|
|
Dave L. Renfro schrieb:
> mueckenh@xxxxxxxxxxxxxxxxx wrote:
>
> > But the set of all lists is countable (as is any quantized
> > or discontinuous set), so is the set of all list entries.
> > Nevertheless, there is no real number missing in every
> > list? So every real number is in at least one of the
> > list? So every real number is one element of a countable
> > set of entries? And there is nothing real really outside
> > of this countable set?
>
> The set of all lists of _what_? Also, "lists" is a term
> that for some people means one thing, and for other people
> it can mean something else.
For me, a list is the physical representation of an injective sequence.
> If you're going to argue about
> details, rather than give a general overview of what's
> going on, you should use the correct mathematical terms,
> such as one-to-one correspondences (i.e. bijective functions).
> Otherwise, the discussion is going to degenerate into
> disagreements over word usage. In fact, you're doing it
> yourself by bringing up "quantized" and "discontinuous",
> two words that, to most people in math, have no direct
> relevance to the discussion at hand.
A list as a physical representation requires some space and is a
quantized object with a finite volume dV. It can be shown that
everything which is not a spaceless entity like a point or a line or an
area belongs to a countable set.
>
> It never ceases to amaze me that people can have so much
> trouble with the main gist (not the details) of the
> argument.
But there are opposite conclusions which can be derived by mathematical
deduction. One of them is the theorem that there is a countable set
which contains more elements than its power set.
Consider the binary tree which has (no finite paths but only) infinite
paths representing the real numbers between 0 and 1. The edges (like a,
b, and c below) connect the nodes, i.e., the binary digits. The set of
edges is countable, because we can enumerate them
0.
/a\
0 1
/b\c /\
0 1 0 1
.............
Now we set up a relation between paths and edges. Relate edge a to all
paths which begin with 0.0. Relate edge b to all paths which begin with
0.00 and relate edge c to all paths which begin with 0.01. Half of edge
a is inherited by all paths which begin with 0.00, the other half of
edge a is inherited by all paths which begin with 0.01. Continuing in
this manner in infinity, we see that every single infinite path is
related to 1 + 1/2 + 1/ 4 + ... = 2 edges, which are not related to any
other path. The set of paths is uncountable, but as we have seen, it
contains less elements than the set of edges. Cantor's diagonal
argument does not apply in this case, because the tree contains all
representations of real numbers of [0, 1], some of them even twice,
like 1.000... and 0.111... . Therefore we have a contradiction:
Card(R) >> Card(N)
|| ||
Card(paths) =< Card(edges)
I find it puzzling that people have so much trouble with the main gist
(not the details) of the
argument.
Regards, WM
|
|