Hamel basis and infinite-dimensional vector spaces

[Buri]

If we could find the Hamel basis for any infinite dimensional vector space, what kind of consequences would this have?

 

[Buri]

This question was asked a while ago and still don't have an answer for it, so just thought I'd bump it up

 

[micromass]

But... we CAN find a Hamel basis for any vector space! (Assuming the axiom of choice)
So I'm not sure what you're getting at here...

Anyway, the existence of Hamel bases for any vector space, implies the following:
- the existence of a non-Lebesgue measurable set
- the exists of a function who's graph is dense in the plane
- the existence of an additive and nonlinear function

But I'm not sure what you want to hear from us...

 

[Buri]

We can find them explicitly? Can we find, explicitly, the basis for the space of all continuous functions? As far as I know, we only know existence by AoC.

 

[Buri]

Munkres in Analysis on Manifolds says, "There is a theorem to the effect that every vector space has a basis. The proof is non-constructive. No one has ever exhibited a basis."

 

[micromass]

Ow, I'm sorry, I didn't understand what you meant.

You are correct, we can not explicitely find a basis for any vector space (we can however, prove that such a basis exists).

There are a lot consequences. Some of them are:
- we can show explicitely that the axiom of choice holds true
- we can show explicitely that the Tychonoff theorem holds true
- we can find a wellordering of $$\mathbb{R}$$
- we can find a set which is not Lebesguemeasurable

Basically, everything which requires the axiom of choice, will be explicitely provable...

 

[Buri]

Wow, really? Would it be worth pursuing something like this? Mathematically, is it worth it? I guess mathematicians who don't accept the AoC would be forced to then lol

 

[micromass]

Well, it won't be worth pursuing, for the simple reason that it cannot be done. Mathematicians have basically proven that we cannot find explicitely a Hamel basis for every vector space. It simply cannot be done.

And what about people that don't accept the AC? Well, they simply have to live with it. They have to live with the fact that you cannot always find a Hamel basis. And there are some other things that they cannot do.
There are however some things that can be done. For example, you can change the axioms of a topological space somewhat. Then things like the Tychonoff theorem suddenly becomes provable! This is the study of frames and locales...

 

[Buri]

Ahh I see. Thanks for answering this question! It'd gone unanswered when I asked it some while back and its finally resolved! Thanks a lot!