Exploring the Group Axioms: What Makes Them an Axiom?

[broegger]

Hi.

I'm reading a simple introduction to groups. A group is said to be a set satisfying the following axioms (called the 'group axioms'):

1) Associativity.

2) There is a neutral element.

3) Every element has an inverse element.

4) Closure.

My questions is simply: why are they called axioms? I thought an axiom was something we take as a starting point, defining it to be true and then deduce something from it (possibly together with other axioms). Why are 1-4 not just the definition of a group?

 

[matt grime]

They are the definition of a a group (modulo the fact that you've omitted to mention the binary operation). A group is something that satisfies these axioms (a model). Note, axioms are not things that are 'defined to be true' . They are just 'things' and in any model of the axioms they are true.

It just depends on how you like to label these things.

 

[mathwonk]

more useful is to think about an example, like the isometries of a cube, possibly orientation preserving, i.e. rotations carrying a cube into itself.

 

[broegger]

But you can't prove an axiom, and 1-4 can be proved (or disproved) for a given set?

 

[Cincinnatus]

In that case you are just proving that whatever set with whatever binary operation satisfies those axioms. You are not proving the axioms themselves.

 

[broegger]

In that case you are just proving that whatever set with whatever binary operation satisfies those axioms. You are not proving the axioms themselves.

Oh, I think I get it now. I guess I was confused about the distinction between the axioms themselves and 'the model' to which they are applied. Thanks, everyone.

 

[broegger]

By the way, does anybody know of a good, relatively accessible, introduction to the subject of mathematical logic?

 

[cogito²]

Robert R. Stoll's Set Theory and Logic is an okay intro set theory text (although it only looks at naive set theory), but an excellent intro logic text. It's also put out by Dover so it's cheap.

edit: Link to book.

 

[broegger]

Thanks, I think I'll pick that one up.