Osius' set theory

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

Elwood Wilkins reminds us that...
> In his 1974 paper Osius showed that there are enough transitive objects
> in the initial topos to define a naive set theory. ...

Gerhard Osius's work was significant in 1974 as one of the ways in which
it was shown that toposes do the same thing as set theory.   In fact Osius's
is, so far as I am aware, the only work that discusses *set* theory -
Benabou, Joyal, Mitchell, Lawvere etc showed that toposes do the same thing
as some form of *type* theory.  Of course, it is the latter that
mathematicians actually use when they claim to be using set theory
as foundations, but set theory - epsilontics as Lawvere calls it - is of
some interest in the study of very high powered induction.

Peter Johnstone included a precis of Osius's paper in his book "Topos Theory"
(Academic Press, 1977). Apart from that, I was unable to track down anything
that built significantly on it.  Indeed,  Osius himself didn't seem very
interested when I wrote to him about it (he now does statistics).

The successor to this paper would therefore seem to be my
     "Intuitionistic Sets and Ordinals", J. Symbolic Logic, 61 (1996) 705--744
This develops, in a symbolic way, the notions of "transitive set" and
"ordinal" in the sense of a carrier equipped with an extensional well founded
relation.  Several qualitatively different notions of ordinal arise
intuitionistically.  I also pick up on Osius's set theory and pose some
questions that suggest ways of adapting it to Grothendieck toposes in general.

What remains of considerable interest (once we have agreed that set theory
is wrong, wrong, wrong) is Osius's categorical notion of recursion.

The equation       f(x)  =   r( { f(y) | y in [=element_of] x } )

he writes as the (3=1, not 2=2) commutative square

                            P(f)
  { y | y in x }   P(X)  --------->   P(A)
       ^            ^                  |
       |            |                  |
       |            |                  |
       |            |                  | r
       |            |                  |
       |            |                  |
       -            |        f         v
       x            X  ------------->  A

which we may of course generalise to a "homomorphism" from any P-coalgebra
to any P-algebra, where indeed P may be any functor instead of the covariant
powerset functor.  The coalgebra admits recursion by definition if there is
exactly one such "homomorphism" to any algebra whatever.

The exercises in Chapter VI of my book "Practical Foundations of Mathematics"
(Cambridge University Press, 1999) explore applications of the commutative
square to recursive functional programs.
      http://www.dcs.qmw.ac.uk/~pt/Practical_Foundations/html/s6e.html

Osius's 3+1 square is about RECURSION - defining functions or programs -
but I have also considered INDUCTION - proving theorems - categorically.

Again this is a property ("well foundedness") of coalgebras.  See Section 6.3
      http://www.dcs.qmw.ac.uk/~pt/Practical_Foundations/html/s63.html
of the book, or my unfinished paper "Towards a Unified Treatment of Induction"
(also known as "On the General Recursion Theorem") which you can get from
my Hypatia page
      http://hypatia.dcs.qmw.ac.uk/author/TaylorP

The final section of the book (s96.html) sketches the way in which this
categorical notion of ordinal can be used to define transfinite iterates of
a functor (for example, internally in a topos).   I hope to use this to
develop categorical notions of the Axiom of Replacement.

Paul Taylor