🚀 go-pugleaf

On May 20, 1:16 am, MoeBlee <jazzm...@hotmail.com> wrote:
> On May 17, 8:08 am, "R. Srinivasan" <sradh...@in.ibm.com> wrote:
>
> > The basic philosophical ideas behind NAFL can be understood without
> > delving into details.
>
> That's not adequate when NAFL is being used in a technical sense to
> comment on details in other systems.
>
> > a NAFL theory cannot
> > be defined as a mathematical object.
>
> Then I'm not very interested in NAFL.
>
> > You can only make the definition
> > precise enough to be able to work within NAFL theories. There are
> > solid reasons for this requirement. One of these is that Godel's
> > incompleteness theorems cannot go through in NAFL,
>
> NAFL or NAFL theories are not even not just a formal theories, not a
> theories even as ones not formally axiomatized; not theories, as you
> say, as mathematical objects. Saying that incompleteness doesn't go
> through for NAFL is like saying incompleteness doesn't go through for
> my grandmother's diary.
>
As I said in a parallel post, patently false and there is no need
whatsoever to formalize the notion of a NAFL theory as a mathematical
object. E.g consider the NAFL version of Peano Arithmetic, call this
NPA. It is true that NPA has infinitely many axioms, which may be
generated by some rule. Here is a proposition about NPA:

NPA is consistent if and only if its axioms are pairwise consistent
(1)

Now (1) will go through in NAFL, but not in classical logic. To see
that (1) is true, we only need to know the rules by which the
infinitely many axioms of NPA are generated. There is no need to
formulate NPA as a mathematical object. That is your misconception.
And as I noted earlier, note that no theorem of NPA will ever appeal
to all of its axioms (the "coding" that enables Godel's incompleteness
theorems to go through in classical PA is not permitted in NPA). (1)
is metatheoretical statement *about* NPA and cannot be formalized
within NPA.

Take a simple analogy. Let us consider rational numbers of the form 1/
n, n=1, 2, ....

We know that if we continue computing the numbers 1/n for n=1, 2,
3, ..., we will always find 1/n > 0 and we will never ever find an n
for which 1/n is less than or equal to 0. To see that this assertion
is true, one doesn't need to postulate the existence of the set of all
rational numbers of the form 1/n. There is no commitment that our
"checking" will ever  be "completable". But  still we know that if we
continue to check, we are always going to find 1/n > 0. That very fact
also guarantees that the process of computing these numbers one by one
will never ever be "complete". It is a classical misconception that
"all" natural numbers can be "exhausted" via the process of computing
them one by one. NAFL computability theory does not accept that; any
such computation will have to use infinite parallelism rather than a
step-by-step procedure.

The notion of consistency of a NAFL theory (say NPA) must be
considered in some similar sense. If we continue to check the axioms
of NPA pairwise, we will never ever find a pair that is inconsistent.
This statement makes sense without any need whatsoever to formulate a
NAFL theory as an object. In the distant future, when NAFL
computability theory is fully developed, one might consider the
possiblity of demonstrating the consistency of NPA by (an infinitely
parallel) computation using NAFL principles (similar to quantum
computation in some non-standard setting).

Regards, RS