Re: Cantor Confusion


Virgil wrote:
In article <1167167820.156449.44540@xxxxxxxxxxxxxxxxxxxxxxxxxxx>,
"Newberry" <newberry@xxxxxxxxxx> wrote:

David Marcus wrote:
Newberry wrote:
stephen@xxxxxxxxxx wrote:
Newberry <newberry@xxxxxxxxxx> wrote:
David Marcus wrote:
Newberry wrote:
It certainly makes it highly couterintutive that there are more
paths
then edges although I do not know if it generates a flat
contradiction.

Yes, it is counterintuitive (depending on your intuition). No, there
is
no contradiction.

Just because a system avoids a contradiction of the type P & ~P does
not mean that it is justified. For example an omega-inconsistent
system
may not produce any P & ~P and would be still unacceptable. Similarly
a
system in which we can prove that

#edges = 2 * #paths

and at the same time that the cardinality of paths is greater than
the
number of edges is unacceptable.

In what system can you prove that #edges = 2 * #paths?

In ZFC. Theorem: In an infinite binary tree there are twice as many
edges as there are paths.
Proof: At the level n the ratio of eges over paths is expressed by
2*2^n - 2)/2^n
For the entire tree we calculated the limit for {n-->oo}
lim{n-->oo} (2*2^n - 2)/2^n = 2. QED

That's fine up to where you write "QED". Why does the limit you
calculated tell us anything about the cardinality of the edges and paths
in the infinite tree?

I was only proving that the ratio of edges to paths in an infinite
binary tree is 2.

But you have proved no such thing.

The number 2 has been claculated for the entire
infinite tree.

Your claim is based on the assumption that

lim n -> oo (f(n)/g(n) = (lim n -> oo f(n))/(lim n -> oo g(n))

It is not.


but that is only valid when both limits on the right hand side exist,
meaning that they are finite.

If you want to assume it where the limits on the right side do not
exist, you will have to prove it in some formal system first.

.

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