FOM: Inductive reasoning

[Stephen G Simpson]

Joe Shipman Wed Jan 19 14:37:19 2000 writes:

 > Technical question here for logicians: in the Boolean-valued model
 > approach to independence proofs, is there ever a way to "project" the
 > algebra of "truth values" into the probability space [0,1] so that
 > (some) statements with intermediate truth values can be assigned a
 > probability strictly between 0 and 1in a consistent way?

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.

 > 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.

-- Steve