[FOM] A computational approach to the countable ordinals
Paul Budnik
paul at mtnmath.com
Mon Aug 11 13:20:35 EDT 2008
hendrik at topoi.pooq.com wrote:
> 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 am not so sure about coding it in LaTeX. I use C++ but the code does
have the ability to generate LaTeX formatted notations. I first worked
on a closely related problem in the early 70's. Seeing the axioms of ZF
as one page computer program for enumerating theorems make me think I
could get an explicit understanding of the combinatorial structures
implicitly defined by those axioms.
>
> I definitely didn't go past the constructive.
>
Just as one can think of the recursive ordinals as the well orderings
definable by recursive processes that are well founded for any infinite
sequence of integers, one can define admissible ordinals at a certain
level as recursive processes well founded for any infinite sequence of
ordinal notations first for recursive ordinals and then for higher
levels previously defined. Of course any particular computable system of
notations will never include all recursive ordinals. A computational
approach in effect replaces the impredictivity of set theory with
explicit incompleteness.
Paul Budnik
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