Re: The consise Cantor's proof ?

From: The Ghost In The Machine (ewill_at_sirius.athghost7038suus.net)
Date: 12/20/04 

Date: Mon, 20 Dec 2004 16:01:24 GMT

In sci.logic, artist formally known as |-|erc
<h@r.c>
 wrote
on Mon, 20 Dec 2004 20:25:40 +1000
<32nnlsF3pc46cU1@individual.net>:
> "The Ghost In The Machine" <ewill@sirius.athghost7038suus.net> wrote in
>> >> >
>> >> > I've posted this 100 times, STOP using T10 as an argument already.
>> >> > 20 times you've brought it up and its nothing to do with the post.
>> >> > The diag of T10 has a CHARACTERISTIC EXPANSION that
>> >> > distinguishes it from the members of the list, regardless if all digit
>> >> > sequences are present. That argument simple does no apply to any
>> >> > of the examples I've been in the last 6 months.
>> >>
>> >> Correct, it does not. TX_10 contains all reals, no bones about it,
>> >> in any practical sense of "contains".
>> >>
>> >> Not that mathematicians are all that practical. :-)
>> >>
>> >
>> > I believe my argument was (6 months ago) it contains all possible
>> > sequences of digits, hence changing digits is not a valid step.
>>
>> So does TX_10.
>
> That's what I'm talking about here! 6 months ago I was using T10 as
> an example right?

Yes. However, TX_10 doesn't contain all real numbers; it doesn't
even contain 1/3.

"Oh, but 1/3 = {.3, .33, .333, .3333, ...}! It's clearly just an
omission of some sort."

Or some such.

>
>
>
>>
>> >
>> >
>> >> >> If you have another list in mind, fine; produce it.
>> >> >>
>> >> >
>> >> > R(x) = UTM(x,y) mod 10
>> >
>> > I take it you forfeit from producing a new real then?
>> > Do you understand the difference between UTM and T10 here?
>> > Sounds benign but for some reason every time I suggest a list
>> > of reals you refute it with T10.
>> > Infinite real numbers computable by a UTM have infinite expansions.
>>
>> Not every UTM creates an infinite real. Consider this UTM, for instance:
>>
>> state 1:
>> on input ' ': write '0' shift right new state 2
>> on input '0': write '1' shift right new state 2
>> on input '1': write '0' shift right new state 2
>>
>> state 2:
>> on input ' ': write '1' shift left new state 1
>> on input '1': write '0' shift left new state 1
>> on input '0': write '1' shift left new state 1
>>
>> This UTM has two problems:
>>
>> [1] it will never halt.
>> [2] it will never write a consistent number.
>
> Some computer programs crash, but nobody's recalled all PCs have they?

It doesn't matter. I can (and do, below) diagonalize the problem.

>
>
>
>
>>
>> As for a non-computable real, I have one (it's not all that original).
>> Consider the real number X represented by [d0] . [d1] [d2] [d3] ...
>> where dx is '1' if UTM(x,x) halts, and '0' if UTM(x,x) does not halt.
>>
>> (I'm assuming UTM(x,y) means a universal turing machine whose state
>> is encoded somehow in number x (the machines stored in some sort
>> of order), and whose input tape is y. The particulars aren't that
>> important.)
>
> Right, UTM is just a particular TM or program that has 2 inputs.
> 1 - the TM it emulates
> 2 - the parameter it feeds to that TM.
>
> In this case the parameter is the digit position you want calculated.
> If the TM halts, you take mod 10 and that's the digit.
>
> The only particulars are encoding 2 parameters on the one tape.
>
>
>>
>> X cannot be computed by a UTM, as the halting problem is undecidable.
>
> Its not well defined then is it?

Isn't it?

"Let h be a TM that takes two inputs: an encoding of the machine, and
an encoding of the tape fed into the machine. h will write 1 in the
first cell if the run terminates, a 0 if the run will never terminate,
after a finite time.

Let j be another TM that is a modification of h; j will write 1 in the
x'th cell if h on input x,x writes a 1, 0 if h on input x,x writes a 0."

How's that?

> I'm thinking of a number between 1 and 10 but I'm not going to tell you,
> I'll call it Omega1to10. Is that noncomputable?
>
> Say you line up all the stars from Sol, Alpha Centurai, ...
> Then if that star explodes, that digit place has a 1, else a 0.
> Since Sol and Alpha Centurai are big stars and should explode,
> the number most likely starts out as 0.11_ _ _ _ _ _..

Stellar cosmology doesn't seem to be your strong suit either. :-P

>
> You DON'T KNOW what the values are, when the star 'halts' you can
> fill in a 1, if the star fizzles to a vapour you can put a 0.
> That doesn't define a real number. The futures not ours to see.
>
>
>>
>> There's one real.
>>
>> A second real might be the comparatively simple one
>>
>> .0123456789101112131415161718192021...
>>
>> which is occasionally referred to as Champernowne's Constant.
>> I would be mildly surprised if any TM can generate this constant,
>> despite its very simple specification, if the alphabet is limited
>> to [0-9] and blank. (If the alphabet is extended the machine
>> might use a "marker" digit to indicate which digit it is currently
>> processing as it generates the next part of the constant.
>> Interspersing blanks in the number -- .0 1 2 3 4 5 6 7 8 9 10 11 ... --
>> is considered cheating. :-) )
>>
>
> Since you can specify the digit required as input to the tape it
> should be trivial. Probably found on some UTM(c, digit) where c < 10^20
>
> The example you were supposed to give was the diagonal of UTM(x,y)mod10

I did that one already, using the halting problem. Did you have
some other in mind?

>
> Herc
> 

-- 
#191, ewill3@earthlink.net
It's still legal to go .sigless.

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