categories: Re: Category Theory and Hereditarily-Finite Sets

[Paul Taylor <pt@dcs.qmw.ac.uk>]

> Hereditarily-finite sets are becoming increasingly more popular
> in computer science research.


Why?  Because some ill-advised first year maths lecturer told you that
the element relation was the foundation of mathematics, maybe?

> "object" is a hereditarily-finite set plus some structure on the set
> and a "morphism" would be a structure-preserving function.

If you're really interested in heredity, so the "structure" is the
element relation, this is a well-founded coalgebra for the covariant
powerset functor.

Coalgebras for the powerset functor were first studied by Gerhard Osius
in JPAA in 1974, although he considered recursion rather than induction.

Well founded coalgebras for general functors (but with some emphasis
on the powerset) are defined in Section 6.3 of my book "Practical 
Foundations of Mathematics" (Cambridge University Press, 1999).

The exercises for that chapter show how various ideas with recursive
programs may be expressed in these terms.  In particular, unary
recursion (with at most one recursive call at each level) is reduced
to tail recursion (equivalent to while programs) together with an
accumulator monoid.

Paul