These notes are not a systematic “Introduction to Set Theory”. I intend them as a
blend of history, intuition, and exposition, with an occasional dash of philosophy.
This is a good place to put in a word or two about philosophy. Two stances dominate for traditional mathematics. The platonist position says that the universe of mathematical objects really exists “out there”, in some sense1. The formalist position says that it’s all a “game with symbols” (governed by precise rules).
Mathematical philosophy is a two-party system in same way as US politics. The full panoply of positions takes up many volumes. We have intuitionism and constructivism, which discard large parts of mathematics as meaningless. Some suggest that mathematical entities are mental objects, things going on inside our brains. Others suggest mathematics is a social construct, akin to a legal system.
Even within platonism and formalism, you’ll find all sorts of nuances and disputes. So-called strong platonism says there is a unique universe containing all mathematical objects. But some prefer a multiverse zoo of mathematical universes.
Formalism is closely associated with Hilbert, but as one writer warned: “We note at once that there is no evidence in Hilbert’s writings of the kind of formalist view suggested by [the intuitionist] Brouwer when he called Hilbert’s approach ‘formalism’.”
For myself, I adopt an incoherent mish-mash of (strong) platonism, formalism, and intuitionism. Platonism provides by far the easiest approach to discussing mathematics. “2ℵ0 = #ℝ because there is a bijection between the set of all infinite bitstrings and the set of real numbers.” It’s pleasant to think of this assertion as just like “The stars on the US flag are in 1–1 correspondence with the US states.”
I know what you’re thinking. “Has he totally forgotten about Russell’s paradox? Has he even studied axiomatic set theory? Does he really believe that 2ℵ0 has a single correct value, even if we may never know what it is?”
I’ll admit that oxygen can get a little scarce in the more rarified parts of set theory. When the air becomes hard to breathe, I take refuge in the formalist gambit, and just claim it’s all a game with symbols. If you ask me what it really means to say that 2ℵ0 = #ℝ, my go-to response: “It’s a theorem of ZFC.” My commitment to platonism is weak.
Syntax (“games with symbols”) seems philosophically safer than semantics (“wrangling infinities”). But even a hard-core formalist has to allow a role for mathematical intuition. Golden retrievers do not understand “Such-and-such is a theorem of ZFC”. At the outermost level, our innate mathematical perceptions take charge.
Let’s see how this plays out with the Continuum Hypothesis (CH). For a constructivist, CH is meaningless, and always was2. For a formalist, CH was resolved by the independence results of Gödel and Cohen: you can’t prove it, you can’t disprove it, there’s nothing more to say3. For a (strong) platonist, 2ℵ0 has a definite value, but alas!—we may never know what it is. Of the three, I prefer the formalist answer.
But usually, when I’m swimming inside ZFC, it’s easier and more fun to imagine that the ZFC universe really exists. I pretend that the axioms are laws of this universe. I encountered ℕ, and the countability of the rationals, and Cantor’s diagonal argument, and lots of stuff like that, long before I learned about ZFC. It made sense at the time, and I’m not ready to throw that away. I trust my intuitions about the platonic universe.
Like I said, an incoherent mish-mash. But very tasty, when sampling meta-mathematical results. Prime example: Cohen’s model for ZFC+¬CH. One marvels at this infinite and intricately constructed mathematical object. The axiom system ZFC is front and center, lending matters a formalist tinge. But the elaborate reasoning establishing the properties of the model—you’ll want to treat that in a platonist, naive-set-theory style. The model feels real.
We can have our cake and eat it too. Just hand-wave at the very end:
Of course, all this reasoning could be written up formally as a theorem of ZFC, if you paid me enough!
(I expounded on my philosophy in my conversation with John Baez.)
[1] Paul Bernays introduced this term in his 1934 lecture, “On Platonism in Mathematics”. He wrote: “Since this tendency [to regard abstractions as real] asserted itself especially in the philosophy of Plato, allow me to call it ‘platonism’.”
[2] A quote from Gödel’s essay, “What is Cantor’s Continuum Problem?”: “First of all there is Brouwer’s intuitionism, which is utterly destructive in its results…Cantor’s conjecture itself receives several different meanings, all of which, though very interesting in themselves, are quite different from the original problem.”
[3] Although you can still ask if CH can be proved or disproved in other axiom systems, like extensions of ZFC.
Naive Set Theory
The term “Naive Set Theory” comes from the book of that name by Paul Halmos. As he says, “In set theory ‘naive’ and ‘axiomatic’ are contrasting words.” (I always thought Halmos coined the phrase, but it turns out that Hausdorff used it in the first edition of his Grundziige der Mengenlehre (Basics of Set Theory) in 1914.)
Modern axiomatic set theory expresses the axioms using first-order predicate logic. Halmos tries to have his cake and eat it too, one reason I prefer other books. That is, he avoids this formal apparatus, while still waving his hands at an axiomatic treatment. I think this leads to more confusion than clarity.
You can do lots of interesting set theory before turning to an axiomatic approach. Hey, Cantor did! Munkres’ Topology puts it this way:
We adopt, as most mathematicians do, the naive point of view regarding set theory. We shall assume that what is meant by a set of objects is intuitively clear, and we shall proceed on that basis without analyzing the concept further.
“But what about the paradoxes?” Well yes, we can’t just ignore them. But it’s easy to avoid them with a little care.
For an introductory textbook, I recommend Kaplansky, Enderton, and Chapter 1 of Munkres:
Irving Kaplansky. Set Theory and Metric Spaces, 1972.
Herbert B. Enderton. Elements of Set Theory, 2nd edition, 1977.
James Munkres. Topology, 2nd edition, 2014.
Diagonal Argument 