[FOM] Feasible and Utterable Numbers

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

Quoting Charles Silver <silver_1 at mindspring.com> Thu, 03 Aug 2006:

>> 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).
>
> 	You've probably said this before, but if so I missed it.   What
> exactly is (are) the appropriate
> restriction(s) on logic to avoid contradiction?  Could you please
> explain this in terms of a natural deduction system with rules such
> as Modus Ponens, Universal Generalization, and the like?

First note that, M is considered not as a primitive predicate.

Second, we can restrict logical derivations to be normal natural 
deduction proofs (or cut free sequent style proof). But it is 
intuitively more appealing to formulate (or think about) the 
restriction as:

* abbreviation of terms is not allowed (and formal derivations should 
be, of course, of feasible length).

See (quite simple) details in my posting to FOM:

http://www.cs.nyu.edu/pipermail/fom/2006-February/009746.html

Please consider this text as a semi-formal presentation of a quite 
FORMAL considerations. The mere intuition is highly insufficient in 
reasoning on feasibility if we pretend to consider this as a 
mathematical concept.


Best wishes,

Vladimir Sazonov



----------------------------------------------------------------
This message was sent using IMP, the Internet Messaging Program.