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From: Vaughan Pratt <pratt@cs.stanford.edu>
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Subject: Re: "Kantor dust"
Date: Wed, 11 Feb 2009 15:51:06 -0800
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Prof. Peter Johnstone wrote:
> Whoa! This simply can't work. Whatever the final coalgebra for N x (-)
> looks like, it must (thanks to Lambek) be isomorphic to N x itself,
> and therefore (since equality for N is decidable) must have lots of
> complemented subobjects {0} x itself, {1} x itself, ... The point of
> the continuity theorem for functions R --> R is that there are toposes
> in which R has *no* nontrivial complemented subobjects [...]
> The only way to get round it (apart from using glue)
> is to replace N by some "nonstandard natural number object" having no
> nontrivial complemented subobjects -- but where you get that from, I
> don't know.

You're assuming product distributes over sums, which would be true for
ordinary product but I specified lexicographic product, with the left
argument as the "high order digit" (converse of the usual convention for
ordinal product in ordinal arithmetic).  Why should {0} x N be a
complemented subobject of N x N  when lexicographic product attaches the
"end" of it to {1} x {0} , which I would expect it will in a topos of
sheaves when participating in a final coalgebra for N x X.

Vaughan