On 4 Mai, 18:30, Dan <dan.ms.ch...@gmail.com> wrote: > > > Either they're both relevant to Cantor's argument, or they're both > > > irrelevant . > > > Cantor's argument is a single-eyed look into the infinite. > > Forall n : d_n =/= a_nn is considered important. > > Forall n : (d_n) is in the list, is not considered important. > > Why should it be considered important under the definition of > equality? If they have at least one different digit , THEY'RE > DIFFERENT.
But there always only finitely many with at least one different digit whereas there are always infinitely many with none different digit.
> Every finite digit appears in the list at least once , but NEVER DO > ALL OF THEM APPEAR AT ONCE , IN THE SAME NUMBER.
Stop shouting. In order to support your "never", you should be able to prove it. But you are not. You can look at a finite domain only, since every line n is at a finite place and is followed by infinitely many more lines.
> > THIS IS REQUIRED TO DISPROVE CANTOR : > exists n , forall m , a_nm = d_m
Of course, in a list that contains all rationals, this is easily proved. A simpler case is the list:
0.0 0.1 0.11 0.111 ...
with the substitution 0 --> 1.
For all a_nn at finite places n (and others cannot be substituted at all) we have all (d_1, ..., d_n) as an entry in the list.
Your assertion that in your anti-diagonal there are more than all digits that are already covered by the list is simply nonsense.
And the proposal of ZeitGeist, that all digits 1 at possible finite indices are in the list but distributed over several lines and not in a single line, can be shouted loudly in a mad house, probably even there raising objections, but not in mathematics..
My list is constructed such that all possible squences of digits 1 are already in the list. No longer sequence is possible (otherwise it would be in the list) and a shorter sequence cannot support Cantor's claim.
Simple as that.
> And would you stop it with the countable language already?
A language is man-made. Men cannot make uncountable language. And they could not convey information by uncountable language.
So if there was an "uncountable language", it was not a language (by the way like finished infinity is not unfinished and therefore is no infinity).
> I find again ,sadly , that in this one circumstance , what cannot be > said must be passed down in silence.
In particular an uncountable language would force you to silence because it is no language. Similarly: An "infinitely complicated law" means no law at all. [§ 125]
If you are interested in some more quotes by Wittgenstein, here you are:
It isn't just impossible "for us men" to run through the natural numbers one by one; it's impossible, it means nothing. [?] you can?t talk about all numbers, because there's no such thing as all numbers. [§ 124]
There's no such thing as "all numbers" simply because there are infinitely many. [§ 126]
Generality in mathematics is a direction, an arrow pointing along the series generated by an operation. And you can even say that the arrow points to infinity; but does that mean that there is something - infinity - at which it points, as at a thing? Construed in that way, it must of course lead to endless nonsense. [§ 142]
If I were to say "If we were acquainted with an infinite extension, then it would be all right to talk of an actual infinite", that would really be like saying, "If there were a sense of abracadabra then it would be all right to talk about abracadabraic sense perception". [§ 144]
Set theory is wrong because it apparently presupposes a symbolism which doesn't exist instead of one that does exist (is alone possible). It builds on a fictitious symbolism, therefore on nonsense. [§ 174]