On 10/13/2010 04:32 PM, Bill Andersen wrote:
>>> ...
>>> 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?
>
> Earlier, Rich said this
>
> 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.
>
> 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. (01)
I commend you on your powers of interpretation; he does indeed seem to
be saying something like that. And, like pretty much everything he has
said about Gödel's theorem and its proof, it is not even remotely true. (02)
> 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, (03)
Correct; specifically, where formula A is the string c1...cn of lexical
items in the language of arithmetic, #c is the number assigned to
lexical item c in the initial encoding, and p(m) is the mth prime, the
the Gödel number of A is exp(p(1),#c1) x ... x exp(p(n),#cn). So long
as one is careful about one's initial encoding, the same construction
can be used to assign numbers to proofs, using the Gödel number #Am of
the mth formula in a given proof instead of #cm. (04)
> the idea being you could then exploit their arithmetic structure to
> tear them apart into their constituents. (05)
Correct, in virtue of the fundamental theorem of arithmetic one can
recover the component formulas of a proof from its Gödel number and the
component subformulas and lexical items of a formula from its Gödel
number. (06)
> 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? (07)
Yes, pretty much, although you could also have non-primes representing
lexical primitives. (08)
-chris (09)
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