I’ve always been a fan of logic puzzles although I would hardly call myself a logician — I actually find them somewhat challenging; but perhaps that’s the point of puzzles, after all.
In Satan, Cantor & Infinity, Smullyan weaves a lengthy fictional narrative into a series of many varieties of logic puzzles — from basic Goodman (always lie / always tell the truth) to very elaborate symbolic logic.
The title and the last chapter of the book share the same name, and it refers to a logic puzzle posed by Georg Cantor (famed mathematician). In this puzzle, Satan allows his denizens to attempt to escape damnation by guessing which number he has pre-selected, chosen from 1 to Infinity. It, among with many others, are imaginatory ways of grasping really elaborate abstract concepts such as “are some infinities bigger than others?”
I will be honest – I did not read this book cover to cover. I read the first part (Goodman principle / knights & knaves) and the better part of the second half, where Smullyman discussed set theory, paradoxes, and infinity. There was a sizable chunk, 40 or 50 pages, that involved a lot of symbolic logic and what appeared to be functional mathematics — I had a hard time comprehending it; in my opinion, it came out of left field and wasn’t introduced very well.
The parts I did read, however, were mostly written well. Many books with logic puzzles like these typically present each one independently, and that was what I had expected of this one as well. Instead, Smullyan has a number of characters that help to tie together the puzzles so that it reads more like a journey or tour instead of enigmatic paroxysms.
There is an assumed requisite amount of mathematical background expected of the reader. Basic knowledge of algebra is a must, discrete math or abstract math would probably be a big help for much of the book (I have had neither). Puzzle enthusiasts may be turned off by the at-times excessive leanings on theoretical math.
I was somewhat familiar with examples such as the Hilbert Hotel (an infinitely large hotel completely filled with an infinite number of guests — what do you do when a new guest shows up? Why, ask the person in room 1 to move into room 2 and tell them to move into room 3, and so on), and I was very loosely acquainted with set theory — but I feel as though I have an improve understanding of set theory after reading this book, particularly Frege, Zermelo, Whitehead and Russell’s work defining axioms through sets. Smullyan did a superb job explaining that part.
I wouldn’t say this is my favorite puzzle book, for reasons cited above, but there were many parts I enjoyed and I don’t regret reading it. Math majors will definitely be more entertained and feel less mentally taxed; non-Math folks should read part I and then skip to Part V, treading lightly the rest of the way.