Das ist von Tonico in sci.mathe.

Bisher keine Antwort des doch so redseligen Angesprochenen:

I'll take advantage of this brand new thread to make clear what I

think (not sure, just "think") is WM's problem with the infamous

sequence 0.1, 0.11, 0.111,...

WM writes down this seq. in the following, Cantorproof's-like, array:

0.1

0.11

0.111

.

He now continues: with a decimal point to the left, let us look at

what we get on the main diagonal:

0.1111, an infinite periodic decimal fraction whose actual value

is 1/9 , where by actual value we can, I hope, agree that we mean

either

1) the value of the converging infinite series 10^(-1) + 10^(-2)

+... , OR

2) the value of the converging sequence 10^(-n) when n --> oo

or some other equivalent thing.

WM continues: let us look now at the sequence formed by the

rightmost digit in every line of the above array, with (again) a

decimal point to the left:

0.1, 0.11, 0.111,... this is the very same sequence we got above,

obviously.

Now begins the tricky part: WM SEEMS to believe that the above

already

disproves Cantor's Diagonal Proof, because we can get on the main

diagonal the SAME number (namely, 1/9) as we get by taking the (limit,

I suppose, of the)

rightmost digits on the right...contradiction!!

If the above INDEED is what WM means (and I can't be sure until he

acknowledges and conffirms this), then his mistake should be clear to

all: first CDP does NOT choose a diagonal but constructs a number OUT

of the digits of the main diagonal, number which can be easily shown

to be different from any element on the given list.

In the above argument NO NEW number was constructed out of the

diagonal so nothing has been proved about CDP at all, let alone a

contradiction.

Second, in this case it is possible to choose the "rightmost" digit

because of the special nature of the list. In general we can't do any

such thing (this won't work with WM because he'll argue, I'm afraid,

that one single instance behaving differently from what the theorem

states is enough to prove it wrong, and he'd be right if it weren't

for my first point above).

Finally, there's another possibility for what WM believes: maybe he

believes that choosing a number in any way using the diagonal can

give

a number that is already contained in the list, and this of course

would be a huge contradiction to CDP.

But this is not the case, since CDP would give in the case of the

particular list that WM uses, the number 0.888., which definitely

isn't in one of the lines of the list nor on the original diagonal of

the list.

I get this number using the Cantorlike procedure to choose the number

from the diagonal (what some call antidiagonal, I believe), say: if

the n-th digit of the n-th number in the list is 1, I then choose 8,

otherwise I choose 0.

Thus, I've implemented CDP in WM's list and I've just proved we get a

brand new number not already contained neither as element of the list

nor as the list's diagonal.

Let's see if WM answers, approves or rejects (part or all of) what I

wrote, and if he makes some repairs to my piece above.

Tonio

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