🚀 go-pugleaf

In article <TORKEL.92Nov22115549@echnaton.sics.se>, torkel@sics.se (Torkel Franzen) makes an good point that "uncountable" seems to have a clear cut
meaning to most working mathematicians, independent of whatever formal system
they may (seem to) be working in (if any). He makes an excellent analogy with
the concept of "complete", in some detail, pointing out that the same applies.

While there may be some room for debate, he presents a very robust and appealing
case.  But then he also goes on to say, in distinction, 

|> On the other hand,
|> such concepts as "formally undecidable" or "definable" do have formal systems
|> as explicit and implicit parameters, and are understood, explained, and used
|> on that understanding.

While I agree totally that this is so with "formally undecidable", I'm not so
sure about the other.

"Definable", when used technically, does of course mean precisely "with respect
to a formal system"; but then Paul Budnik's point was that (precisely speaking)
this is also true for "uncountable".  But Torkel Franzen was making the point
that "uncountable" also corresponds to a clear-cut mathematical idea that is
independent of any formal system; and I suggest that "definable" is similar.

I suspect that many working mathenmaticians have a vague but firmly held idea
that there are certain identifiable "definable" things in some platonic sense,
just as much as there are uncountable things.  While they may not be able to
give quite such a good account of them as they could of uncountable, they might
make a fair stab at it, if pressed.  It would almost certainly be based on the
simple idea of sets being given in (unrestricted-looking) comprehension terms
   { x | P(x) } .

Of course the property P can only be made precise within a formal system.
But then, as said above, the same remark applies to uncountable.

Does this idea ring any bells with anyone else; that "definable" may have some
meaning in mathematical "reality" (independent of a formal system); that is
*at least as much* as does "uncountable".  
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              Bill Taylor              wft@math.canterbury.ac.nz
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             Quantum particles: the dreams that stuff is made of.
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