Naive Set Theory
by Paul R. Halmos · first published 1960
83% recommend14% neutral3% critical
Why HN recommends it
Short, readable, yet rigorous introduction to axiomatic set theory (ZF, the axiom of choice and its equivalents); suited to self-learners who want set theory notation and proof practice before further math.
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…deep (much deeper than the idea of a fixed point) -- that's why it took someone until the early 20th century to formalize it. For an awesome (and mind-blowing) explanation of the paradox, see Halmos' Naive Set Theory (botom of page 6): http://sistemas.fciencias.unam.mx/~lokylog/images/stories/Al...
on “Russell's paradox and the Y combinator” ↗…(Kiselev) * Calculus Made Easy (Thompson) * How to Count without Counting (Niven) * Introduction to Probability Theory (Hoel) * The Little Schemer (Friedman) The proceed to more advanced texts like: * Naive Set Theory (Halmos) * Linear Algebra Done Right (Axler) * Geometry Revisited (Coxeter) * Infinitesimal Calculus (Keisler) * Concrete Mathematics (Graham) * Information Theory, Inference and Learning (MacKay)…
on “How I faced my fears and learned to be good at math” ↗Set theory: You should definitely look at Naive Set Theory by Halmos. Despite the name, it's totally rigorous development of ZF set theory with discussion on the meaning and equivalent statements of AC.
on “Ask HN: Math books like SICP?” ↗…https://www.zuj.edu.jo/download/linear-algebra-2nd-edition-k... Halmos, Finite Dimensional Vector Spaces George E. Forsythe and Cleve B. Moler, Computer Solution of Linear Algebraic Systems Paul R. Halmos, Naive Set Theory, Van Nostrand, Princeton, NJ, 1960. More has been done since this book, but this book is a gorgeous introduction to axiomatic set theory. So, even people who want to dig into the latest work would do…
on “Ask HN: Older textbooks/papers you consider classics still worth studying today?” ↗"Naive Set Theory" by Paul R. Halmos.
on “Ask HN: Best Mathematics book for complete noobie?” ↗…for quantification[8], which combine to create first order predicate logic[9]. From their you can either go to second-order logic or to set theory in order to begin talking about collections of things. Naive Set Theory[10] is a good introductory book, although you can jump straight in to ZFC set theory[11] for an axiomatic approach. Relational algebra, which will be familiar in a loose sense to anyone who has ever…
on “Ask HN: How to improve my abstract thinking?” ↗Here's my list: Basic 1. Chapter Zero (Carol Schumacher) 2. Naive Set Theory (Paul Halmos) Linear Algebra 1. Finite-Dimensional Vector Space (Paul Halmos) 2. Linear Algebra Done Right (Steven Axler) Real Analysis 1. Real Mathematical Analysis (Charles Pugh) 2. Introduction…
on “Ask HN: Good Reading/Immersion in Mathematics” ↗…writing in mathematics, it is not true that unclear writing is glorified over clear exposition. To the contrary, authors like Paul Halmos are rightly celebrated as great expositors, e.g. see his books like Naive Set Theory, Finite Dimensional Vector Spaces, Measure Theory etc. and his expository writing such as "How to Write Mathematics" and "How to Talk Mathematics": [PDF] http://www.math.uh.edu/~tomforde/Books/Halmos-How-To-Write.p...…
on “The pitfalls of jargon and the damage it does to mathematics” ↗…linear algebra book or resource. I never watched them, but I’ve heard that Gil Strang’s MIT 18.06 lectures are good. http://www.youtube.com/course?list=ECE7DDD91010BC51F8 For #3, try to find Halmos’s book Naive Set Theory. It has a very nice and accessible explanations.
on “What are imaginary numbers?” ↗…freely available books by Robert Ash: http://www.math.uiuc.edu/~r-ash/ - in particular, his misleadingly-named "Complex variables" (with W.P. Novinger) is a short, rigorous book on complex analysis. "Naive Set Theory" by Paul Halmos is now available for something like $12. (I have not read the following books.) For number theory, maybe the book by George Andrews? It's very elementary and very cheap, and looks top…
on “Ask HN: Exploring math without a teacher?” ↗…Theory" by Hodges "Reinforcement Learning: An Introduction" by Sutton and Barto "Neuro-Dynamic Programming" by Bertsekas and Tsitsiklis "Set Theory An Introduction To Independence Proofs" by Kunen "Naive Set Theory" by Halmos
on “Bret Victor's Bookshelf” ↗…names (so you can recognize, write and pronounce them without being confused) -- just learn the ones that show up in your reading, and (2) set theory notation plus some basic set theory (read Halmos' Naive Set Theory book up to the point that it becomes confusing). Others have mentioned it, but be aware that it is common for one field to have multiple notation conventions. Often the notations originate from seminal…
on “Ask HN: How can I learn to read mathematical notation?” ↗…independently. For people just starting out and finding this study guide pretty overwhelming, let me re-emphasize two of Dr. Smith's suggestions for starting points: Smullyan's First Order Logic and Halmos's Naive Set Theory. These two books will get you to the same depth of understanding that 95% of working mathematicians need or apply in their day-to-day work, and it's a fact that 95% of modern mathematical theories are…
on “Teach Yourself Logic: A Study Guide [pdf]” ↗…Churchill -written for engineers- is a very down to earth introduction with lots of examples and applications to resolution of integrals and PDEs. In the case of set theory I cannot recommend enough Halmos' "Naive set theory". It's completely rigorous yet easy to read, and lays out all the basics of ZF, constructing both ordinals and cardinals. For probability Khinchin has some very thin fun books, written for some classes…
on “Ask HN: Mathematicians, what textbooks are best for learning these math topics?” ↗The Little Schemer by Friedman and Felleisen, and Naive Set Theory by Paul Halmos
on “Ask HN: What, in your opinion, are the greatest and most useful textbooks?” ↗