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Abstract
This solid volume with the ominous black cover and eerie glowing disc lettered with inscrutable strings of characters such as “DN d Q, ” “LSDJ d, ” and “L2LDJQ ± ” is the fruit of over 30 years of Ross Brady’s logical labours. And it is worth the wait. Since the early 1970s, Brady has been interested in strong theories of sets, in which the comprehension axiom, to the effect that every open sentence φ(x) (in the language of first order logic with the binary predicate ‘∈ ’ for membership) determines some set. That is, theories in which the comprehension axiom (ca) (∃y)(∀x)(x ∈ y ↔ φ(x)) holds completely unrestrictedly. This means that according to such theories, there is a universal set (take φ(x) to be x = x, or perhaps (∃z)(x ∈ z) if we have no identity in the language) and each set a has a complement (take φ(x) to be x � ∈ a). This is not a cumulative hierarchy of sets familiar to us in set theories in the vicinity of zf. This is Frege’s (and our) naïve notion of sets, where every “property ” determines an extension. This causes trouble as is well known from the paradoxes of Burali-Forti, Russell and Curry. We can form troublesome extensions: (ca) assures us that there
