In article <email@example.com>, WM <firstname.lastname@example.org> wrote:
> On 1 Mai, 06:47, Dan <dan.ms.ch...@gmail.com> wrote: > > > Yes, but the list can be given by a formula only. I did not exclude > > > that. I said: A real number can be listed by a terminating decimal, by > > > a periodic > > > decimal, or by a formula supplying each of its decimals. > > > It is not possible to list irrationals *by writing their decimals*. > > > > I never claimed it was possible, nor was it necessary. > > It is necessary for Cantor's proof. Unless every digit exists on the > diagonal, the diagonal number is undefined in an undefined list. But > that is Cantor's claim: forall n : a_nn =/= d_n.
But, in order to get every digit of the diagonal, no real need be expanded beyond finitely many places. > > As I have shown
What WM claims to have shown has usually not been shown to the satisfaction of anyone other than WM himself, and even if shown adequately would not imply what WM claims it implies. > > > > A formula > > still determines each decimal in a unique way, just as writing the > > decimals would. > > Yes, but there are only countably many formulas. > > > > > A formula is not writing the decimals. > > > > A formula allows me do get whichever decimals I need for what I'm > > doing > > and after each of them there are infinitely many further decimals.
For the diagonal argument to work, it is only necessary to be able to expand the nth decimal in a list of reals to its nth decimal place.
Since this is theoretically possible, even though practically very difficult, Cantor's argument remains valid.
> And > in a ratinals-complete Cantor-list there are infinitely many lines > showing exatly that entry that you have excluded from the first n > lines of the list.
Not outside of Wolkenmuekenheim, > > > > , and deduce properties about them in general . > > A formula is partly like a vending machine (with infinite storage) , > > and the n'th decimal is like the n'th type of soda. > > When I order the n'th decimal , I don't expect the vending machine to > > spew out all the sodas like crazy (write down all digits) . > > I just want my n'th soda. Nothing more , nothing less. When I need the > > m'th soda , I'll order the m'th soda . > > And I will show you the same as for the n-th. There is no chance to > come to an end.
But there is proof that an anti-diagonal different from every listed real exists.
> > Mathematics is not possible with heat.
Then cool off, and learn how to so it. > > > Let's reiterate how a proof by contradiction works : > > No. The claim that all rationals can be enumerated is already nonsense > (since for every enumerated q_n there are infinitely many not > enumerated - you can enumerate every rational but not all). The claim > that all reals cannot be enumerated is correct (infinity is simply > infinite), but interpreted on the basis of the first claim it is > nonsense too.
When one has ennumerated every rational, unless that somehow also has left some other rationals, one has ennumerated all of them.
At least everywhere outside of Wolkenmuekenheim. > > To require an enumeration of the reals in order to contradict Cantor, > is a highly successful trick
And one that someone as incompetent at mathematics as WM could not refute even if it was wrong. > > You demand a contradiction of astrology by an astrological argument? I > conradict nonsense by showing that it is nonsense.
While we agree that WM produces a lot of nonsense, he has shown himself to be incompetent at creating proofs of any but the simplest of things,
Astrology is sane by comparison to WMytheology. --