[FOM] Feasibility

[Vladimir Sazonov]

Quoting Matt Insall <montez at fidnet.com>:

> Dear Vladimir,
> Your original posts, if I recall correctly, indicated that you consider
> ``infinity'' to be less than or equal to 2^1000. Now, it is known that no
> natural number is infinite, so this indicates to me that you do not believe
> that numbers larger than 2^1000 exist. You seemed to give ``feasibility''
> as a justification for this conclusion. I do not agree that feasibility of
> notation should be used as the basis for determining what exists and what
> does not exist, but if we momentarily adopt your position, that feasibilty
> of _notation_ should be used in this metaphysical way, to determine what we
> consider to exist, then the main question becomes not just feasibility of
> the notation you prefer, but also why should one prefer such notation. 

Dear Matt, 

I actually do not know what do you mean by existence.
Anyway,...

When I use the term "feasible", I actually mean the relation 
of natural numbers to the real world via counting some real 
physical objects like the written symbol "|". This leads to 
considering unary notation system. Of course, physical "size" 
of these bits (which you suggest to discuss) does not actually 
matter because of the fact that even the number of electrons 
is less than 2^1000, nothing to say about the number of such 
physical macro objects as "|" whichever way we will represent 
them (on a sheet of paper or in computer memory). 

Is it now clear what do I mean? (Non-)efficiency of unary 
notation system playes no role here. (It would be an issue 
when considering calculations. Then decimal or binary notation 
system will come to the scene, of course.) A natural and 
simple relation to the real world - this is the point at 
this moment. If numbers are intended to count some objects, 
let them count real (macro) objects of our world. Those numbers 
which "count" are considered as feasible. It looks plausible 
that 2^1000 is not feasible in this sense. At least we, human 
beings, will hardly be able to count till this number, to make 
so many derivation steps in mathematical proofs, etc.  

No question, the idea of feasible numbers is extremely 
vague - it is unclear where is the "border line" between 
feasible and non-feasible, although it is sufficiently clear 
and certain that 2^1000 is non-feasible. But let us try to 
put this idea in a formal framework: write down axioms 
governing feasible numbers and try to elaborate what the 
appropriate underlying logic could be so that these axioms 
would be meaningful and consistent (in a suitable sense)? 
Say, postulate that (i) the "set" F of feasible numbers is 
closed under +1 (and even under +), once there is no clear 
boarderline between feasible and non-feasible, but that
(ii) 2^1000 is not in F. 

What I assert, there is really some formal framework (see 
http://www.csc.liv.ac.uk/~sazonov/papers/lcc.ps) where these 
axioms are consistent (actually, feasibly consistent.) 
Once it is a formal framefork, everything deduced about 
feasible numbers here is quite rigorous. Thus, we started 
with a highly vague idea (but the idea which is clear enogh 
to initiate first steps of its formalization) and came to 
mathematically rigorous style of reasoning on feasibility. 
I believe that the resulting formal approach (actually initiated 
by Rohit Parikh for some other version) gives a new insight. 

To give more details how can it be done means actually to 
present the wole paper. Thus, let me stop here. I do not 
think it makes sense to continue the discussion without 
invoking more formal aspects. (One such aspect appeared 
in posting of Arnon Avron.) I will also be busy with some 
other activity and, unfortunately, will hardly be able 
to reply actively. 

> 
> PS: Have a happy holiday, if this is to you a holiday season, as it is for
> us in the USA. (Have a happy season, in any case.)

Thanks! Merry Christmas and happy New Year to you and 
everybody here in FOM. 

Vladimir Sazonov




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