Today's basics topic was suggested to me by reading a crackpot rant sent to me by a reader. I'll deal with said crackpot in a different post when I have time. But in the meantime, let's take a look at axioms.
What is an axiom?
If you want to do any kind of formal or logical reasoning, or any kind of inference, you need to start with some set of known facts. There is simply no way of performing inference starting from absolutely no knowledge. Axioms are the set of known facts that are accepted as
basic primitive unproven facts: all proofs are ultimately built upon the inference rules of some logic combined with an initial set of axioms.
In math, we tend to try to find minimal sets of axioms: we prefer to take as little for granted as possible, and then build from that basis using logical inference; but sometimes we'll go off on a philosophical vein for a while, either denying the necessity for any unprovable axioms (Descartes), or freely accepting axioms that seem reasonable, and seeing where they lead (Chaitin).
For example, in basic planar Euclidean geometry, we typically start with some basic math, and five geometric axioms: all other statements in Euclidean planar geometry are supposed to be provable (up to Gödel's limits) starting from these five axioms:
- Given any two points, they can be joined by exactly one line.
- Given any finite, non-zero length line segment, it can be extended infinitely into
exactly one line.
- Given any line segment, there is exactly one circle with one endpoint of the segment as the center, and with the other endpoint on the circle.
- All right angles are equivalent modulo translation, rotation, and mirroring.
- Given a line l and a point p which is not on l, there is exactly one line that passes through
p but never intersects l.
The last one of those is really the interesting one, because it's the one which doesn't really
look like an axiom. Throw out any of the others, and you get an incomplete or inconsistent
geometry; throw out the fifth one, and you get valid geometries that are just different.
Being a tad more fundamental, there are a set of 9 axioms that form the basis of the ZFC system
of math - that is, Zermelo-Fraenkel set theory with the axiom of choice. Most modern math is built on the ZFC axioms. One thing worth pointing out about the ZFC axioms is that despite the fact that I just said that there are 9 axioms, ZFC is actually an infinite set of axioms: depending on the exact presentation, at least one of the ZFC axioms is actually an axiom schema: a template for an infinite series of related axioms. (Any complete axiomatization of set theory - and therefore any version of math built on set theory - must be infinite.)
For example, in the classic formation of ZFC, the axiom of replacement is actually a schema for an infinite series of axioms. The axiom of replacement says :
(∀ x: (∃!y : P(x,y))) → (∀A: (∃B:(∀y: y∈B⇔(∃x∈A:P(x,y)))))
Or in (confusing) english, "Given any set x: if there exists exactly one set y such that a predicate P is true for the pair of X and Y, then given any set A, there must be a set B where a set is a member of B if and only if there is some x in A where P holds for both x and y."
The thing to note there is that P is a free symbol: it can be instantiated by any predicate. But ZFC doesn't include the ability to quantify over predicates. So this isn't really a single axiom - it's a schema that generates a set of axioms, one for each possible value of P.
Whew! I hope I got that right; the ZFC axioms are extremely easy to foul up by misplacing a paren. I'm sure some kind commenter will correct my if I blew it!
We often talk about axiomatizations of some field of math: for example, when I wrote about the natural numbers, I showed an axiomatization of them using the Peano axioms. An axiomatization of a formal system is a reduction of that system to a basic set of axioms from which the other facts about that field can be derived using inference.