Re: A consideration concerning the diagonal argument of G. Cantor

Tim Little wrote:

On 2008-05-16, Julio Di Egidio <julio@xxxxxxxxxxxxx>
wrote:
So, let's restate it:

Within N* (that is, N U {oo}):
n = oo => [1/n, 1] = [0, 1]

With a common definitions of operations in N*, yes
1/oo = 0 and so
[1/n, 1] = [0, 1] when n = oo.


Thanks, that's something!




Within N:
n -> oo => [1/n, 1] = (0, 1] -> [0, 1]

Your choice of notation is strange.

Is the first clause "n -> oo" supposed to be a limit?
Of what?

Is the "=>" intended to represent logical
implication? It is
certainly not true that [1/n, 1] = (0,1].


I think: possibly not!

Notation:

'=' for "equivalence"
'=>' for "implication"
'->' for "limit" -- by finite induction

Then, over N*:

n = oo => [1/n, 1] = [0, 1]

Then, over N(*):

n -> oo => [1/n, 1] = (0, 1]

The syntax of the open interval is saying: the lower bound tends to 0 as some unknown p(n); although, that p(n) is explicit within the LHS of the equivalence.


I hope it is at least al little bit clearer.

In the meantime, I am in a struggle to transfinite induction...


-LV



Regarding the second arrow (which also appears to be
used as a limit),
whether [0,1] is the limit or (0,1], or even whether
a limit exists,
depends upon what sort of limit you are using - which
you haven't
defined either.


- Tim
.