Re: An uncountable countable set


Virgil schrieb:

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.

OR, as is actually the case, the endless sequence of 1's fraction
0.111... is distinct from every finite truncation of it AND every digit
of it CAN be indexed by a natural number.

Or 10 is smaller than eleven and 10 is larger than eleven.

"To be different" means for all unary representations of n
An : 0.111... - n =/= 0
Every digit can be indexed by a natural number means that there is no
digit, which is different from all unary representations of natural
numbers. By construction of 0.111... this means further that every
digit at position n and all digits at smaller positions can be indexed.
This means there does not exist any digit which differs from all unary
representations.
not forall n: 0.111... - n =/= 0
En : 0.111... - n = 0.
Or is your logic unable to transform "no sequence of 0.111... differs
from all sequences 0.111...1" into "there is at least one sequence
0.111..1 which is equal to 0.111..."?

Regards, WM

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