Probability Measure over sentences of ZFC 
Possible Duplicate:
Continuous Strictly Positive Measures on Countable Boolean Algebras 

Let B be the Boolean Algebra of sentences in the language of set theory modulo provable equivalence in ZFC. That is to say, for each sentence $\sigma$ in the language of set theory, let $[\sigma]$ be the set of sentences in the language of set theory provably equivalent to $\sigma$. The set of equivalence classes $[\sigma]$ under the obvious operations is the Boolean Algebra in which we are interested.
Is it possible for there to be a countably additive probability measure $\mu$ on $B$ such that $\mu$ assigns every non-zero element of $B$ strictly positive probability? Must there be such a probability measure $\mu$? (Note: I added the condition of countable additivity when some commenters correctly pointed out that every countable Boolean algebra admits a strictly positive probability measure.) 
 A: This Boolean algebra is atomless (assuming ZFC is consistent).
Namely, let $\varphi$ be a sentence whose equivalence class is $\not=0$.
Then ZFC+$\varphi$ is consistent.  By the incompleteness theorem, ZFC+$\varphi$+$\neg$Con(ZFC+$\varphi$) is consistent as well.
It follows that the equivalence classes of $\varphi\wedge\neg\mbox{Con(ZFC}+\varphi$) and
$\varphi\wedge\mbox{Con(ZFC}+\varphi$) are both below the one of $\varphi$ and both positive.
Luckily, there is just one countable atomless Boolean algebra, namely the algebra of clopen subsets of the Cantor space ($2^\omega$).
This algebra sits inside the measure algebra of the Cantor space (with the product measure coming from the measure that assigns $1/2$ to the singletons $\{0\}$ and $\{1\}$).
So there is your countably additive measure (countably additive with respect to sums that actually exist in the Boolean algebra).  
Edit:  As Emil Jerabek points out, this is actually not the case.  The problem is that the countable BA is not a complete subalgebra of the measure algebra, i.e., there are infinite
subsets of the countable BA that have a sup in the countable BA but that sup does not agree with the sup taken in the big measure algebra.
So, it seems this argument only gives a finitely additive measure on this BA.
