[FOM] Feasible and Utterable Numbers

[Mirco Mannucci]

---- V.Sazonov at csc.liv.ac.uk wrote:


> Closure properties of feasible numbers in the spirit of semi-sets are
> the most crucial and delicate ones to formalise (without a
> contradiction!). The ordinary approach to semi-sets via non-standard
> models of arithmetic is insufficient here.
>

I completely agree that they are, but who mentioned non-standard models of arithmetics?
Certainly NOT I. What I suggested here, is simply that instead of partitioning the (finite) set of all available
terms in crisp equivalence classes, one can consider instead "fuzzy equivalence relations".
The reason being that defining a "number" (aka an equivalence class in my perspective) as feasible only
if an effective reduction to the the SSSS...0 form has been carried out, maybe a bit too restrictive.

After all, most people would agree that 2^100 is a feasible number, even though not many have actually  DONE the
computation. What I think we mean in this case is that there is a high degree of confidence that such a calculation
COULD be successfully completed, if  needed.

Anyway, a model of ultrafinite arithmetics based on  fuzzy equivalence relations will appear in a forthcoming
paper of mine, "Life without Omega, deconstructing the Term Model".

No non-standard models of arithmetics, I promise.

>
> >
> > a "number" is utterable iff there is at least one of its denoting
> > alternatives that is small.
> >
> > The natural question is, of course: what does it mean small in this context?
> >
> > The answer, it seems to me, is that there are several "theories of
> > utterability",
> > depending on your choice for smallness. For instance, as it has been
> > suggested by
> > Simpson, I could state that small=feasible complexity (or length). A number
> > is utterable iff one of its denoting terms has feasible length (or Kolmogorov
> > complexity, if you choose the Kolmogorov Tao).
>


> ???


Sazonov seems to be puzzled here, but I thought (perhaps erroneously) that it was pretty clear.
Quite simply, look at a "number".  This is a collection of terms that have been proven to
be equivalent to one another. Now, assume that you have a map f from  terms to terms
(there are NO numbers here, only terms), which represents some measure of
complexity of the input term.

 f: t1---------------------------------> t2

For instance, the most trivial measure is length (Kolmogorov complexity
is just a more sophisticated measure  from this viewpoint. Others may also
be considered).

Going back to our "number", suppose that one of its denoting terms gets mapped into a
feasible term (i.,e. into a term whose equivalence class is feasible).
Then I say that my number is utterable. One of its description is small.

The bad thing about unutterable numbers is that, unlike unfeasible numbers, they cannot even be
NAMED (however, we can PROVE that they exist, much in the same way as we know that most reals are
trascendent, even though we can only name a handful of them).

The good thing is that (unlike Sazonov in his "Feasible Numbers", as well as other authors
that rely on the supposed "size" of nature) I do not believe that either utterability or feasibility
are absolute notions.

They are, in my opinion, CONTEXTUAL.

Thus, something that from some viewpoint and given available resources is unutterable now,
may become utterable at a later stage (same applies to feasibility).

Indeed, the beginning of all this stuff is a paper written by one of the
greatest mathematicians of all times, Archimedes of Siracusa.

The paper's  title is "The sand reckoner".

The topic is "prove that there is an utterable number greater than all the particles of sands of Sicily".

We are still playing this game.....

Best to all

Mirco Mannucci