Set Theory
goodmath | May 19, 2007 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.
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…
goodmath | May 16, 2007 So, what's set theory really about?
We'll start off, for intuition's sake, by talking a little bit about what's now called naive set theory, before moving into the formality of axiomatic set theory. Most of this post might be a bit boring for a lot of you, but it's worth
being a bit on the pedantic side to make sure that we're starting from a clear basis.
A set is a collection of things. What it means to be a member of a set S is
that there's some predicate PS - that is, some way of describing things via logic - which is true only for members of S. To be a tad more formal, that means that…
goodmath | May 14, 2007 While I've been writing about the Surreal numbers lately, it reminded me of some of the fun of Set theory. As a result, I've been going back to look at some old books. Since I've been enjoying it, I thought you folks would as well.
Set theory, along with its cousin, first order predicate logic, is pretty much the
foundation of nearly all modern math. You can construct math from a lot of
different foundations, but axiomatic set theory is currently pretty much the dominant approach. (Although Topoi seem to be making some headway...)
There's a reason for that. Set theory starts with some of…