[FOM] Feasible and Utterable Numbers

[V.Sazonov@csc.liv.ac.uk]

Quoting Mirco Mannucci <mmannucc at cs.gmu.edu> Sat, 22 Jul 2006:

>
> If one wants to make the feasible numbers subset closed under some operation,
> "fuzzy equivalence classes of terms" may be considered, in the spirit
> of semi-sets and related approaches (something I shall gloss over here,
> as it is not the focus of this posting).

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.


>
> 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).

???

> I could also follow an entirely different road and say that small means
> much smaller than the size of its standard "successor" description.

See my paper "On feasible numbers" where feasible numbers are divided 
into three (increasing) classes of small, S, (potentially) medium, M, 
and (potentially) large feasible numbers, F. All these (semi)sets are 
downward closed.

It is interesting that medium numbers M are closed under successor and, 
nevertheless, are < 1000 so that there is no formal contradiction (due 
to an appropriate restriction on logic). Note that feasible numbers are 
also closed under x+y and bounded by 2^1000; therefore feasible numbers 
can also be called "additive". There can be considered also a wider 
class P of "polynomial" numbers which are closed also under 
multiplication and bounded by 2^(2^1000). This sequence (S,M,F,P) can 
be continued trying to "exhaust" the potential infinity of the row of 
natural numbers. The main point of these considerations is that all of 
this is formalised and consistent. However, the underlying logic should 
be appropriately restricted to avoid formal contradictions.


Kind regards,

Vladimir Sazonov


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