FOM: Boolean-valued models

[Joe Shipman]

Simpson:

>>The ``algebra of truth values'' is the complete Boolean algebra that
is being considered, right?  And ``projections'' are homomorphisms,
right?  So the question seems to amount to: Does there exist a
homomorphism of (some Boolean subalgebra of) some complete Boolean
algebra onto the measure algebra of a non-trivial probability space?

If this is the question, then the answer is, yes, of course.  For
instance, consider the complete Boolean algebra M for adding a random
real to the universe.  M is the standard atomless probability measure
algebra.  Each Boolean value (i.e., each element of M) is an
equivalence class of measurable sets and can be assigned a probability
equal to its measure.

But I'm sure Joe Shipman knows all this very well, better than I in
fact, so maybe I am misunderstanding the question.<<


Yes, I didn't mean "is this possible for some Boolean Algebra of truth
values", I meant for the particular Boolean algebras used in the
independence proofs, in such a way that the relevant independent
statements went to something between 0 and 1, rather than simply going
to 0 in in the independence half and 1 in the consistency half?



Simpson:

>>  > Is there a Boolean-valued model in which the axioms of ZFC have
 > value 1 but CH has a value strictly between 0 and 1?

Again, doesn't this question have the following easy affirmative
answer?  By Cohen/Solovay, let B1 and B2 be complete Boolean algebras
such that [[ CH ]]_B1 = 1 and [[ CH ]]_B2 = 0.  Let B = B1 x B2.  Then
[[ CH ]]_B = (1,0).  The Stone space of B is the disjoint union of the
Stone spaces of B1 and B2.  There is an obvious homomorphism of B into
the 4-element Boolean algebra {0,1} x {0,1} which carries an obvious
probability measure, so you can consistently assign CH a probability
of 1/2.<<

I figured this out immediately after posting my previous reply.  It
seems an unsatisfactory solution, though, since you are just patching
together a universe where CH is "true with probability 1" and one where
it is "true with probability 0".  I'm looking for a plausible way to
think about mathematical propositions having a probability between 0 and
1 that is an OBJECTIVE probability.  I will take home a book on
boolean-valued models tonight and see if I can get a better intuition on
this.

At any rate, this does suggest that we don't need to relax the
requirement that the set of propositions with probability 1 is logically
closed in order to model nondeductive reasoning.

-- Joe Shipman