Re[2]: [tied] Re: Numerals query again

At 12:24:52 PM on Sunday, November 9, 2003, Harald
Hammarstrom wrote:

>>>> 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.

There's a bit of that in it, but I think that it merely
contributes to the more basic problem that the predicate is
ill-defined in the first place.

> 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. [...]

Probably better known (at least in English) as the paradox
of the unexpected hanging, thanks to Martin Gardner's
extensive discussion of it in one of his books. I don't see
it as a paradox on induction; it can be rephrased in a way
that completely removes the inductive element. The problem
here is more akin to that of the liar paradox, in that both
involve self-reference, though in this case it's of a more
complicated kind.

Brian