Quote by Demystifier
So basically, Ballentine does not believe in the quantum Zeno paradox because he does not believe in collapse.

I don’t have the Ballentine’s book (and Ballentine’s no relative of mine :) ), so the following is mostly based on his Comment quoted by DevilsAvocado in post 7 in this thread.
It does not look like “Ballentine does not believe in the quantum Zeno paradox”, he says “The quantum Zeno effect is not a general characteristic of continuous measurements.” I understand this as follows (and I may be wrong): the quantum Zeno paradox exists or does not exist depending on the specific characteristics of the actual measurement.
Furthermore, the authors of the article he (mildly) criticizes write (
http://tf.nist.gov/general/pdf/905.pdf ) in the reply to his Comment: “Ballentine states that “collapse of the wave function” is not necessary to quantum mechanics”. We agree. However, we feel that the explanation given in our article, which invokes von Neumann’s “collapse” postulate, is useful for giving a simple explanation of our experiment.”
So it looks like there is agreement that collapse is not necessary.
Quote by Demystifier
But it seems that he does not understand that effective collapse can almost be "explained" by modern understanding of decoherence, and it seems to be because he is not aware of the importance of decoherence.
The reason for such a suspicion comes from another part of his (otherwise great) book:
Sec. 9.3  The Interpretation of a state vector
Subsection  The measurement theorem for general states
After Eq. (9.13) he writes:
"The terms with alpha_r1 notequal alpha _r2 indicate a coherent superposition of macroscopically distinct indicator vectors ... It is clear that the nondiagonal terms in (9.13) cannot vanish ..."
But it seems to me that someone who were familiar with decoherence would immediately recognize that they CAN vanish, due to decoherence. Nevertheless, he does not even mention decoherence  at this place at which a "Modern Introduction" to QM should.
Any comments?

I cannot be sure Ballentine knew about decoherence in 1998, when his book was published (he knew about it in 2005 though :) 
http://pra.aps.org/abstract/PRA/v72/i2/e022109 ), but in the text you quoted he seems to argue that collapse is, strictly speaking, incompatible with unitary evolution, and I believe he’s right. Furthermore, it seems there is no positive experimental evidence of collapse (see the quote from Schlosshauer’s article at
http://www.physicsforums.com/showpos...0&postcount=41 ).
You mentioned decoherence. But, as far as I understand, decoherence is a result of influence of environment, i.e. of something external with respect to the experiment, so one can talk about “effective collapse”, but that does not contradict the fact that, strictly speaking, there is no collapse (otherwise unitary evolution is wrong). So I think I fully understand Ballentine’s thrust against collapse.