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.
As I have shown, forall n : there are infinitely many lines contining d_1, ..., d_n. The counter argument is, that the supremum d is not a line. But this supremum is not defined in the list either, because the argument concerns only digits d_n at finite positions. After that everything is possible. No d defined.
> 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. 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.
, 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. > > > > The most accepted cosmological model predicts the universe is > > > spatially infinite and roughly homogenous in large scales. > > > The realm we can ever use is finite. > > Even if that were true,
It is true, since light velocity cannot be exceeded without time reversal. If time reversal was possible, we would have noticed it.
> scientists use observations about this 'finite > realm' to predict that the universe as a whole must be infinite.
That has not been confirmed. Opinions are shaky. But it is completely irrelevant for my argument. > > > Entropy is increased elsewhere. > > > > Information is never truly 'erased' . > > > Switch off your pocket calculator.
> A highly naive answer, coming from a 'physicist'.
Simple answers are best. Better than "inaccessible cardinals".
> 'Erased information' is simply transferred in the environment, in a > form such that it's recuperation would be impractical to us (it no > longer appears as information to us , but as 'heat' ) .
Mathematics is not possible with heat.
> 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.
To require an enumeration of the reals in order to contradict Cantor, is a highly successful trick, obfuscating many intelligent minds, but nevertheless not better than shit.
You demand a contradiction of astrology by an astrological argument? I conradict nonsense by showing that it is nonsense.