Gracious Living

The Zermelo-Fraenkel Axioms for Sets
October 29, 2010, 13:17
Filed under: Math, Set Theory | Tags: , ,

I’m going to change tacks a bit here, leave topology behind for a second, and talk about sets from a less naive, more higher math perspective.  An axiom, by the way, is an assumed truth or defining property.
Most people are at least a little bit familiar with Euclid’s axioms for geometry; without actually saying what points or lines or planes are, they define (more or less) the essential relationships between them, and the true statements of geometry are all logical consequences of the axioms.

Table, Chair, Beer Mug

A statement of geometry?

Points, lines, and planes are supposed to be atomic, not relying on any previous definitions, so that their meanings are entirely given by the axioms, and not our preconceived notions of what they “look like.”  In the immortal words of David Hilbert, we must always be able to replace the words “point,” “line,” “plane” by “table,” “chair,” “beer mug” and still have valid proofs.

Well, the Zermelo-Fraenkel axioms (or ZFC; the C stands for “choice,” the last axiom) do essentially the same thing for set theory, with the ideas “set” and “element” taken as atomic.  So we can think of “sets” as being a kind of thing and “is an element of” as being a kind of “relation” between two sets, but other than that we shouldn’t be using the standard “collection-of-things” visualization that sets were first introduced with.  Rather, the axioms will formally give us the idea that “piles-of-things” model so well.  We’re also going to go forward without “things” to pile: all sets will be sets of other sets, and we only know that a set exists if we can build it from the axioms.

Below the jump, I introduce the axioms.  They’re usually given as sentences of formal logic, using mathematical symbols, but I’d rather say them in plain English.

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October 25, 2010, 13:38
Filed under: Math, Set Theory | Tags: , ,

A relation from A to B is just a subset of A\times B.  For example, R=\{(a,b)\in\mathbb{R}\times\mathbb{R}:a^2=b^2\} is a relation from \mathbb{R} to \mathbb{R} that includes, say, (0,0),(4,4),(4,-4),(-4,4),(3,-3), and so on.  We generally write aRb instead of $(a,b)\in R$.  Except in very few circumstances, we actually only care about three kinds of relations.

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October 24, 2010, 17:31
Filed under: Math, Set Theory | Tags: , ,

This will probably be review for a lot of readers, but I just want to make sure we’re all on the same page before we really start.  These terms get used a lot a lot in higher math.  In the 1900’s, sets were seen by luminaries like Bertrand Russell as being the salvation of math, since you could describe basically any mathematical structure in terms of sets.  Nowadays, mathematicians tend to believe that the “guts” of math really aren’t that important, and category theory, which instead describes the relations between structures, has become more foundational.  By the way, category theory is just about the coolest possible thing, but its coolness only really becomes apparent once you have a bedrock of examples to work with.  I’ll start talking about it much later.  For now, let’s talk about sets.

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