Axioms of Vector Spaces

[Dragonfall]

Why are they called "axioms"? Shouldn't they be called "definitions"?

 

[Hurkyl]

Why are they called "axioms"? Shouldn't they be called "definitions"?

Axioms are the traditional name for the elements of a generating set of a mathematical theory.

 

[HallsofIvy]

And they are NOT, in any sense, "definitions". They may well give some properties of "undefined terms" but it is essential to mathematics that "undefined terms" remain undefined so that we can apply the same theory to many different situations.!

 

[tiny-tim]

Why are they called "axioms"? Shouldn't they be called "definitions"?

Hi Dragonfall!

One person's axiom is another person's theorem …

If a theory needs, say, five axioms, A1 to A5, from which you can easily prove a simple theorem, T, then often you can start with A1 to A4 and T as the five axioms, and prove A5 as a theorem.

We have to arbitrarily promote certain theorems to "axiom" status, to get us started.

(btw, I think you're thinking of axioms like "on a plane, for any two points there is defined a line joining them" …

this is written as a definition, but you can equally write it as a theorem … "on a plane, there is a unique line joining two points")

 

[jbunniii]

Why are they called "axioms"? Shouldn't they be called "definitions"?

The axioms themselves are not definitions. They are properties that an arbitrary object may or may not have. We then DEFINE a vector space as a particular type of object that happens to have those properties.

 

[Dragonfall]

I mean that every vector space exists as pure sets under ZFC already. You just need to define them, not postulate their existence.

 

[HallsofIvy]

You titled this "Axioms of Vector Spaces". The axioms for vector spaces have nothing to do with postulating the existence of vector spaces!

 

[jbunniii]

I mean that every vector space exists as pure sets under ZFC already. You just need to define them, not postulate their existence.

ZFC just gives you a set of points without any additional structure, e.g., the ability to "add" two points together, or to "multiply" a point by a scalar from some field. The additional structure comes from the vector space axioms.

Furthermore, you need both a point set and a scalar field to construct a vector space. In general you can't take just ANY point set and ANY scalar field and mash them together into a vector space. For example, if S = {0, 1} and F = the real numbers, then (S,F) is not a vector space because it does not contain every real multiple of 1.

[Edit]: Notice also that you cannot take a point set with, say, 6 elements, and turn it into a vector space, no matter how you define addition and multiplication, and no matter what field you choose. The reason is that every finite vector space is isomorphic to F^n for some finite integer n, meaning it must have exactly |F|^n elements. But there is no field with six elements.

 

[Dragonfall]

You titled this "Axioms of Vector Spaces". The axioms for vector spaces have nothing to do with postulating the existence of vector spaces!

Axioms are propositions that are taken as true without proof. It makes no sense to think of the "axiom of additive closure" as an axiom, as a set with addition may or may not satisfy it. You are DEFINING what a vector space is!

ZFC just gives you a set of points without any additional structure, e.g., the ability to "add" two points together, or to "multiply" a point by a scalar from some field. The additional structure comes from the vector space axioms.

You define "add" and "multiply" in ZFC as functions, or ordered tuples. This "additional structure" already exists in ZFC. No need for further axioms. Just definitions.

 

[quasar987]

I too have never understood why the defining properties of a vector space are sometimes called the "axioms" of a vector space. That use of the word "axiom" does not seem to agree with the definition of the dictionary. http://dictionary.reference.com/browse/axiom

 

[tiny-tim]

I too have never understood why the defining properties of a vector space are sometimes called the "axioms" of a vector space. That use of the word "axiom" does not seem to agree with the definition of the dictionary. http://dictionary.reference.com/browse/axiom

From that dictionary:

Logic, Mathematics. a proposition that is assumed without proof for the sake of studying the consequences that follow from it.

How does that not agree?

 

[Dragonfall]

Write one of these "axioms" out in full formally, and see for yourself.

 

[quasar987]

From that dictionary:


How does that not agree?

Well, what's the proposition in this case?

 

[matt grime]

A vector space is defined to be something satisfying the axioms of a vector space. Amongst other things one of the axoims is that x+y=y+x for all x,y. The 'proposition' would be - if x,y are in V then x+y=y+x.

 

[matt grime]

Axioms are propositions that are taken as true without proof. It makes no sense to think of the "axiom of additive closure" as an axiom, as a set with addition may or may not satisfy it. You are DEFINING what a vector space is!

Yes, something that satisfies the axioms. Moreover, axioms are *not* taken as true without proof. That is a fundamental misunderstanding. An axiom is just a statement. It may or may not be true when applied to some special case. The canonical example is the parallel postulate in geometry.

You define "add" and "multiply" in ZFC as functions, or ordered tuples. This "additional structure" already exists in ZFC. No need for further axioms. Just definitions.

You're confusing a model with the axioms. I can, in your terms, define the real numbers using ZFC I'm sure, yet there are models of ZFC where the real numbers are not a set. I may also want to make my definition of a vector space independent of the axioms of the underlying set theory.

 

[Cantab Morgan]

I'm obviously over my head here, because I've had to read this discussion about a dozen times before I even understood the question, but I'd still like to offer the following...

It seems like Dragonfall wants to restrict the word "axioms" to mean the axioms of set theory itself, like the axiom of choice for example. That's pretty deep for folks like me who are content with naive set theory.

But mathematicians often use the word "axiom" when constructing interesting sets that have some structure beyond that of a simple set. Despite the objections, I feel that the term is useful here too, for a couple of reasons.

First, one can mix and match these "axioms" to construct interesting, different structures. For example, groups are semi-groups with additional properties, captured as the additional axioms obeyed by a group that are not obeyed necessarily by a semi-group.

Second, we still have many of the same worries when talking about set-theoretic axioms, as we do when talking about these properties of certain sets. Namely, we'd like them to be minimal (no theorems included as redundant axioms) and consistent.

For example, there's an additive identity of a vector space. But is it unique? Yes. Does that uniqueness deserve its own axiom/property or can it be proved from the others? You see what I'm driving at is that seeking a minimal set of axioms is like seeking a minimum set of properties of a vector space. The use of the word axiom here does not feel forced or unnatural.

And when it comes to consistency, it would be possible to write down a set of properties for some structure, and yet have no set that could obey them all. That's like coming up with a system of axioms that are mutually contradictory.

I'm curious to know whether Dragonfall would consider the "axioms" of geometry, like the parallel postulate and so forth, as legitimate axioms, or merely the definition of a particular geometry (Euclidean, Poincare, etc).

 

[Dragonfall]

A vector space is defined to be something satisfying the axioms of a vector space. Amongst other things one of the axoims is that x+y=y+x for all x,y. The 'proposition' would be - if x,y are in V then x+y=y+x.

You're confusing a model with the axioms.

You're right, I was confusing a model with the axioms. I realized that after remembering Skolem's theorem, for some reason.

I'm curious to know whether Dragonfall would consider the "axioms" of geometry, like the parallel postulate and so forth, as legitimate axioms, or merely the definition of a particular geometry (Euclidean, Poincare, etc).

Funny you brought that up, since I had almost forgotten about classical geometry and its "axioms". I'm not sure what I would have thought of them.