On Oct 13, 2010, at 17:04 , Christopher Menzel wrote: (01)
> On 10/13/2010 02:52 PM, Bill Andersen wrote:
>> On Oct 13, 2010, at 01:56 , Rich Cooper wrote:
>>
>>> Hi Duane, yes, iterators in software were what I tried to convey
>>> there. There is no function that will iterate the primes. By pairing
>>> each prime in ascending order with any other iterated set, you create
>>> unique prime keys for each element of that set, keys that cannot be
>>> factored.
>>> Thanks for your inputs,
>>>
>>> -Rich
>>
>> Hi Rich
>>
>> Here's a fixed precision implementation of a prime iterator, along the
>> lines Chris Menzel described.
>>
>> Enjoy
>
> Clearly, Bill, your code must be flawed. ;-)
>
> It's really hard to figure out where the disconnect is here. It is so
> completely obvious that the simple algorithm I provided iterates the
> primes (i.e., lists them in order) that Rich simply must be attaching an
> entirely different meaning to "iterates" than the rest of us. Does his
> talk of keys suggests that maybe he is getting database stuff confused
> with basic computability? (02)
Earlier, Rich said this (03)
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. (04)
>From what I could gather, he was trying to claim that somehow, under Gödel's
>construction, the non-theorems were the only things represented as prime
>numbers. As I understand Gödel's method, and as you pointed out earlier, the
>business of the use of primes for arithmetizing syntax had nothing to do with
>the theoremhood (or not) of the resulting sentences (or proofs) so
>arithmetized. I can't recall how the construction went exactly, but it seems
>to me that the point was that all sentences (proofs) were represented by
>non-primes, the idea being you could then exploit their arithmetic structure
>to tear them apart into their constituents. So you'd have some non-primes
>that represent non-sentences (proofs), some non-primes that represent
>sentences (proofs), and primes that either represent primitives in the
>construction or fail to represent any legal syntactic entity. Is this about
>right? (05)
> Either way, some face time with an
> introductory text on the latter subject is in order. I'd suggest
> Boolos, Jeffrey, and Burgess, _Computability and Logic_. (06)
Yes. We all could benefit from more such face time. (07)
> -chris
>
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Bill Andersen
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