[FOM] A computational approach to the countable ordinals

[hendrik@topoi.pooq.com]

On Thu, Aug 07, 2008 at 10:50:44AM -0700, Paul Budnik wrote:
> I have always been sceptical of completed infinite totalities in set 
> theory. At the same time I consider questions like "Does a computer have 
> an infinite number of outputs?" to be objectively true or false. This 
> led me to the idea that the only mathematical questions that are 
> objectively true or false are those logically determined by a 
> recursively enumerable sequence of events. This includes most of 
> standard mathematics, but excludes questions like the continuum 
> hypothesis. I published a philosophical paper about this, What is 
> Mathematics About?  (http://www.mtnmath.com/pom.html), in Paul Ernest's 
> online journal.
> 
> This philosophy has led me to suspect that a computational approach to 
> the ordinal numbers can ultimately go beyond the recursive ordinals 
> provably definable in ZF. Writing code to define an expandable notation 
> system allows one to do computer experiments on notations systems that 
> are not otherwise possible.  It also helps immensely in keeping track of 
> all the details. I have put some effort into this project starting with 
> the Veblen hierarchy and its generalizations.  I have two questions for 
> this group.

I played with ordinal notations in the 70's, developing 
algorithms for addition, multiplication, and I think I may have tried 
exponentiation, too.  The notation(s) resembled the Veblen hierarchy, 
but were different in details.  I was being minimalist at the time, and 
thought building addition, multiplication, and exponentiation into the 
notation was excessive.  Maybe it's time to dig up that stuff and code 
it in Latex?

> 
> I have reached a point where the best way to proceed seems to be to use 
> ordinals greater than the Church Kleene ordinal to generalize the 
> functional hierarchies used to construct recursive ordinal notations. I 
> know the countable admissible ordinals have been used to construct 
> recursive ordinal notations using collapsing functions. but I wonder if 
> anything like I am suggesting has been attempted?

I definitely didn't go past the constructive.

-- hendrik boom