Srivastava on Naive Set Theory

In a previous book note, and also in the Appendix to the Guide, there is a review of Shashi Mohan Srivastava’s earlier A Course on Mathematical Logic. Parts of that book, as I say, could make useful supplementary/revision reading. So I was interested to see that he now has another short introductory text, An Introduction to Naïve Set Theory and Its Applications (Springer, 2024).

What counts as “naive” set theory according to Srivastava? Quite a lot; basically, it’s set theory for applications, while not worrying too soon about a formal axiomatization of ZFC. Except that Srivastava is wedded to the idea that a set, properly speaking can only contain other sets as members. After giving a ZF-style extensionality axiom, he writes

Consider the collections of all boys of a class and that of all girls in the class. Are these two collections sets? Note that these two collections, say A and B, respectively, contain the same sets because none of them contain any set. But A and B are not the same collections. Hence, A and B do not satisfy the extensionality axiom and so are not sets.

This seems off-beam to me. Sets (including in mathematical applications!) are usually allowed to have non-sets as members, and the corresponding natural set theory allows urlements. An alert student reader should be puzzled!

Things do go quickly from this perhaps inauspicious(?) start. We are looking at ‘Some more applications of Zorn’s Lemma’ by p. 28. But standing back from the details, this means that, as I see it, the book falls between two stools. On the one hand, it isn’t a replacement for the sort of gentle introduction to elementary applicable set theory which we get in e.g. the opening chapter of Munkres’s topology text (or in the opening chapters of Tim Button’s contribution to the Open Logic project). On the other hand, it isn’t a replacement for a fully-developed entry-level text like Enderton’s classic (which more illuminatingly interweaves “naive” informal set theory with a formal axiomatization).

Readers who have learnt some set theory from elsewhere might be intrigued, though, by Chapter 5 which gives an approachable treatment of the Banach Tarski “paradox”. (So that might get a mention in the next iteration of the Guide!)

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