Is the style of _Scott 1967_ outdated in discussing continuum hypothesis in a probability space? Here is the article I am considering: Scott, D. Math. Systems Theory (1967) 1: 89(see here), an article named A proof of the independence of the continuum hypothesis.
He shows that CH is not true in a model of set theory--something like a special probability measurable space $(\Omega,\Sigma, P)$ where $\Omega=[0,1]^I, I>\omega_1$.
Of course on that time the methods such as forcing was not written the same as that of today and therefore the style of the article may be out of fashion.
So here are my 3 questions


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*Is this presentation of this article out of fashion?

*Some people I know thought this article was quite confusing and is not helpful. Is it really a confusing and strange article? If so, why is it strange?

*Is the such an article misleading and inappropriate for beginners of set theory to study?


Just give me some ideas so I would be assured.
 A: Scott's article is trying to give an intuitively accessible presentation of the independence of CH. To do this, he makes a few choices:


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*The structure he produces, in which CH fails, is not a model of ZFC, but rather a model of a theory which just treats objects of "type 2 or lower" - numbers, reals, and functionals. So the statement "He shows that CH is not true in a model of set theory" is not actually correct.

*Rather than develop forcing as a general method, he focuses on presenting a single "natural" example of a forcing extension - the objects in this structure don't come from names in the usual sense but rather "random variables."

*The structure he produces in the end is Boolean-valued, allowing him to sidestep the complexity of building a generic filter (and not have to restrict attention to countable models).
A cursory examination of the literature will show that the first two bulletpoints are quite unusual: presentations of forcing (like Kunen) tend to build models of ZFC (or its large finite fragments), and focus on the general method rather than giving precedence to any specific model. So the answer to your first question is definitely affirmative in this regard.
The third bulletpoint, however, winds up being more significant. We can think of the first two bulletpoints as aimed at non-logicians (and recall that Scott's paper appeared in the journal "Mathematical Systems Theory," not a logic-specific journal - see also the very end of the conclusion of the paper). The third, however, is a deep observation: that forcing, originally developed in terms of partial orders, can be recast in terms of Boolean-valued models. This was the real innovation of Scott and Solovay here, and is definitely an important part of modern forcing. While many (most?) texts introduce forcing via partial orders - Jech's book being an exception here, as I understand it - the Boolean models approach becomes quite useful in proving general results about forcing, even if one mainly thinks of forcing extensions in terms of partial orders (as I do). So the third bulletpoint is a fundamental aspect of forcing, and in that regard Scott's article certainly isn't "out of fashion."
Pedagogically, the relevant question here I think is:

Is Scott's focus on a simpler kind of theory and on a distinguished natural model helpful?

This is necessarily somewhat subjective, but I think it is not. Let me point out two serious drawbacks:


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*The first bulletpoint makes it extremely hard to use this approach to prove a genuine consistency-over-ZFC result: one would have to argue that the "low-level" theory one is using captures exactly the restriction of ZFC to that level, and this is in general a very hard task.

*Names, though a bit more technical, are also incredibly flexible objects. Even if we view the random variables approach as more concrete (personally I don't - I actually find names easier to think about!), it's much more limiting.
Basically, my take is: Scott's article is a very nice proof that CH is independent from a certain theory, and it does an excellent job of building interest in forcing. However, the exposition he gives is specific enough that it lacks the full power and flexibility of forcing; if one is really interested in forcing then one needs to go further, and Scott's article won't replace the basics. It may make the Boolean models approach easier to learn, but I'm not sure that's the case, and since the partial orders approach is almost always nicer for cooking up forcing extensions (the Boolean models approach shines most in the analysis of those models) you should really learn that one as well (I personally strongly advocate learning and mastering it first).

Of course one can work out how to generalize Scott's argument, but at that point one is really just laboriously redeveloping forcing via Boolean algebras, and should really just learn the full approach at once.
