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 n1 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)
_________________________________________________________________
Message Archives: http://ontolog.cim3.net/forum/ontologforum/
Config Subscr: http://ontolog.cim3.net/mailman/listinfo/ontologforum/
Unsubscribe: mailto:ontologforumleave@xxxxxxxxxxxxxxxx
Shared Files: http://ontolog.cim3.net/file/
Community Wiki: http://ontolog.cim3.net/wiki/
To join: http://ontolog.cim3.net/cgibin/wiki.pl?WikiHomePage#nid1J
To Post: mailto:ontologforum@xxxxxxxxxxxxxxxx (017)
