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.)