Naive set theory is fun, and as we saw with Cantor’s diagonalization, it can produce some incredibly beautiful results. But as we’ve seen before, in the simple world of naive set theory, it’s easy to run into trouble, in the form of Russell’s paradox and a variety of related problems. <\/p>\n
For the sake of completeness, I’ll remind you that Russell’s paradox concerns the set R={ s | s ∉ s}. Is R∈R? If R∈R, then by the definition of R∉R. But by definition, if R∉R, then R∈R. So R is clearly not a well-defined set. But there’s nothing about the form of its definition which is prohibited by naive set theory!<\/p>\n
Mathematicians, being the annoying buggers that they are, weren’t willing to just give up on set theory over Russell’s paradox. It’s too beautiful, too useful an abstraction, to just give up on it over the self-reference problems. So they went searching for a way of building up set theory axiomatically in a way that would avoid problems by making it impossible to even formulate the problematic statements.<\/p>\n
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To some extent, from a modern viewpoint, it can seem a little silly. The problems of naive set theory involve self-reference issues. And most modern math geeks have absorbed the fact that self-reference is inevitably<\/em> a problem. That’s the simplistic understanding that most of us have of Gödel’s incompleteness theorem: that in any formal system, whether it’s set theory or something else, if it’s expressive enough to be a complete system, there’s some<\/em> way of forcing it to allow the formulation of self-reference statements that introduce problems like Russell’s.<\/p>\n There’s two things to remember when you think that:<\/p>\n\n
\nThe late 19th\/early 20th century mathematicians didn’t know about incompleteness,
\nand believed that you could create a perfect universal mathematics. So trying
\nto form a perfect axiomatization of set theory was a very natural thing for them
\nto try to do.<\/li>\n
\nform. They’re ubiquitous. Naive set theory is a beautiful idea, but isn’t a
\nparticularly well-formed mathematical theory. Axiomatizing it, so that it has
\na solid formal basis is<\/em> a useful thing to do. It means that until you
\npush it out to its limit, that it’s a well-formed valid mathematical theory. So
\neven recognizing that we can’t complete avoid the kinds of issues that permeate
\nnaive set theory, we can reduce<\/em> their impact, and most of the time,
\navoid them entirely.<\/li>\n<\/ol>\n