Re: limitation to the halting proof

From: George Greene (greeneg_at_greeneg-cs.cs.unc.edu)
Date: 06/01/04 

Date: 01 Jun 2004 15:54:09 -0400

I apologize if this gets confusing arguing vs. 2 different people
who were already arguing with each other.

 : > you claim the uncountable irrationals are not able to be put on
 : > paper

Of course not; even worse, not a single one of them
(let alone all of them) "is able to be put on paper"
in the usual sense.

 : > as opposed to 2, that they go 'inbetween' the numbers we
 : > can *write* hence *see*.

That's ridiculous; nobody is claiming that; there are a lot
of *numbers* you can "see" DESPITE the fact that you can't
"see" the decimal expansion of them. Like any transcendental number
with a geometric representation (like Pi).

 : > you use a faulty premise of open
 : > diagonalisation

The burden of proof that it is faulty is on you.

 : and when its shown to be nonsense

You have never shown any such thing.
At best you have shown that the concept has consequences that
you personally are too stupid to understand.

The Ghost In The Machine <ewill@aurigae.athghost7038suus.net> writes:
 : I've yet to see a working proof that it's nonsense. Granted, every
 : number we ever write down, every number we think of, every number
 : we work with in the practical world is in fact rational,

No, it isn't.

 : given the nature of floating point arithmetic in modern computers and
 : the simple finiteness of the representations used.

Well, if you are working with finite bit-strings, yes, but who says
that's all you have to "work" with?

 : Or, they are
 : symbolic algebra -- e.g., "sqrt(2)"

Exactly.

 : -- which we can manipulate
 : using various rules (sqrt(2)^2 = 2, sqrt(2) + sqrt(2) = 2*sqrt(2),
 : sqrt(2) / sqrt(2) = 1).

And in addition to those irrationals (with their INFINITE digit-
expansions), there are also [FINITARY] rules for manipulating INFINITE
sequenes, which is how we can also deal with some easy transcendentals.

 : pi and e are also interesting transcendentals
 : which we discuss symbolically; there's a few others.

Exactly.
Therefore, the claim that everything we work with is rational is
just wrong.

 : Or, if one wants to get ridiculous and write down a number on
 : every atom in the Universe that one can see from the Earth,
 : one only has so many atoms, regardless of representation.

Of course, but the point is, you don't HAVE to do it THAT way.
There are other ways. There are any number of functions over
an infinite domain that have finitary definitions.

 : > you have no mathematical opposition to countable functions and
 : > countable reals is just a step further.

It's not a question of "mathematical opposition".
It's a question of "mathematical competence". YOU have to
DEFINE what YOU mean by "countable reals". UNTIL YOU SHOW
SOME AXIOMS, nothing you say matters.

 : > UTM(n), from n e N IS the countable reals,

No, it isn't.
In the first place, UTM(n) is almost meaningless.
As generally conceived, the UTM takes TWO n's as arguments.
UTM(m,n) is what MACHINE m does with INPUT-NUMBER n.
Applying UTM to itself is going to require re-encoding (m,n) as a
single number. More to the point, ONE single individual TM
will, at best, encode only ONE real number. You cannot encode the
entire family of "countable reals" (whatEVER they may be) with only
one TM, not without more complexity than you personally know how to manage.

 : > it doesn't miss a beat, not all n halt thats
 : > a petty copout requirement of YOUR argument, now even that is invalid.

It's not a "copout requirement". IT'S A FACT.
SOME TM'S DON'T HALT, for some inputs. Get over it.

 : It's not even countable. It's finite.

Presumed antecedent of Ghost's "it" here is "the UTM", but
the factual antecedent of Herc's "it" was UTM(n), which is
something different and arguably should not be being talked
about at all, since he can't define it.

 : Of course mathematics does
 : not always deal in the concrete and the graspable -- one interesting
 : question is whether the digit expansion of pi is normal, for
 : example. Of course pi's digit expansion can never be written
 : down in full, which means we have to talk about something that
 : is by necessity incomplete.

No, we don't.
There are finite TM's that exactly represent Pi digitally.
If you were to choose some other framework then you might
be able to get an even more finitary representation. The
problem THERE has more to do with the definition of "normal"
than of "pi". Normality is not fundamentally a numeric property.

 : But mathematicians do that all the time. N = {1, 2, 3, ... }
 : for example, has more than 3 elements, and most people infer
 : that N = {1} union {x + 1: x in N} [that's two Peano axioms
 : right there] solely from this representation.

Sure, but that's conventional, which is another way of saying
"indefensible". The fact that most people do something can
only be a reason why something is wrong, not why it is right.

 : Of course one can be led astray

Precisely as I just said.

 : So we are forced to talk about a set R whose elements we can
 : never write out in full, exhaustively list, or compute using
 : a UTM, even were we to run said UTM until the heat death of
 : the Universe (current theory suggests about 50 billion years
 : or so).

We are not "forced"; WE defined the set! WE could've refused to!

 : In any event, Cantor has *two* proofs. The first proof relies
 : on R having no gaps between numbers, no matter how one
 : cuts them. [*]

Please. This is so NOT about R.
The relevant proof is the proof that every set has a subset that is not
a member of it.

 : http://en.wikipedia.org/wiki/Cantor%27s_first_uncountability_proof
 :
 : The second is the diagonalization proof:
 :
 : http://en.wikipedia.org/wiki/Cantor's_diagonal_argument
 :
 : Both have the same problem, in that they rely on talking about
 : infinite things without explicitly writing them out in full.

This cannot be a problem if the proof doesn't even DEAL with infinity!
THE WHOLE point about the ACTUAL proof is that it works JUST as
well for FINITE sets! IT DOESN'T GIVE A *** whether anything
is or isn't infinite!

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