His first proof of this is set out in eleven numbered steps, with duly noted uses of the Leibniz axiom and modus ponens. Similar examples of formality can be found on pages , , and elsewhere in volume 2, as well as volume 1. The preface of volume 2 explains that the book weaves a middle path between those set theory books that merely build set-theoretic tools for use elsewhere and those that aim at research in set theory.
But I question this assessment. Many of the topics constituting what I take to be the beginnings of the subject appear only very late in the book. Perhaps it was the elaborate formal development of the early theory that has pushed this basic part of set theory to the end of the book. Furthermore, many other elementary topics, which one might expect to find in a set theory text aimed in part at graduate students, do not appear in the text at all. Large cardinals are not mentioned beyond the inaccessible cardinals.
Certainly ultrapowers are one of the most powerful and successful such tools, whose fundamental properties remain deeply connected with logic. Because his account of forcing is based on countable transitive models, some set theorists may find it old-fashioned.
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