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Re: [ontolog-forum] HOL decidability [Was: using SKOS for controlled val

To: "[ontolog-forum]" <ontolog-forum@xxxxxxxxxxxxxxxx>
From: Christopher Menzel <cmenzel@xxxxxxxx>
Date: Tue, 12 Oct 2010 23:11:32 -0500
Message-id: <4CB53174.90008@xxxxxxxx>
On 12/10/2010 10:18 PM, Rich Cooper wrote:
>> ...
>>> Not so fast!  I'm sure you remember that the set of primes is
>>> infinite, and that there is no (known) function that can iterate
>>> them. 
>> Of course there is.  For any given number n>1, it is easy to test
>> whether n is prime.  For a particularly crude algorithm, for each i<n
>> (i>1), look for a number j<n (j>1) such that ij=n.  If you fail to
>> find such an i, then n is prime.  To construct a list of the primes,
>> apply the above procedure to each number in turn, starting with 2,
>> adding the primes you discover to the list as you go.  This informal
>> procedure is easily expressed formally as a recursive function; one
>> typically demonstrates this in the first week or two of a course on
>> computability.
> RC: I didn’t say there was no way to calculate them;     (01)

The procedure above simply identifies and lists them.  I'm not sure
what you mean by calculating them.    (02)

> I said there is no function that *iterates* them.      (03)

Doesn't "iterate" mean "list"?  If it doesn't what does it mean?    (04)

> And your algorithm above iterates *integers* till it starts to factor,
> then continues to factor forever without returning if the number is
> unfactorable (i.e., if its prime).      (05)

Uh, no it doesn't.  If n is prime, the procedure simply halts when i
reaches n-1 and fails to find a j<n such that ij=n.    (06)

>>> Godel showed that there are theorems which though true cannot be
>>> proven,  
>> No, he didn't.   First of all, theorems are by definition statements
>> that have been proved. 
> RC: From Google:
> Definition:A proposition that has been or is to be proved on the basis
> of certain assumptions Context:In Book 1 of Elements, Euclid's
> proposition 41 is the theorem "if a parallelogram has the same base with
> a triangle and is in the same parallels, then the parallelogram is
> double the triangle."
> school.discoveryeducation.com/lessonplans/programs/conceptsInGeometry/    (07)

Uh, right.  Theorems are statements that have been proven.  Isn't that
what I said?  That's why "theorem which...cannot be proven" is an oxymoron.    (08)

>> CM: So what you are trying to say is that Gödel showed that there are
>> arithmetical statements which, though true, cannot be proven.  But
>> that's not true either.  Provability is relative to a system and *any*
>> arithmetical statement can be proven in some system or other -- just
>> take that statement as an axiom.
> RC: The “some system or other” is the whole point.  That is the part
> that you can’t iterate over for every observed case.  There are
> primes, remember, and you will encounter them in iteration of
> supersets of the primes.    (09)

Word salad.    (010)

>> CM: Here's what Gödel proved: Given any consistent, decidable set S
>> of axioms in the language of arithmetic** capable of proving a
>> certain minimal amount of arithmetic, there will be statements in the
>> language that are neither provable nor disprovable *in S*.  From this
>> it follows that, for any such set S, there are statements that are
>> true in the natural numbers but which S cannot prove.
> RC: The plurality of Systems, as you called them, are ordered in pairs
> with the integers (a superset of primes), but not with the primes.
> That is how Godel constructed his proof.  So there are true theorems
> that can’t be proven (factored) and false theorems that can’t be
> disproved (factored), because in association with the integers, they
> occasionally designate a prime – which by definition can’t be
> factored.       (011)

Abject nonsense.    (012)

>>> and theorems which though false cannot be disproved, all based
>>> on the primes.
>> CM: This is just nonsense.  The key to Gödel's result, for a given
>> system S, is the "arithmetization" of S's syntax, which refers to any
>> method of encoding the terms and formulas of S (and, indeed,
>> sequences of such) as numbers.  This one to represent the proof
>> theoretic apparatus of S in S itself as actual functions and
>> relations on the numbers. 
> RC: I didn’t say anything about defining S in S itself, you did, and
> your formulation of Godel is not the only way to formulate it.  So by
> enabling self reference, it’s nice that you have moved ahead of the
> conversation, but not relevant to the issue at hand, which neither
> requires nor disallows self reference.      (013)

*sigh*  You toss precise technical vocabulary about with only the
vaguest inkling of its true meaning, making claims that are not only
syntactically tortured but semantically DOA.  I'd hoped that perhaps a
clear and forthright accounting of your errors and confusions in regard
to mathematical logic combined with a dash of mild ridicule might get
you to hit the books instead of shooting your mouth off in ignorance of
the subject.  I see now I was mistaken.  In the face of correction, you
will simply dig yourself a deeper hole.    (014)

I will therefore cease replying to you, no matter the topic, as it is
clear that discussion with you is a waste of time and bandwidth, at
least in regard to technical matters on which you feign expertise.  I
hope at least that folks in this forum have been properly alerted to
this and I apologize to them for not seeing the pointlessness of my
efforts sooner.    (015)

-chris    (016)

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