FOM: Books on categorical foundations: a baker's dozen
Todd Wilson
twilson at grianne.engr.csufresno.edu
Fri Feb 27 19:03:00 EST 1998
An interesting recent book on categorical foundations that was ommitted
in Vaughan Pratt's baker's dozen (Fri, 27 Feb 1998 19:15:58) is
A. Joyal and I. Moerdijk, Algebraic Set Theory, LMS Lecture
Note Series 220, Cambridge University Press, 1995.
Starting from a set-up more general than topos theory (technically a
"Heyting pretopos") and a notion of *small map*, which is simply a
collection of morphisms that satisfy a few simple axioms that are
based on a clear, underlying conception (namely a map with small
fibers), set theory is developed as a kind of (universal) algebra.
The cumulative hierarchy is a free algebra under the operations of
small union and successor (x |-> {x}). The ordinals are the quotient
of this free algebra gotten by requiring successor to be montone. A
different kind of ordinal algebra (the "Von Neumann ordinals") arises
by requiring the successor instead to be inflationary. The existence
of these free algebras in models of the above setup are shown to
exist in the presence of a subobject classifier. The blurb on the
back of the book says that it "provides a uniform description of
various constructions of the cumulative hierarchy of sets in forcing
models, sheaf models and realizability models".
--
Todd Wilson
Computer Science Department
California State University, Fresno
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