[FOM] Eliminability of AC

[Bas Spitters]

On Monday 25 February 2008 23:45:26 Larry Stout wrote:
> Two major theorems in undergraduate mathematics need some form of
> choice:
> 	Every vector space has a basis.
> 	Tychanoff's theorem:  The product of compact toplogical spaces is
> compact.
> Will they do for examples from "ordinary mathematics"?

These are standard examples, but they do not motivate the use of the axiom of 
choice very well.
* When using the first lemma to obtain a theorem, the next result one wants to 
prove is that the result does not depend on the choice of the basis. In 
general, the use of a basis for a vector space should be avoided. Concretely, 
results depending on the basis of a Hilbert space do not tend to generalize 
to operator algebras.
* The use of the axiom of choice in Tychonoff's theorem is only needed when 
working with topological spaces instead of working with locales (pointfree 
topology). It may be argued that the category of locales is more pleasant 
than the category of topological spaces.
We discussed these issues recently:
http://cs.nyu.edu/pipermail/fom/2007-December/012350.html

Bas