Whats the difference between a definition and an axiom?

[michonamona]

What is it?

For example

A function f: A-->B is called injective if, for all a and a' in A, f(a)=f(a') implies that a=a'.

What is keeping this definition from being an axiom?

 

[╔(σ_σ)╝]

Axioms are, as they say, rules of the game. For example the real numbers have axioms that tell you how to multiply, add etc. However, definitions are simply definitions (sorry about the circular comment) they are not rules but are explanations of certain properties.

In your example this is a definition because it states what you should interpret a bijection to mean.

I will also like to add that axioms are what we build everything else from; while definitions are usually tools used to make ideas and notions precise.

 

[JonF]

Axioms are asserted unproved propositions.

In your example, did that statement "assert" anything? i.e. say soemthing exist, or that an object has some property? Or did it just name something.

 

[radou]

As JonF said, you can define anything you want, but it doesn't need to be interesting or useful mathematically, while axioms contain certain mathematical truths which are fundamental.

 

[Hurkyl]

What is keeping this definition from being an axiom?

Nothing, really. The difference is more pedagogical than meaningful. Using the word "definition" tends to suggest that the new concept is being constructed out of concepts you're already familiar with.

 

[JonF]

Well you can't define anything, take Russell’s paradox for example, but most things you'd want to for sure.

 

[JonF]

Nothing, really. The difference is more pedagogical than meaningful. Using the word "definition" tends to suggest that the new concept is being constructed out of concepts you're already familiar with.

For it to be an axiom wouldn’t it have to be “there existed” or some such thing instead of “is called”

 

[logarithmic]

Nothing, really. The difference is more pedagogical than meaningful. Using the word "definition" tends to suggest that the new concept is being constructed out of concepts you're already familiar with.

I'm not sure I agree. The reason why his example is a definition is because it just says we'll just call a function with this property injective. He isn't asserting anything new other than naming some object.

Whereas an axiom is like a theorem that isn't proved. E.g "there is an infinite set".

Although sometimes parts of what forms a definition is called an axiom just for flavor. For example, some may say that the distributive law is one of the axioms of a field. But really it's just part of the definition of a field: "A set that satisfied these 10 properties, one of which is the distributive law, we'll just call a field". Those properties are sometimes called axioms, but I don't really think they are axioms, at least not like the ZFC axioms.

 

[gomunkul51]

An interesting read:
http://en.wikipedia.org/wiki/Axiom

 

[Hurkyl]

He isn't asserting anything new other than naming some object.

For typical purposes, there is no difference between:

• Defining the predicate "f is injective" in terms of evaluation, equality and such
• Extending the language to include "injective" as a primitive notion, and asserting an axiom relating "injective" to evaluation, equality, and such.

 

[JonF]

I could be totally off on this, but I thought definitions were constructable out of the primitive concepts and were short hand ways of saying volumes worth symbols.

 

[Hurkyl]

If you fix one particular presentation of a theory, yes. But there are many ways to present a theory.

e.g. Hilbert presented Euclidean plane geometry by taking "point", "line", "between", "incident" and "congruent" as primitive notions, and asserted various axioms.

The (FAPP) same theory can be presented by taking "0", "1", "+", "*", and "<" as primitive notions, asserting the complete ordered field axioms, and then using these primitive notions to construct the Cartesian plane in the usual way.

 

[JonF]

Okay, I guess this is where my confusion comes from. From what I understand if we take primitive concepts and axioms together they have to imply the existence of something. Whereas a collection of definitions should avoid doing that at all cost.

Are you saying in many cases we could swap some axioms with definitions if we add existence to the definitions and everything would work out about the same? If so I get that.