Re: Borel set

On 2007-05-24, in sci.math, Dave L. Renfro wrote:
This is a point I've been making for many years in sci.math.
If it's uncountable choice that someone is concerned about,
then they should say so. Speaking of the axiom of choice,
I think Jacques Hadamard argued that, from a plausibility
standpoint, arbitrary choice is more acceptable for existence
than explicitly defined choice, since there is more freedom
for the object to exist (i.e. there are fewer requirements
that we have to satisfy to know that the object exists),
or something like this.

The justification for the axiom of choice -- for me, in any case -- is that
it is a mathematical expression of the idea that sets are arbitrary
extensional collections. On a conception of sets based on definability it is
indeed not at all obvious that choice is justified, though I don't see how
it would justify the *failure* of choice either. (Of course, we might be
able to work out whether or not choice holds for some mathematically defined
notion of definability, but that's another matter).

--
Aatu Koskensilta (aatu.koskensilta@xxxxxxxxx)

"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
.