[FOM] Reply to Eray re "No other object is X"

[Gabriel Stolzenberg]

Dear Eray,

   Re:

> On 3/21/06, Gabriel Stolzenberg <gstolzen at math.bu.edu> wrote:
> >    But a classical mathematical can't stop there.  He feels a need
> > to say that |x| is defined for all x.  On the other hand, in a
> > constructive mindset, one does not feel such a need.  Nothing is
> > missing.
>
> Excuse my ignorance. Is this also why in many definitions, the
> properties relating to an object of type X, P1, P2, .., PN are given,
> and there is an additional property that says "No other object is X"?
> I am just trying to understand, because (naively) if I give an
> algorithm that enumerates all X, there is no need to make the claim
> of negating the rest. Can you please expand on these claims a little
> further?

   I'm afraid I don't have anything helpful to say about this.  I
wasn't thinking of |x| viewed classically as part of a more general
phenomenon, although I can see why it occurred to you to look at it
that way.  It fits.  Except that I would replace "Nothing else is X"
with "Everything X has these properties."

   Question.  In order to prove that your algorithm enumerates all X,
don't you, in effect, have to show that nothing else (besides what the
algorithm enumerates) is X?

   I'm sorry I can't be more helpful.

   With best regards,

    Gabriel