[FOM] A computational approach to the countable ordinals
Paul Budnik
paul at mtnmath.com
Thu Aug 7 13:50:44 EDT 2008
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.
Have there been other attempts to write code for ordinal notations as a
research tool to create stronger notations systems? I know this has been
done to support automated proofs.
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?
Paul Budnik
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