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I am an individual who enjoys independently studying mathematics, who has developed a decent degree of mathematical maturity by means of doing so over the years.

At the moment, I am particularly interested in foundations (logic and set theory, mostly). I use several ‘standard’ treatments of the respective introductory material (e.g., Enderton, Van Dalen, Suppes), but have recently come across books of a somewhat different breed; for instance Fraenkel’s “Abstract Set Theory” and Robert Wolf’s “Tour through Mathematical Logic.”

I find the friendly exposition and “survey” nature of such books very illuminating of the subject matter as a whole, but can’t- for some reason- help but feeling guilty that I’m perhaps not spending the time on the more rigorous texts, such as those mentioned above.

I do feel more confident and aware of the approach and direction of the more standard treatment books after reading corresponding material in the books of latter style; my question to more experienced mathematicians: is this appreciably helping me as an investment in my time, or should I simply buckle down with the heavier resources and postpone ‘the big picture and how it all ties together’ until later?

emacs drives me nuts
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Cyrus
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    Nothing you enjoy is a waste of time. – CyclotomicField Jul 15 '22 at 20:27
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    Aside from plain enjoyment and learning more about the overall landscape of a field, historical context, "big picture", etc. is that the less technical texts tend to be good introductory material you can show students, friends & family, and so on who don't have as much technical background or don't aim to learn about the technicalities in the first place. I found Robert Wolf's "Tour through Mathematical Logic" especially good for this and enjoyed reading it myself (with logic as my area of expertise). – Hayden Jul 15 '22 at 20:50
  • As @CyclotomicField writes, "Nothing you enjoy is a waste of time." I'd modify it a bit, to say nothing you enjoy is a waste of time especially if it involves scholarly or cultural endeavors... and that most certainly includes mathematics. – David G. Stork Jul 15 '22 at 21:09
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    I would recommend both kinds of books because the non-rigorous books, especially books meant for folks in other disciplines, often contain insights that are lost in a rigorous presentation. – John Douma Jul 15 '22 at 21:10
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    I consider Fraenkel's book and Wolf's book very different. Wolf's book is an overview that doesn't go very deeply, mathematically or historically, into any particular topic, whereas Fraenkel's book is one of my go-to books for details rarely found anywhere else (e.g. see footnote 1 on p. 77, see "One might suspect ...." near the bottom of p. 118 through the top third of p. 119; see Theorem 5 on p. 147; see footnote 3 on p. 229), and topics often only very briefly mentioned at all in standard texts (e.g. arithmetic of linear order types, detailed historical notes and references in footnotes). – Dave L. Renfro Jul 15 '22 at 21:23
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    As for "rigorous", I don't see that being lacking in Fraenkel's book, at least not unless you equate "rigorous" with "axiomatic-based and everywhere dense with machine-readable-only symbolic forests". In this sense, Fraenkel's book is written in the "post rigorous stage" as described in Terence Tao's essay There’s more to mathematics than rigour and proofs. – Dave L. Renfro Jul 15 '22 at 21:26
  • @JohnDouma that’s a fantastic point, thanks for your response – Cyrus Jul 16 '22 at 02:31
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    @DaveL.Renfro thank you, too, for your insights. I agree with you regarding both books; and also, admittedly, somewhere became lost in the mindset that “symbolic forests” were the ‘true’ mark of math. Thanks for coaxing me out of that! – Cyrus Jul 16 '22 at 02:32
  • I'd also like to add, looking at things more rigorously does have benefits other than the usual claims about "true math" - after looking at some more rigorous text on math, some nameless concepts/guesses that I had asked people in the past were given names by the text, and I was able to not only clearly communicate about them, but more importantly now I could point to concrete reasons why those ideas worked. – C7X Jul 16 '22 at 06:10
  • Just to be clear regarding my previous comment, I consider as friendly non-rigorous expositions books such as Infinity of the Mind by Rucker, In Search of Infinity by Vilenkin, Playing with Infinity by Peter, Infinity by Lieber, To Infinity and Beyond by Maor, etc. Fraenkel's book belongs to (continued) – Dave L. Renfro Jul 16 '22 at 10:00
  • what I suppose we could call "non-axiomatic rigorous introductory", which would include books such as Theory of Sets by Kamke, The Continuum and Other Types of Serial Order by Huntington, The Theory of Sets and Transfinite Numbers by Rotman/Kneebone, Naive Set Theory by Halmos, Set Theory by Hausdorff, (continued) – Dave L. Renfro Jul 16 '22 at 10:01
  • Cardinal and Ordinal Numbers by Sierpinski, Set Theory by Kuratowski/Mostowski, etc. And then there are books I suppose we could call "axiomatic introductory" (by introductory, I mean forcing techniques are not included, although there might be an introductory overview chapter on forcing at the end), which includes most advanced undergraduate level set theory texts written since the 1960s or so. (continued) – Dave L. Renfro Jul 16 '22 at 10:01
  • I'm not sure I would consider these to be more rigorous. Instead, they simply include an axiomatic development, often at the expense of many of the topics one can find in some of the non-axiomatic books I mentioned. Of course, this is needed if one is going to be a specialist in set theory, but for virtually everyone else who simply uses set theory in a nontrivial way (e.g. proofs by transfinite induction), I think the axiomatic development plays more of a role of "mathematical appreciation" about a certain subject (in this case, set theory) than actual tools and methods that (continued) – Dave L. Renfro Jul 16 '22 at 10:01
  • one would later use in one's own work (unless it's set theoretic topology or some such that requires heavy use of axiomatic set theoretic methods). – Dave L. Renfro Jul 16 '22 at 10:01
  • Given all I've said, I should elaborate on my comment about symbol forests. Although there are some books that go too far for my tastes (Example 1 and Example 2), I didn't express my earlier comment very well. What I meant was that rigor (in the sense of being mathematically sound, not in the sense of being advanced and/or difficult) is not the same thing as naively putting everything into symbolic form (the latter you'd do in formalized mathematics). – Dave L. Renfro Jul 16 '22 at 10:23
  • Regarding my first comment above where I wrote see "One might suspect ...." near the bottom of p. 118 through the top third of p. 119, at the time I wrote this I searched for where I had posted about this in MSE (I remembered having posted something) but I couldn't find anything then. By accident just now I came across what I was thinking of, namely comments I made to this answer. The fact that what I posted were comments is why I couldn't find it before, because google searches tend to ignore MSE comments (but not always, for some reason). – Dave L. Renfro Jul 20 '22 at 16:50

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