[FOM] Finite Set Theory
Vladimir Sazonov
V.Sazonov at csc.liv.ac.uk
Sun Feb 19 11:02:46 EST 2006
Quoting Harvey Friedman <friedman at math.ohio-state.edu> Sun, 19 Feb 2006:
> I have had several occasions to explain what set theory is to "ordinary
> people" of a variety of age groups.
>
> I have found it very useful and effective to start with finite set theory
> only - restriction attention to very small sets.
>
> It works very well, and everybody is quite satisfied. Nobody feels confused
> or uneasy.
>
........
> 17. Again, this goes over reasonably well. And then I say that we can rework
> all of the above even allowing such infinite sets.
>
> Now, where would FOM subscribers have issues with this development?
I would say that everything is quite clear and *sufficiently*
convincing, at least on that level of consideration based on the common
sense of "people from the street" or even those mathematically trained
or just the ordinary professional mathematicians. In fact, I think that
the problem of set-theoretic paradoxes has been successfully resolved
by mathematical practice - virtually everybody work in the framework of
ZFC and are quite happy.
Another story - people concerned with more subtle foundational
questions. Say, for me, the meaning of powerset operation even for the
case of HF (hereditarily finite sets) and even for not so huge concrete
finite sets is doubtful. But I would like to stress that these my
doubts do not mean any pretension on imposing restrictions to
mathematical practice (which generally consists in developing
*arbitrary formalisms as "tools of thought"*). Rather, I hope on some
extension of this practice by new ideas and formalisms around the
(vague) concept of feasibility as reflecting practical (say,
computational) reality where these imaginary powersets are not
realisable.
In particular, a feasibly finite set theory could be in principle
developed where the powerset 2^1000 would not exist (or would be an
infinite class). Here 1000 is understood as a set consisting of one
thousand elements. (Also note that a set x could be quite small, but
yet non-feasible because of the cardinality of its transitive closure
TC(x).)
This approach is like the complexity theory, but with declaring
something too complex (like the exponential time or space) as a kind of
infinity, thereby moving complexity problems to the foundations of
mathematics. Of course, mathematics here is understood wider than just
ZFC. What should not be changed, at least in the ideal, is the highest
standard of mathematical rigour - its main and distinguishing, even the
definitive feature.
A preliminary example of considerations in this direction is done, for
example, in my paper "On bounded set theory"
http://www.csc.liv.ac.uk/~sazonov/papers/cong-fin.ps where, in
particular, a BST is considered whose definable operations = provably
recursive operations = exactly PTime computable ones over HF. Recall
that PTime is also considered as a version of feasible computability.
But I discussed above on feasibility and typically use this term in a
much more strong sense.
Vladimir Sazonov
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