Define a blackboard that holds all significant facts.

On the blackboard will be several numbers, most likely adding to a large sum.

A significant fact is the total of the numbers on the blackboard.

This number is 'well defined' to coin a sci.math phrase, but its impossible to put it
on the blackboard!

Is there always some numbers missing from the blackboard? Some integer that the board
cannot represent? Could the board hold all numbers? Maybe it could, we have NOT
proven that the board is incapable of storing a complete set of numbers just because
'the sum of those numbers' is 'well defined' but missing from the board.

------------------------------------------

Same drill with omega, all sequences are on a computable list, but not omega.

The reason is because the list is made by computer programs, ALL OF THEM.

Input 0123456789....
________________________
TM1 011101010101101000..
TM2 101000101011010100..
TM3 000000001111010101..
...

In this model, each TM only outputs 0 or 1, for some unary input.

omega = "the sequence that represents Halt(TMn, n)"

omega looks "well defined" just like the sum_of_all_numbers_on_the_blackboard was "well defined".

But just like the sum was significant and had to be written on the board,
omega is a program itself, and has to be on some row of the model it references!

omega = "for some x, TMx's output being the sequence that represents Halt(TMn, n)"

If omega is well defined, then so is antiomega.

antiomega = "for some y, TMy's output being the sequence that represents !Halt(TMn,n)"

when antiomega parses its own TMnumber as a parameter, it fails.
antiomega(y) = "at y,y TMy(y)'s output being !Halt(TMy,y)"

If TMy(y) halts, then the value loops, and vice versa, its a contradiction in the formulation.

Since antiomega is an impossible formulation, omega is impossible!

Therefore "for some x, TMx's output being the sequence that represents Halt(TMn, n)"
is not a valid definition of a sequence.

BUT THE SEQUENCE OF HALT VALUES *IS* WELL DEFINED you all say!

No it's not. "The sequence of halt values of all programs except the program generating that sequence"
may be well defined, but Omega is only defined from a groundless platonic perspective, and as such
it doesn't refute the possibility of all sequences being countable.

Herc
--
"If all the girls who attended the Yale prom were laid end to end, I
wouldn't be a bit surprised." - Dorothy Parker

From:	examachine at gmail.com
Subject:	Re: Why we cannot compute omega
Date:	23 Jan 2005 17:49:25 -0800

I think you don't know what the halting probability is.

--
Eray

From:	\|-\|erc
Subject:	Re: Why we cannot compute omega
Date:	Mon, 24 Jan 2005 12:35:49 +1000

wrote in message
> I think you don't know what the halting probability is.
>
> --
> Eray

What *is* the halting probability?

Herc

From:	The Ghost In The Machine
Subject:	Re: Why we cannot compute omega
Date:	Mon, 24 Jan 2005 04:00:18 GMT

In sci.logic, |-|erc

wrote
on Mon, 24 Jan 2005 12:35:49 +1000
<35j58cF4ju9snU1@individual.net>:
> wrote in message
>> I think you don't know what the halting probability is.
>>
>> --
>> Eray
>
> What *is* the halting probability?
>
> Herc
>

It's the probability that a given machine will halt, of course.
Since there's a 1-1 and onto mapping from N to the set of machines
(if a number doesn't correspond to a machine, close the ranks)
there are some minor issues here, but the problems are the same
as with Chaitin's Omega; it is not a computable number.

However, it should be a reasonably well-defined one.

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

From:	examachine at gmail.com
Subject:	Re: Why we cannot compute omega
Date:	23 Jan 2005 20:00:19 -0800

Omega.

--
Eray

From:	\|-\|erc
Subject:	Re: Why we cannot compute omega
Date:	Mon, 24 Jan 2005 14:10:38 +1000

wrote in ...
> Omega.
>
> --
> Eray
>

Is the halting probability different to the defn of Omega I gave?

Herc

From:	rupertmccallum at yahoo.com
Subject:	Re: Why we cannot compute omega
Date:	23 Jan 2005 19:55:56 -0800

You can prove in ZFC that omega exists.
Which formal system would you prefer to use?

From:	\|-\|erc
Subject:	Re: Why we cannot compute omega
Date:	Mon, 24 Jan 2005 13:59:16 +1000

wrote in message
> You can prove in ZFC that omega exists.
> Which formal system would you prefer to use?
>

Go ahead!

Herc

From:	rupertmccallum at yahoo.com
Subject:	Re: Why we cannot compute omega
Date:	23 Jan 2005 20:11:35 -0800

|-|erc wrote:
> wrote in message
> > You can prove in ZFC that omega exists.
> > Which formal system would you prefer to use?
> >
>
> Go ahead!
>
> Herc

Right, well, a real number is defined to be a set of rationals,
nonempty, bounded above, with no maximal elements, and such that if x
is in the set then every rational y less than x is in the set. We need
to prove the set corresponding to omega exists. Well, we can prove Q
(the set of rationals) exists and then by the power set axiom and the
axiom of separation we can prove that any set of the form {x in
Q|phi(x)}, where phi is a formula in the first-order language of set
theory, exists. And it's easy to show that the set corresponding to
omega is of this form. That's an outline of how the proof would go.

From:	\|-\|erc
Subject:	Re: Why we cannot compute omega
Date:	Mon, 24 Jan 2005 14:26:19 +1000

NOW he uses English!!

Herc

wrote in >
> |-|erc wrote:
> > wrote in message
> > > You can prove in ZFC that omega exists.
> > > Which formal system would you prefer to use?
> > >
> >
> > Go ahead!
> >
> > Herc
>
> Right, well, a real number is defined to be a set of rationals,
> nonempty, bounded above, with no maximal elements, and such that if x
> is in the set then every rational y less than x is in the set. We need
> to prove the set corresponding to omega exists. Well, we can prove Q
> (the set of rationals) exists and then by the power set axiom and the
> axiom of separation we can prove that any set of the form {x in
> Q|phi(x)}, where phi is a formula in the first-order language of set
> theory, exists. And it's easy to show that the set corresponding to
> omega is of this form. That's an outline of how the proof would go.
>