[FOM] 'Recursive notations' for countable admissible level ordinals

Paul Budnik 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.

Paul Budnik

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