Re: ** says: Definition: sum{i in N} i = 0

In article <1183884971.653384.22820@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
WM <mueckenh@xxxxxxxxxxxxxxxxx> wrote:

On 7 Jul., 23:31, Virgil <vir...@xxxxxxxxxxx> wrote:

The question is only this: Can removal of the removable discontinuity
yield another result than
lim_[x-->0] sinx/x = 1?

The question is, what is your justification for attempting to remove the
discontinuity in the first place?

To find the one and only true value.

What guarantees that there is a "one and only true value"?

Without an axiom system to rely on, one cannot prove that any such thing
exists. And in any of the axiom systems I am aware of, it doesn't.

For sin(x)/x the answer would be so as to have an everywhere continuous
and differentiable function.

But for your 'P(x)/K(x)', there is no justification at all, since there
is a direct way to find what the value should be "at oo".

I go the most direct way to find the value it has "at oo" - not the
value it should have.


Without many unjustified and unstated assumptions on WM's part, there is
no such thing as a 'value it has "at oo"'.

On the other hand, given any of the standard set theory axiom system, in
which all assumptions are explicitly stated, one can prove that the
ratio of number of paths to number of levels in an infinite binary tree
is infinite.



On the other hand, the continuity of the paths guarantees that the
result is the only one possible, in mathematics.

That may be what transpires in in WM's MathUnRealism, but it does not in
actual mathematics, since by direct analysis, one can show that "at oo"
the ratio of paths to nodes must be infinite:

That argument is analog to: Contra has shown that sin0/0 = 100.
Therefore l'Hospital is wrong in this case.

Only in WM's MathUnRealism does any such analog exist.
In order to argue that sin(0)/0, which simplifies to 0/0, MUST BE equal
to 1, WM must also argue that 0/0 MUST ALWAYS BE equal to 1.

Then according to WM, every derivative must always equal 1 at all
points, and f(x) = |x| must have a derivative at x = 0, and lots more.

None of which occurs outside WM's MAthUnRealism.
.