[FOM] Ultrafinitism

[hendrik@topoi.pooq.com]

On Tue, Feb 10, 2009 at 05:36:00PM +0200, Alex Blum wrote:
> Jean Paul Van Bendegem presents a putative counterexample to a 
> generalization of mathematical induction. He writes, in part:
> 
>  "(a) I can write down the numeral 0 (or 1, does not matter),
> (b) for all n, if I can write down n, I can write down n+1 (or the 
> successor of n),
> hence, by mathematical induction,
> (c) I can write down all numerals."
> 
> 
> The properties of numbers in mathematical induction hold of numbers 
> irrespective of how they are named.  Since a number may be named  in a 
> notation  which  could  never be completed,  (b), even if true, need not 
> be true, and thus the predicate 'I can write down the numeral' is 
> inappropiate for mathematical induction.

In fact, there are well-known notation systems for numbers
(lambda-expressions for Church numerals, for example) which have 
short names for unimaginably huge numbers, but have only unimaginably 
huge notations for many much smaller numbers.  "In principle", of 
course, they could all be written down.

I suspect that Jean-Paul van Bendegem did not have such a notation in 
mind, although it probably does satisfy the premisses of his 
argument.  But the huge swaths of numbers with only unimaginably large 
notations do provide a formal analogue to the difference between what we 
can imagine doing and what we can in fact do, which seems to be the 
motivation behind ultrafinitism.

These huge swaths of infeasibility would seem to be are analogous to the 
huge swaths of ordinals without notations at all in the more aggressive 
ordinal notation systems.

--hendrik