Re: An uncountable countable set


Virgil schrieb:

0.111.. is not in the list, then it must have more digits than can be
indexed (and hence, can exist).

They are satisfactorily indexed by the infinite set of finite natural
numbers, N.

You just proved that there are infinitely digit positions which are not
indexed by natural numbers (*all* of which are given in the list).

Having infinitely many does not require that any one of them be
infinitely large.

And each of the infinitely many naturals is only finitely large.

Either the diagonal number 0.111... is not distinguished from all
finitely large numbers of the list
0.
0.1
0.11
0.111
....
then Cantor's proof fails.

Or 0.111... is distinguished from all finitely large numbers of the
list
0.1
0.11
0.111
....
then the digits of 0.111... cannot all be indexed by natural numbers.

Either or! That means there is no actual infinity - neither nor.

Regards, WM

.

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