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THE book for studying Boolean-valued models of set theory
If you are studying to become a set theorist these days (2001 or later), you will probably learn forcing by way of partial orders. This is the quickest way to go, and for most purposes, the most useful in applications. It's also the approach that is taken in Kunen's book on set theory, which is still pretty much the standard textbook.

Nonetheless, you may one day find that you need the Boolean algebra approach. This was the approach developed by Scott and Solovay after Cohen's somewhat inscrutable approach to proving the consistency of not-CH had been adequately digested. The Boolean algebra approach isvery elegant and algebraic, and the theorems are often bettermotivated than they are in the partial orders approach.

When I was a graduate student, I studied both approaches,but eventually forgot much of what I learned about Boolean-valued models because in practice I always relied on the other approach.

Recently, however, in some research I've been doing in an extension of ZFC in an extended language, I found that some of the usual assumptions one can make in doing forcing over ZFC models were no longer applicable in the new setting. Without going into the details, the consequence was that I had to do forcing over non-wellfounded models of ZFC and examine the properties of the resulting forcing extension, perhaps iterating the process omega many times. The only way to do this has been to work with Boolean-valued models M^B of the non-wellfounded ground model M, prove the desired properties within M^B, then collapse with a generic ultrafilter, and go on to the next model.

Well, after that long-winded introduction, my point is this: Bell's treatment of Boolean-valued models is outstanding. I have several of Bell's books and his talent as an expositor is his relentless attention to detail. He does no hand-waving. If you need to face the details of Boolean-valued models, Bell's approach is the right way to go.

In the first chapter he develops the theory enough to prove that all ZFC axioms hold in V^B. In Chapter 2 he shows how to do independence proofs in Boolean valued models -- illustrating with CH and developing the usual results about chain conditions and distributivity, never once working with a 2-valued forcing extension. The third chapter reveals some of the elegance of the Boolean algebra approach in its development of the proof of the consistency of not-AC using group actions. Chapter 4 shows how to get the usual results about forcing involving 2-valued models by considering (V^B)/U, where U is a generic ultrafilter. Chapter 5 is a special chapter about cardinal collapsing, introduced because the Boolean algebras introduced before always preserved cardinals and cofinalities. Finally, chapter 6, which was added in the 1985 edition, treats iterated forcing. This chapter contains details that appear nowhere else and are very handy if you need to deal with such things.

As ever, Bell has done a thorough job in his treatment of this subject. It is the right reference for Boolean-valued models of set theory. ... Read more

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Chapters: Sigma-Algebra, Boolean Satisfiability Problem, de Morgan's Laws, Propositional Calculus, Logical Conjunction, Logical Disjunction, Logical Connective, Boolean Ring, Majority Function, Exclusive Or, Sheffer Stroke, Negation, Boolean Function, Propositional Formula, Boolean Algebras Canonically Defined, Introduction to Boolean Algebra, Laws of Form, Canonical Form, Karnaugh Map, Truth Table, Interior Algebra, Relation Algebra, Boolean Logic, Boolean-Valued Model, Bent Function, Bitwise Operation, True Quantified Boolean Formula, Field of Sets, Boolean Prime Ideal Theorem, Minimal Negation Operator, Functional Completeness, Boolean Data Type, Quine-mccluskey Algorithm, Complete Boolean Algebra, Residuated Boolean Algebra, Logic Alphabet, Free Boolean Algebra, Two-Element Boolean Algebra, Zhegalkin Polynomial, List of Boolean Algebra Topics, Logical Nor, Shannon's Expansion, Stone's Representation Theorem for Boolean Algebras, Monadic Boolean Algebra, Parity Function, Logic Redundancy, Davis-putnam Algorithm, Petrick's Method, Robbins Algebra, Circuit Minimization, Evasive Boolean Function, 2-Valued Morphism, Algebraic Normal Form, Implicant, Boolean-Valued Function, Absorption Law, Logical Matrix, Correlation Immunity, Conditioned Disjunction, Implication Graph, Consensus Theorem, Lupanov Representation, Boolean Domain, Modal Algebra, Symmetric Boolean Function, Boolean Conjunctive Query, Derivative Algebra, Reed-Muller Expansion, Chaff Algorithm, Boolean Expression, Switching Function, Product Term. Source: Wikipedia. Pages: 454. Not illustrated. Free updates online. Purchase includes a free trial membership in the publisher's book club where you can select from more than a million books without charge. Excerpt: In propositional logic, a propositional formula is a type of syntactic formula which is well formed and has a truth value. If the values of all variables in a propositional formula are given, it determines a...More: http://booksllc.net/?id=1557634 ... Read more

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Chapters: Group, Boolean Algebra, Field, Ring, Ideal, Monoid, Semigroup, Algebraic Structure, Kleene Algebra, Magma, Lattice, Special Classes of Semigroups, Module, Infrastructure, Biordered Set, Semigroup With Involution, Pseudo-Ring, Semiring, Semigroup With Two Elements, Cancellative Semigroup, Complete Heyting Algebra, Moufang Polygon, Action Algebra, Exponential Field, Mv-Algebra, N-Ary Group, Monogenic Semigroup, Nowhere Commutative Semigroup, Completely Regular Semigroup, Trivial Semigroup, Empty Semigroup, Lindenbaum-tarski Algebra. Source: Wikipedia. Pages: 253. Not illustrated. Free updates online. Purchase includes a free trial membership in the publisher's book club where you can select from more than a million books without charge. Excerpt: In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity and invertibility. While these are familiar from many mathematical structures, such as number systemsfor example, the integers endowed with the addition operation form a groupthe formulation of the axioms is detached from the concrete nature of the group and its operation. This allows one to handle entities of very different mathematical origins in a flexible way, while retaining essential structural aspects of many objects in abstract algebra and beyond. The ubiquity of groups in numerous areasboth within and outside mathematicsmakes them a central organizing principle of contemporary mathematics. Groups share a fundamental kinship with the notion of symmetry. A symmetry group encodes symmetry features of a geometrical object: it consists of the set of transformations that leave the object unchanged, and the operation of combining two such transformations by performing one after th...More: http://booksllc.net/?id=19447 ... Read more

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Boolean Algebras Canonically Defined. Boolean algebra (structure), Boolean ring, Boolean logic, Complete Boolean algebra, Abstract algebra, Universal algebra, Free Boolean algebra, Boolean- valued model, Algebraic structure, Foundations of mathematics ... Read more

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A simple, straightforward guide to digital circuitry for students and hobbyists. Lisa A. Doyle has revised this popular reference to cover the latest memory chips and packages, microprocessor basics, alphanumeric codes, demultiplexers, LCDs and much more. In addition, the best parts of the first edition have been retained: thorough descriptions of digital codes, conversions, and number systems; explanations of the basics of logic; and examples of the application for digital circuitry in TVs, stereos, communications, automobiles and computers. ... Read more