[FOM] S4 + ZFC

[Kreinovich, Vladik]

> Michael Carroll asked: 

> what happens if we add modal operators to the object 
> language of ZFC?

> I haven't been able to find any research along these lines. I wonder 
> whether it has been pursued and found fruitless ...

Addition of modal operations to math have been successfully used in
applied mathematics, when instead of knowing the exact value of an
object (number, function, etc.), we only know a set of possible values
of this object, In this case, "necessarily P" means that P holds for all
objects from this class, "possible P" that it holds for some. 

This is a very widely spread language in interval mathematics, which
deals with interval uncertainty. Of course, due to the applied character
of this application, the emphasis is usually on algorithms and
efficiency. 

Modal interval analysis is a well-developed area with many practical
applications, developed mainly by a group in Girona, Catalonia; for an
intro see

http://ima.udg.edu/SIGLA/X/mod_interval/Interval.html

See http://mice.udg.es/ for numerous applications; the Mice center
website has a large bibliography. I am sending a copy of this message to
Josep Vehi and Miguel Sainz, leaders of this lab. 

In our paper (cited below) we explore the possibility to extend modal
methods beyond intervals

Bernadette Bouchon-Meunier and Vladik Kreinovich,
  "From Interval Computations to Modal Mathematics:
  Applications and Computational Complexity",
  ACM SIGSAM Bulletin, 1998, Vol. 32, No. 2, pp. 7-11.
http://www.cs.utep.edu/vladik/1996/tr96-47.pdf

See also the corresponding chapter in our book 

Vladik Kreinovich, Anatoly Lakeyev, Jiri Rohn, and Patrick Kahl,
  "Computational complexity and feasibility of
  data processing and interval computations",
  Kluwer, Dordrecht, 1997.

Yours

Vladik