably many of them do describe
paths. An alternate way of describing an infinite path is as an infinite
binary sequence of left o right branchings. It should be clear even to
someone as mathematically handicapped as WM that there is an obvious
bijection between the set of paths of a Complete Infinite Binary Tree
and the set of all infinite binary sequences of left o r right
And Cantor has shown that the st of all infinite binary sequences is
uncountable. For every different subset of nodes which is a path there
is an infinite binary sequence different from that of any other path and
every infinite binary sequence determines a path different from that of
any other infinite binary sequence.
> > I believe the proofs that no set can be surjected to its power set are
> > valid, and will continue to believe it until I see an EXPLICIT
> > surjection from some set to its power set.
> You could also require an explicit bijection of the set of unicorns to
> the naturals - otherwise you are convinced that there are uncountably
> many unicorns.
Since unicorns are not a necessary part of the world of mathematics, I
will leave all conjectures about them to WM.
> > > You
> > > will never gather more than countably many infinite paths.
> > Then we will never access all of the Complete Infinite Binary Tree, even
> > though it does exist.
> Where? Outside of WMytheology!