On Thu, 14 Sep 2000, Francois Lamarche wrote:

Has anybody ever proved the following result?

In the category of posets the initial algebra with the following operations:

1- a constant 0

2- an inflationary successor s : x \le sx

3- sups of omega-Chains

is the least non-denumerable ordinal Omega.

The least non-denumerable ordinal is usually called omega_1, and even though the result you are after is not directly stated in the book A. Joyal, I. Moerdijk, Algebraic Set Theory, LMS Lect Notes 220, Cambridge University Press, 1995, this book is clearly relevant to your question. Joyal and Moerdijk, in a tour de force, construct a number of free algebras for various kinds of successor operation and "small"-indexed sups, for a quite general notion of "small" that includes the kind of cardinality restriction to which you refer. --------------------------------------------------------------------------- Todd Wilson A smile is not an individual product; Computer Science Department it is a co-product. California State University, Fresno -- Thich Nhat Hanh ---------------------------------------------------------------------------