> >> Being the smallest nonspecial number makes N special.
>
> > Does it?
>
> > Actually I suspect there may be a fault in the schema. Is
> > being special decidable?
>
> The real problem is that 'special' is not well-defined.
> It's the old heap of sand problem. If a collection of n
> grains of sand constitutes a heap, surely so does a
> collection of n-1 grains, but then ...
I would rather say that the paradox arises from absense of
separation of meta and object properties i.e that something
can become 'special' by virtue of being part of the proof. This is
reminiscent of the liar paradox and a bit like e.g claiming it's
impossible not to have principles because if someone claims he/or she
has no principles, that itself constitutes a principle. Ideally you
will want to keep separate regular principles from meta-principles and
so on.
I find another paradox on induction quite intriguing namely:
The teacher in a high school class say to her students that they
will have an unpreparered test next week. Unprepared in the sense
that the students won't know what day the test is until they come to
school and the teacher says it's the day of the test. Then, a student
reasons as follows: She can't have the test on friday because then we
would know on thursday evening that the test will be on friday (because
the test hasn't been yet and there's only one day left) and that would
invalidate the unprepared-premise. Being 100% sure the test won't be
on Friday, one can do the same reasoning to show that it can't be on
Thursday either because then the students' would know it on wednesday
evening and so on. So the teacher can't have the test at all!
best wishes
Harald