Re: Cantor Confusion

In article <1162562518.437541.264190@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
mueckenh@xxxxxxxxxxxxxxxxx wrote:

Virgil schrieb:


If real numbers are to be represented by sequences then any two
sequences which are "arbitrarily close" represent the same number.

Therefore Cantor's proof is invalid.

Non-sequitur. "arbitrarily close" means that the distance between them
is smaller that any positive real number, which makes them equal, even
in Robinson's non-standard analysis.


With increasing length of the
list, the difference introduced by exchanging the diagonal becomes
smaller and smaller. For an infinite list it vanishes at all.

For which term of the sequence has it vanished entirely?


In Cantor's list there are those unique representations required.

Not so. Even in decimal, Cantor's diagonal rule allows for certain
rationals having dual representation.

Which two numbers could that be?


WM misses the point, as usual.
It is not two numbers at all, but one number with two representations.

Like 1.000... and 0.999... in decimal arithmetic.

And this will be the case for every rational q in a base b
representation for which there exists any n in N with q*b^n in N.
Which is a set of rationals dense in the set of all reals.


Therefore I do not understand why you say "Numbers are fixed entities".
They are merely defined by sequences.

The sequence 1 + 1/2 + 1/4 + 1/8 + ...+ 1/2^n + ... "defines" a fixed
number. That that number has other representations does not make that
number into a variable quantity.

The sequence 1 - 1/3 + 1/5 - 1/7 +-... does not define a fixed number.

What sort of motion does WM ascribe to that number?
.

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