FOM: actual infinite

[Vladimir Sazonov]

Thomas Forster wrote:
> 
> It has always seemed to me that one ought to be able to give
> a satisfactory answer to the old puzzle about how finite beings
> can apprehend actual infinities by use of IT ideas like lazy
> evaluation.   


I do not know which relation lazy evaluation has to this puzzle. 
But I do not see any puzzle at all. Whichever (sufficiently coherent) 
fantasy we have, mathematics allows to approach it by choosing 
an appropriate formalism, say ZFC. We are finite beings, our 
fantasies (even about infinity - whatever it is) are finite and 
formalisms are also finitary. What is the problem? Finitary formalisms 
imitate our finite fantasies about anything. This "anything" should 
not exist at all, except in our fantasies. Of course, our fantasies may 
have some relation to reality, may be indirect. But again, we can 
approach (approximate) any kind of reality with the help of mathematical 
formalisms. What is the problem? How formal thinking is related with 
informal? 

There is some additional explication of this done by Mycielsky 
(in JSL somewhere in 80th). I already recalled this some time ago. 
He showed that ANY first-order theory T (PA or ZFC, etc.) has 
corresponding "isomorphic" (in a natural sense essentially the 
same as T) first order theory T' such that any finite number of 
axioms of T' has a FINITE model. 

Thus, if a theorem on some infinite cardinals is provable in 
ZFC then in ZFC' essentially the same proof uses, of course, 
a finite number of axioms and all we need in this concrete proof 
about infinite cardinals may be interpreted in a finite model. 

However, this finite model may be nonfeasibly huge and the question 
of grasping a huge finite by feasibly finite human beings may 
be repeated. In a sense, this is not better than infinity. Thus, 
this way or other we come to the problem of feasibility in 
mathematics or to the problem of building a feasible version of 
mathematics or mathematics of feasible objects (numbers, etc.). 
How it can be done is a normal research question. 
It is not easy, but not an unresolvable puzzle. 

What is the problem - how to reduce, say, ZFC to a theory of feasible 
objects. As I remember, Essenin-Volpin suggested some solution. 
But I was never able to understand this as a rigorous mathematical 
enterprise. (However, I like some of his papers on feasibility - 
those written in Russian long time ago. They provoke the thought.) 
To resolve this problem, if possible at all, the concept of feasibility 
should be first investigated mathematically/rigorously/formally 
as deep as possible. 


I'm sure I can't be the only person who has tho'rt
> this.  There must be a literature on it.  Can anyone point
> Franklin Pacheco (and me - i have a semester off next year, when
> i'll have time to think about things like this) at it?
> 
>          v best wishes
> 
>                Thomas


-- 
Vladimir Sazonov                        V.Sazonov at csc.liv.ac.uk 
Department of Computer Science          tel: (+44) 0151 794-6792
University of Liverpool                 fax: (+44) 0151 794 3715
Liverpool L69 7ZF, U.K.       http://www.csc.liv.ac.uk/~sazonov