🚀 go-pugleaf

In article <1992Apr7.124948.5325@vax.oxford.ac.uk>,
loader@vax.oxford.ac.uk (R Loader) writes:
>IMHO, extending the underlying logic of set theory is unlikely to give
>interesting set theorys. On the other hand, perhaps one could introduce new set
>theoretic principles involving modal operators?

If one extends the underlying logic of set theory by adding the S4 modal
axioms, one obtains theorems such as:
    If necessarily (x is in y) then (x is in y);
    If (x is in y) then possibly (x is in y);
    If possibly possibly (x is in y) then possibly (x is in y);
and so on. If one adds the definitions
    The interior of x is the set of all y such that necessarily (y is in x);
    The closure of x is the set of all y such that possibly (y is in x);
then in virtue of the theorems it follows that
    x is a subset of the closure of x;
    The interior of x is a subset of x;
    The closure of the closure of x equals the closure of x;
and so forth.  That is, interior and closure defined in this way will have
many (all?) of the usual properties of interior and closure as defined in
topology. (And, in general, the interior and closure of x will be distinct
from each other and from x itself. We will have "more" sets than before,
I think, which might impact the independence proofs.)

It seems to me that this *does* introduce new set theoretic principles
involving modal operators, and does so by changing the underlying logic. So
I fail to grasp the contrast between the two approaches that R. Loader
describes.

I think the real problem lies elsewhere: it's not clear to me what we would
mean by *the* interior of a set. It would seem that there would have to be
some one topology that grew naturally out of set theory.  If there is one,
I don't know what it is.  Do you?

--
Michael Carroll             br.mjc@forsythe.stanford.edu
Research Libraries Group