[FOM] Feasible and Utterable Numbers

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

Hi Vladik!

Thanks for your clarifying comments.

> Thus, if we make a computational cell as small as a particle, we still
> have no more than 10^88 particles performing 10^40 steps - to the total
> of no more than 10^128 steps overall.
>
> This translated to 2^430 computational steps which is slightly larger
> than 2^100.
>
> In short, your argument remains valid if you add one more 0, and talk
> about 2^1000 instead of 2^100.


Note that I usually mention just 2^1000 to be guaranteed from any 
physical objections. But, in fact, this is not so important for me.

I do not know whether it is realistic in a remote future to make a 
computational cell as small as a particle and to construct a computer 
(comparable with the size of the World or, say, a Galaxy) which would 
reliably save and manipulate in its memory with binary strings of the 
length like 2^100. Let us better think about our contemporary real life 
and real computational practice. Let us also think in terms like "which 
is the precision of calculations sufficient for projecting a reliable 
nuclear reactor?". Then I think (and hope that experts could confirm) 
that postulating 2^100 = infinity is more than enough. In some cases 
2^50 = infinity could be probably quite good. (I hope it is clear what 
I mean.) So, what is the "best" upper bound for feasible numbers 
(postulated as closed under successor and even sum operations) may 
depend on some real circumstances, and the very concept of feasibility 
is extremely vague. Nevertheless, it can be appropriately formalised 
and therefore be subject to mathematical methods of rigorous reasoning. 
Even if some such a version of feasibility theory (with not so high 
postulated upper bound for feasible numbers, say 2^30= 1,073,741,824) 
will happen to be contradictory with some quite lengthy, but 
practically feasible proof of contradiction available for the 
contemporary computer, this theory will still make a sense if we would 
restrict ourselves to not so long proofs. But in the case of 
postulating only 2^1000=infinity no formal contradiction is possible 
(as soon as the ordinary mathematical formalisms like PA or ZFC are 
reliable). This is what I really wanted to say and demonstrate.


Kind regards,

Vladimir


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