Editing Naive Set Theory (book) (section)

{{Short description|1960 mathematics textbook by Paul Halmos}}{{Infobox book
| image             = NaiveSetTheory.jpg
| author            = [[Paul Halmos]]
| pub_date          = 1960
| caption           = First edition
}}

{{Italic title}}
:''See also [[Naive set theory]] for the mathematical topic.''
'''''Naive Set Theory''''' is a [[mathematics]] textbook by [[Paul Halmos]] providing an undergraduate introduction to [[set theory]].<ref>Review of ''Naive Set Theory'' by H. Mirkil (April 1961), ''[[American Mathematical Monthly]]'' 68 (4): 392, {{doi|10.2307/2311615}}.</ref> Originally published by ''Van Nostrand'' in 1960,<ref name="Rieger"/> it was reprinted in the [[Springer-Verlag]] [[Undergraduate Texts in Mathematics]] series in 1974.<ref>{{cite book | last1 = Halmos | first1 = Paul | title=Naive set theory|year=1974 | series=Undergraduate Texts in Mathematics | publisher=Springer-Verlag |isbn = 978-0-387-90092-6 | doi=10.1007/978-1-4757-1645-0 | mr=0453532}}</ref> 
It is on the list of 173 books essential for undergraduate math libraries.
<ref>{{cite web |url=https://maa.org/the-basic-library-list-the-basic-library-list-maas-recommendations-for-undergraduate-librariesthe-basic-library-list/ |title=The Basic Library List: MAA’s Recommendations for Undergraduate Libraries| date=4 November 2022|access-date=24 May 2025}}</ref>

While the title states that the set theory presented is 'naive', which is usually taken to mean without formal [[axiom]]s, the book does introduce a system of axioms equivalent to that of [[Zermelo–Fraenkel set theory|ZFC set theory]] except the [[Axiom of foundation]].  It also gives correct and rigorous definitions for many basic concepts.<ref name="Rieger">Review of ''Naive Set Theory'', L. Rieger, {{MR|0114756}}.</ref><ref>Review of ''Naive Set Theory'', Alfons Borgers (July 1969), ''[[Journal of Symbolic Logic]]'' 34 (2): 308, {{doi|10.2307/2271138}}.</ref>   Where it differs from a "true" [[axiomatic set theory]] book is its character: there are no discussions of axiomatic minutiae, and there is next to nothing about advanced topics such as [[large cardinal]]s or [[forcing_(mathematics)|forcing]]. Instead, it tries to be intelligible to someone who has never thought about set theory before.

Halmos later stated that it was the fastest book he wrote, taking about six months, and that the book "wrote itself".<ref>{{citation | title=Paul Halmos: celebrating 50 years of mathematics | editor1-first=John H.|editor1-last=Ewing | editor2-first=Frederick W.|editor2-last=Gehring | publisher=[[Springer-Verlag]] | year=1991 | isbn=0-387-97509-8 |at=Interview of Halmos with Donald J. Albers, p.&nbsp;16|url=https://books.google.com/books?id=Xnv5fXu1vFoC&pg=PA16}}.</ref>