A universal property for the ordinal Omega?

A universal property for the ordinal Omega?

[Francois Lamarche]

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.


This seems to be a pretty natural question to ask, so it must have been
done before.



Francois Lamarche

Re: A universal property for the ordinal Omega?

[Todd Wilson]

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.

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Todd Wilson                           A smile is not an individual product;
Computer Science Department           it is a co-product.
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