[FOM] 'Recursive notations' for countable admissible level ordinals
paul at mtnmath.com
Fri Oct 16 12:30:11 EDT 2009
I am extending the ordinal calculator
(http://cs.nyu.edu/pipermail/fom/2009-September/014061.html) to include
notations for what I call admissible level ordinals. These are countable
ordinals >= the Church Kleene ordinal or the second admissible
ordinal. There is an outline of the approach I am taking at
http:/www.mtnmath.com/ord/admis.html This includes speculation about the
next step beyond admissible level notations which may have the
potential to go beyond what is provably definable within ZF. I would
appreciate feedback on this.
For admissible level ordinals I am using a recursive analogue to
recursive notations for recursive ordinals. This notation system ranks
all notations in the system, but of course it cannot enumerate notations
for all smaller ordinals.
These admissible level ordinal notations allow something like nested
ordinal collapsing. This is possible at any level in the countable
admissible hierarchy with a notation in the system. The next step may be
to go beyond iteration on ordinal notations and iterate the well founded
functional hierarchies used to construct these notations.
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