[FOM] S4 + ZFC

Dana Scott dana.scott at cs.cmu.edu
Sat Aug 25 00:26:40 EDT 2007

An easy way to model modal set theory is to use the
Boolean-valued models.  Any complete Boolean algebra
(cBa) gives a model for (classical) ZFC in the well
known way.

Consider then the following fact:

    Theorem. Any complete Heyting algebra (cHa) can be
    embedded into a cBa in such a way as to preserve finite
    meets and arbitrary joins.

A proof can be found in Peter Johnstone's excellent book,
"Stone Spaces".  We recall that a cHa is a complete lattice
where finite meets distribute over arbitrary joins.

Let H be a cHa, and let it be embedded into a cBa B.
Think of B as a powerset for a moment to fix ideas.
What is the image of H in B?  It is a family of subsets
closed under finite intersections and arbitrary unions.
Well, that is just a topology on the underlying universal
set of B.  Topology is just the McKinsey-Tarski way of
interpreting S4:  The interior operation of the topology
is the algebraic interpretation of the necessity operation.

The trick here is to see that complete lattices not only
give us a semantics for quantifiers, but we can expand the
semantics to higher-order logic, even to set theory.  And the
Heyting-part of the interpretation can be an arbitrary cHa.
If we think of H as a sublattice of B (with respect to finite
meets and arbitrary joins), then the interior operation on
B is defined abstractly just as we do in concrete topologies:

	I(x) = \/{ h in H | h =< x }.

Two extreme cases, given B, would be H = {0, 1}, a very
strong S5 logic, or H = B, a very trivial logic of necessity.
There are usually many topologies inbetween -- if B is non
trivial.  And we have to be careful to note that H does
not determine B uniquely.

So much for generalities.  What might be interesting is to
discuss and contrast special cases -- especially those that
are not mere topologies on powersets.  Different examples might
have the same first-order logics but different set theories.

BTW, we can also have towers of cHa's where each is embedded
in the next right up to B.  These would model theories of
weaker and weaker necessities.  Would they be interesting?

      Dana S. Scott
      1149 Shattuck Avenue
      Berkeley, CA 94707-2609
      Tel: (510) 527-5287

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