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As the title suggests, apart from Jech's Set Theory, are there any other noteworthy books or notes which cover forcing through Boolean algebras? All other texts I've seen use the poset approach (Kunen, Weaver, etc).

I'm going through the chapter on Jech's book, and having somewhere else to look for when I get stuck would probably be useful.

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    Having learned forcing from Jech, the Boolean valued model approach is very enlightening, but nearly impossible to actually use. If you look at Jech further down the road, he also falls back to the poset based approach. – 2017-02-14
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    @AsafKaragila yes, I've read elsewhere that applying it is a nightmare, but having read the poset treatment in Kunen, it all seems very mysterious, "this is a name, this is a valuation, (???), here's the independence of CH". Afterwards, Kunen says the Boolean valued model approach is easier to motivate, so I thought checking it out might give me some intuitive insights on the whole thing. – 2017-02-14
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    I don't know about that, actually. – 2017-02-14
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    @AsafKaragila I'm sorry, what do you mean? – 2017-02-14
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    I'm not sure I agree with Kunen on that. – 2017-02-15
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    See https://www.amazon.es/Set-Theory-Boolean-Valued-Models-Independence/dp/0199609160. – 2017-02-15
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    Perhaps you are using Kunen's 1980 book. Though it is a great book, his [2011 rewrite](https://www.amazon.com/Set-Theory-Studies-Logic-Mathematical/dp/1848900503) contains the same material and lots of discussion of finer points added, new exercises and new hints for the old ones, etc. And you can use his [Foundations](https://www.amazon.com/Foundations-Mathematics-Studies-Logic-Mathematical/dp/1904987141) as a companion just in case of need. – 2017-02-15

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Try Rosser's book, Independence Proofs.

Also, once you understand that these methods involve semiregular topologies, look at the section on the relatively prime integer topology in Counterexamples in Topology by Steen and Seebach as well as the embedding of minimal Hausdorff topologies into semiregular topologies in Extensions and Absolutes of Hausdorff Spaces by Porter and Woods. With regard to the latter, you will find that there are three kinds of extensions to H-closed spaces which need not be the same. This will seem somewhat disconnected from set theory. However, in so far as set theory presumably encompasses the "philosophy of the infinite", this three-fold multiplicity seems related to the fact that the free orthomodular lattice on countably many generators can be embedded into the free orthomodular lattice on three generators.

I had been led to these resources by my own curiosity and Jech's presentation of Boolean-valued forcing. If formulating a complete Boolean algebra from a given separative partial order involves topologies formed from regular open sets, then one should be asking questions of what role semiregular topologies play in foundations. One cannot do this without learning about instances of semiregular topologies elsewhere in mathematics.

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    "This three-fold multiplicity seems related to the fact that the free orthomodular lattice on countably many generators can be embedded into the free orthomodular lattice on three generators." How so? (It may be that I'm just lacking knowledge here - I only know of two types of natural extensions to closed H-spaces, the Fomin and Katetov extensions, and I'm unaware of what role if any orhtomodular lattices really play here.) – 2017-02-16