The title of Paul Halmos' book Naive Set Theory suggests that it treats set theory naively rather than axiomatically. However, the book constructs set theory from well-chosen axioms and in this sense the book develops set theory axiomatically except that it does not define the notion of a set formally. Also, Russell's paradox is eliminated, which is in general part of a set theory developed in the "naive" way.
As there are several types of axiomatic set theories, I am curious which type of set theory is treated in Naive Set Theory. I would think it is the Zermelo-Fraenckel set theory with axiom of choice. The book assumes the axiom of choice namely in $\S15$.
Any comment is appreciated.
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$\begingroup$As you have surmised, Halmos develops ordinary ZFC from axioms.
Note that the fact that he "does not define the notion of a set formally" is not an exception from doing things axiomatically -- on the contrary, it is the very core of the axiomatic method (from a modern perspective) that you don't bother to attempt to define what the objects you're speaking about are. Instead we have axioms that tell us how those objects behave, and that is all we need to know in order to prove things about them.
The title of Halmos's book is something of a misnormer, at least relative to how the terminology has come to be used. Usually, "naive set theory" means the inconsistent theory assumed by Cantor and the rest of the 19th century, in which every definable collection is a set, leading to Russell's paradox and others. But that is certainly not what Halmos is doing.
What Halmos calls "naive" seems to be simply the fact that he develops set theory without first presenting a theory of formal logic -- so he has explicit axioms for set theory, but the way he reasons from those axioms are by the ordinary quasi-formal English prose that is used in the rest of mathematics too, rather than the usual formal-logic game of symbol sequences and exact mechanical rules for "what is a valid proof".
$\endgroup$ 5 $\begingroup$An interesting point is that Halmos doesn't ever introduce the axiom of regularity, which he alludes to in a cryptic statement (Chapter 2, the "axiom" of specification) about how sets which contain themselves don't seem sensible but might exist--well, they don't if you introduce the axiom of regularity. Probably because it is not necessary in the vast majority of mathematical work, Halmos chose to omit it completely. That is probably another reason Halmos used the term "naive." He didn't even bother to introduce all of ZFC because the text is merely intended to give the working mathematician all of the set theory necessary in normal settings; I think he is merely warning the reader that this book doesn't serve as the basis for an honest-to-goodness course in set theory, as that isn't the stated purpose for the book. In short, the book develops ZFC minus regularity.
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