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From: William Hughes on 27 May 2010 10:32
On May 27, 1:10 am, Transfer Principle <lwal...(a)lausd.net> wrote:
> On May 24, 5:39 pm, William Hughes <wpihug...(a)hotmail.com> wrote:
>
> > On May 24, 9:04 pm, Herc7 <ozd...(a)australia.edu> wrote:
> > > You are selecting specific diagonals based on the list.
> > > The probability of fitting a random diagonal to the list of computable
> > > numbers resulting in a different set of numbers is 0.
> > There are however an infinite number of counterexamples.
>
> Hmmm. Let's look at Herc/Cooper's post again.
>
> > > The _probability_ of fitting a _random_ diagonal to the list of computable
> > > numbers resulting in a different set of numbers is 0.
>
> (emphasis mine)
>
> Probability? Random?

There are _however_ an infinite number of counterexamples
(emphasis added). My comment was not intended to contradict
the probability statement.

It is true that the set of counterexamples may have measure
0 in the usual Lebesgue measure. Indeed, I suspect
this, though I cannot see a simple way of demonstrating it.

However, the set of counterexamples is non-empty and uncountable.

[Note that the idea that an infinite number of reals can have
Lebesgue measure 1 already requires uncountability. If you insist on
trying to make Herc's statements consistent you will need a
theory of probability in which a countable infinity of elements,
each with the same probability, has a probability of 1.]

- William Hughes
From: Daryl McCullough on 27 May 2010 11:15
LauLuna says...

>You are overlooking the possibility that 'this cannot be proven by
>Roger Penrose' express no proposition, that it be paradoxical just
>like the Liar sentence.

But it obviously is *not* paradoxical. Assuming it is true does
not lead to a contradiction. Assuming it is false does not lead
to a contradiction. That's very different from the liar paradox,
in which both assumptions lead to contradictions.

>Penrose will surely know that if he proves it, then it is not true; so
>he must know he cannot prove it;

No, that doesn't follow. That only follows if you *also* assume
that Penrose never proves anything false. (There is actually
a technical difficulty in applying the notion of "what is provable"
to a human being, as opposed to a formal theory, and that's this:
for a formal theory, if the theory proves A, and B logically follows
from A, then the theory proves B. Humans don't live long enough
to prove all the logical consequences of what they've already proved.
And furthermore, once a human spots a contradiction in what they've
done, they will typically try go back and fix their assumptions to
eliminate the contradiction. They don't just go ahead proving
everything.)

>in fact he has proved that the sentence cannot be proven by Roger Penrose;

No, he hasn't. He's proved that *if* he never proves anything false,
*then* he will never prove that sentence. He hasn't proved that he
will never prove that sentence.

On the other hand, let's assume the following facts about
Penrose:

(1) Penrose believes that he never proves anything false.
(2) Penrose proves everything that is a logical consequence
of things that he believes.

In that case, since "this cannot be proven by
Roger Penrose" is a logical consequence of
"Penrose never proves anything false", then
Penrose *will* prove the sentence "this cannot
be proven by Roger Penrose". It follows that
the sentence is both false and provable by Penrose.

So for the "Penrose statement" we have the following:

1. If Penrose never proves anything false, then the Penrose
statement is true, but not provable by Penrose.

2. If Penrose can prove the Penrose statement, then the
Penrose statement is false (and Penrose proves false statements).

It's exactly the case with Godel's theorem: The theory is either
unsound (proves false things) or incomplete (fails to prove true
things).

--
Daryl McCullough
Ithaca, NY

From: Transfer Principle on 27 May 2010 15:52
On May 25, 10:11 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote:
> Transfer Principle <lwal...(a)lausd.net> writes:
> > Notice that Herc/Cooper claims to have proved it,
> Notice that Herc/Cooper claims to be both Genesis Adam

Back to Genesis Adam again?

I definitely do _not_ accept Herc/Cooper's claim that
he is Genesis Adam, since, to say the least, Adam is
_dead_ according to Genesis (for having eaten the
forbidden fruit). How can someone be a specific
person mentioned in a book if according to that book,
that person is dead? (Unless that book is wrong, of
course but Herc/Cooper also states belief that the
book is true.)

Therefore, just because I'm entertaining one of
Herc/Cooper's claims (about Cantor), it doesn't mean
that I necessarily accept any of his other claims
(especially not those about Adam).

Indeed, I convinced Herc/Cooper to start this thread
about Cantor in order to avoid discussing Genesis on
sci.math, but Genesis and Adam have still turned up
in this thread.
From: Transfer Principle on 27 May 2010 16:07
On May 25, 12:21 pm, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote:
> Don Stockbauer <donstockba...(a)hotmail.com> writes:
> > Now, now, now.  Let's not fall victim to the "Argument Against the
> > Man" (er, Person, nowadays).
> In this instance the information that Herc appears to be clinically
> insane is salient. Recall the nature of lwalker's valiant quest. He
> wants to vindicate people who get called (mathematical) cranks

I made a resolution that I would avoid five-letter
insults such as the word "crank" unless at least one
other poster in the thread has already used the word.

Here Aatu has used the word "crank," and in the plural,
to boot. This means that use of the word "crank," in
both singular and plural, is now open to me.

Also, Aatu refers to Herc/Cooper as "insane." I don't
accept that most so-called "cranks" are "insane." Of
course, much of the evidence supporting Aatu's claim
of his insanity are non-mathematical (Genesis Adam and
all that).

So I should reject all of Herc/Cooper's mathematics
just because of his insane claim to be Adam? This
issue has also come up with Archimedes Plutonium and
his Atom Totality theory. I'd like to consider some
interesting mathematical theories yet be able to
separate these "cranks'" mathematical ideas from their
more controversial theories about Adam and Atom.

Of course, if only Herc/AP would stick to alt.atheism
and sci.physics, respectively, when discussing their
non-mathematical ideas, it would be easier, for me as
well as everyone else, to read about their ideas for
math here on sci.math.
From: |-|ercules on 27 May 2010 18:29
"Aatu Koskensilta" <aatu.koskensilta(a)uta.fi> wrote in
> "|-|ercules" <radgray123(a)yahoo.com> writes:
>
>> Aatu Koskensilta wrote: All mathematical proofs can be formalised,
>> after all.
>
> This is hardly something I've claimed to have a proof of. It's just an
> observation.

You said it was a statement of incontrovertible fact, you just snipped that!


>
>> One must be skeptical when the proof that all informal proofs can be
>> made into formal proofs, is itself just an informal proof.
>
> Why?

You can't see how ironically stupid your informal proof is that ANY informal proof can be formalized?

Formalize the proof that all proofs can be formalized!

DUH!

Not only are you unaware of what you are saying, you lie, cheat and steal to cover your flamboyant stupidity.

Herc
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