On May 1, 11:18 pm, WM <mueck...@rz.fh-augsburg.de> wrote: > On 1 Mai, 18:27, Dan <dan.ms.ch...@gmail.com> wrote: > > > They have similar formulas,but behave in different ways . > > You would be correct in affirming that b_inf = 1/9 is not part of the > > list . > > Of course. It is not part of the list, because it cannot be written as > decimal number. Proof: All decimal numbers that can be written in this > form *are already in the list*. And there is no reason why 1/9 should > be missing, if it could be a decimal fraction. > > > If it were part of the list , then a_inf would be a well-behaved > > natural number , but it isn't . > > It is neither well behaved nor a natural number. The sequence 0.111... > does not exist other than by a finite name or formula. > > > Limits only work properly with real numbers, not natural numbers . > > Yes, that is true. But (and please read this very attentively!): > Cantor's argument requires the existence of the complete sequence > 0.111.... in digits: > > You can see this easily here: > > The list > > 0.0 > 0.1 > 0.11 > 0.111 > ... > > when replacing 0 by 1 has an anti-diagonal, the FIS of which are > always in the next line. So the anti-diagonal is not different from > all lines, unless it has an infinite sequence of 1's. But, as we just > saw, this is impossible. > > Regards, WM
I see no significant difference between referring to a mathematical object by a formula and referring to it by 'writing it down' . Writing it down , when possible , is just another formula for it. "1296" , and "36 * 36" , are both references to the same object , and neither of them is "more true of a name" for the object than the other .
Just like "1296" is "36 * 36" , and we can substitute one for another in mathematical expressions , and maintain their truth , so is "1/9" the same as "0. the infinite sequence of 1's" . That we cannot truly write down the name of the second expression is of no relevance. The most important point is that 'names for the same object , whether finite or infinite, are fully interchangeable'
Since "0. the infinite sequence of 1's" is the same as "1/9" then
"The third digit of 0. the infinite sequence of 1's" is the same as "The third digit of 1/9" is the same as "1" .
Same object . Different names .Since we treat all possible names of the object the same , whether it be "a formula for the object" , or an "enumeration of its digits" , we never run into problems. And in every name for an object we can recover every other name, if we so wish .