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Topic: Matheology � 258
Replies: 104   Last Post: May 5, 2013 2:26 PM

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 dan.ms.chaos@gmail.com Posts: 409 Registered: 3/1/08
Re: Matheology § 258
Posted: May 1, 2013 12:47 AM

> 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. A formula
still determines each decimal in a unique way, just as writing the
decimals would.

> A formula is not writing the decimals.

A formula allows me do get whichever decimals I need for what I'm
doing , 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 .

> > 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.

realm' to predict that the universe as a whole must be infinite.

> Entropy is increased elsewhere.
>

> > Information is never truly 'erased' .
>
> Switch off your pocket calculator.
>
> Regards, WM

A highly naive answer, coming from a 'physicist'.
'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' ) . It's never
erased .
http://en.wikipedia.org/wiki/Landauer%27s_principle

Let's reiterate how a proof by contradiction works :

Proving forall y , (non (P(y)) :
You claim to have x such that P(x) holds .
I take x , and generate a counterexample to your claim (thus proving
it false) .

Then, you claim to have z such that P(z) holds .
I take z , and generate a counterexample to your claim (thus proving
it false) .

....................................
If my counterexample-generating procedure is universal (I can refute
all your possible claims ) , then I win , but the
example-refutation game can go on forever . forall y , (non (P(y))
is proved .

On the other hand , if you claim , P(k) holds for some exemplified k ,
and I cannot generate a counterexample , then you win .
forall y , (non (P(y)) is disproved .
exist k , P(k) is proved .

Now then, let's see how the Cantor game works :
You claim that the reals (between 0 and 1 for example) are
denumerable .
Therefore , you must present a list L_1 (or a formula for a list )
than contains all the reals . "exists L_1 , contains_all_reals( L_1)"

But, I take your list L_1 , and use the antidiagonal to construct a
real number R_1 , that is not in the list .

You can then build a better list , L_2 , that contains all the members
if L_1 , in addition to R_1 , the counterexample for L_1 .

But, again, I take your list L_2 , and use the antidiagonal to
construct a real number R_2 , that is neither in L_2, nor in L_1 .

And the game goes on forever .

This means I win . There is no list (or formula for a list) that
contains all real numbers , therefore , I dub the real numbers 'un-
listable' .

Date Subject Author
4/29/13 Virgil
4/29/13 mueckenh@rz.fh-augsburg.de
4/29/13 Virgil
4/30/13 dan.ms.chaos@gmail.com
4/30/13 mueckenh@rz.fh-augsburg.de
4/30/13 dan.ms.chaos@gmail.com
4/30/13 mueckenh@rz.fh-augsburg.de
4/30/13 Virgil
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5/1/13 Virgil
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