[FOM] Expanded ordinal calculator is available
paul at mtnmath.com
Tue Jan 19 11:14:43 EST 2010
Version 0.2 of the ordinal calculator has been released. In addition to
bug fixes it adds support for notations for countable ordinals greater
than the ordinal of the recursive ordinals. These are used in a form of
collapsing to define large recursive ordinals.
An analog of recursive ordinal notations is constructed for larger
countable ordinals using recursive functions on incomplete domains. For
recursive ordinals a recursive function on the integers ,
`limitElement', is defined for each notation alpha. The union of the
ordinals represented by alpha.limitElement(n) for every integer n is the
ordinal represented by alpha. For larger ordinals a similar purpose is
served by function `limitOrd'. This accepts ordinal notations of a
given type (such as recursive ordinal) as input. The domain of
`limitOrd' is necessarily incomplete within the computational model. In
writing programs to evaluate ordinal expressions beyond the recursive
ordinals one is compelled to build incompleteness in from the ground up.
There are programming language constructs like C++ subclasses and
virtual functions that facilitate this.
A form of collapsing is used that `freezes' the ordinal notation system
at a given point. This allows the embedding of this frozen structure to
expand the notations for recursive ordinals. This leads to notations for
recursive ordinals a bit beyond the Bachmann-Howard ordinal. This
approach can be generalized further. It is documented in the paper "A
Computational Approach to the Ordinal Numbers: Documents ordCalc0.2".
Documentation, complete source code and Windows installer are available
at www.mtnmath.com/ord and https://sourceforge.net/projects/ord
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