Hi Bill,
Nicely analyzed; my comments are below,
-Rich
Sincerely,
Rich Cooper
EnglishLogicKernel.com
Rich AT EnglishLogicKernel DOT com
9 4 9 \ 5 2 5 - 5 7 1 2
<snip/>
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.
BA:> 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.
Yes, and the factors designate those 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?
That's correct - the primes, not being factorable, designate unique
expressions, and cannot be factored into those "legal syntactic
entity" elements.
The discussion started with the concept that HOLs are more expressive
than just the combined FOL primitives can express. The primes are not
factorable into those FOL primitives, but are unique expressions which cannot
be reached through sentences composed of combinations of AND, OR, NOT and
ground instances.
Thanks for making this clearer, Bill.
-Rich
> 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_.
Yes. We all could benefit from more such face time.
> -chris
>
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