Re: THERE ARE oo DIGITS IN < 0 . 1 2 1 2 1.. >
From: |-|erc (H_at_r.c)
Date: 01/26/05
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Date: Wed, 26 Jan 2005 12:25:22 +1000
<bobbysim@gmail.com> wrote in
> |-|erc wrote:
> > --
> > In 100 years cardinality and incompleteness theory will only be
> offered under modern history as cult beliefs
> > Veni, vidi, vamos.
>
> I'll bite...
> Why's that Herc?
It's all invented and its all wrong!
"you can't prove me" does not make mathematics incomplete, its not a true statment with no proof.
HOW DO YOU KNOW ITS TRUE? with a system that doesn't know what its proving? wrong.
"this has no proof in any system" is a meta godel statment that by defn has no godel proof.
oo people each flip oo coins, can you make a new sequence?
the anti-diagonal has no maximum bound to the number of flips that have been tossed.
since it has been tossed to oo coins (given an oo list), its not a new sequence. one person
cannot outperform oo other people at the same task, simple resources problem.
<5% new material!>
****************************************************************
WHY WE CANNOT COMPUTE OMEGA
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!
Are 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)"
<n ranges from 1 onwards>
<Halt(TMn,n) is just Halt(n,n) where n represents TMn in some parser>
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)"
<n ranges from 1 onwards>
If omega is well defined, then so is antiomega.
antiomega = "for some y, TMy's output being the sequence that represents !Halt(TMn,n)"
<n ranges from 1 onwards>
<!Halt does the opposite of the Halt function, if its parameter halts it will loop indefinately, and vice versa>
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.
**********************************************************
True = [ proven | provable ]
Mathematicians don't need the word true.
For "I think its true" say "I think its provable".
For "G is true" say "G is proven"
That is a fact! That is true, that is provable.
I won't be a hypocrite here, on with the proof!
True is not a ball, a cat, a dog, an object you can look at.
True is an INTERPRETATION, a perspective.
Interpretation only occurs when a subject is interpreting the object.
Without the subject there is no TRUE. Our universe could
be completely devoid of intelligent life, planets might
still orbit stars but that would not be a true fact, because
there would no one to dispute it, and no language to represent it.
[edit, add the part people were supposed to deduce]
Hence, there is only TRUTH TO WHOM. Nothing is itself TRUE,
something can only be TRUE TO THE OBSERVER. If someone
states X is TRUE, then he must KNOW its true, its an abrivaition
for "I know X is true". TRUTH and KNOWLEDGE are the 2 ends
of the same piece of string.
***************************************
> >> >Truth is what's proven.
> >>
> >> Can you prove that?
> >
> >yes
> >truth
> >absolute truth
> >substantiated truth
> >derived truth
> >proven truth
> >proof
> >
> >which line is this sequence of equivalent statements is erronous?
>
> I don't accept any of those as equivalent. But if you claim that
> they are equivalent, and that truth is the same as provability,
> then you should be able to prove their equivalence.
Truth : absolute truth
{absolute truth} - {truth} = {}
{truth} - {absolute truth} = {partially true} = {}
absolute truth : substantiated truth
substantied_true -> absolute_true (truth has been justified}
absolute_true -> substantiate_true (what is known)
absolute_true <-> substantiated_true
substantiated truth : proven truth
sub·stan·ti·ate ( P ) Pronunciation Key (sb-stnsh-t)
To support with proof or evidence
proven truth : proof
THEREFORE Truth <=> Proof
*******************************************
Unfortunately, despite the illusion of rigor, mathematicians have
a longstanding habit of being gullible. Any clever lie to a mathematician
will be slotted into whatever paradigm is incomplete into which it
may hold true. To fool a mathematician one starts with the mere
subset relation. proven(facts) C true(facts). Then you make a
LIARS statement, factX <=> true(factX) = ~proven(factX).
The solution today? Mathematics in incomplete, we can't prove
factX but we *know its true*. It makes my blood boil and
'in this consistent (no LIAR statements allowed) domain' just adds
bamboo under the toenails.
Do it properly without inventing fantasy formal systems that don't know
what they're proving! true(facts) = proven(facts) U provable(facts).
factX <=> true(factX) = ~proven(factX)
factX <=> proven(factX) U provable(factX) = ~proven(factX)
factX <=> t U p = ~t
factX <=> t = ~t (extension to case p={})
factX <=> CONTRADICTION
"this statement has no proof" is a contradiction in terms, no different to "this is false".
Herc
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