Subject: categories: Re: Two topos questions

From: [email-masked]
Date: Thu, 03 Nov 2005 11:18:38 -0500

What I was talking about 15 Jan 1997 was
(not hoping for an axiom of infinity without assuming one, but)

the fact that most of the mathematical uses of the rig N of natural
numbers do not work in a topos, if one interprets that rig to mean the one
characterized by Dedekind recursion.

1. starting with characteristic functions of subobjects, then adding and
multiplying them for various combinatorial calculations
2. applying the least number principle
3. measuring the fiber dimension of a bundle of linear spaces

all require the inf-completion of N, also known as the semicontinuous
natural numbers. It contains the truth-value object omega and is contained
in the semicontinuous reals (themselves indispensible for norming internal
Banach spaces, and constructible simply as one-sided Dedekind cuts).

Yet another way to picture these objects in the case of a Grothendieck
topos E is to consider the sheaf of germs of continuous maps from E to the
appropriate locale : the order topology (not the discrete one) on N, the
order topology (not the interval topology) on nonnegative reals.

Is any more known now as opposed to 9 years ago about the mathematical
applications of finiteness to variable and cohesive sets ? The fact that
K-finiteness is appropriate for some applications and that its theory
resembles the classical theory for constant discrete sets should not
distract us from the achievements of geometers in using coherence,
Notherianness,etc., nor from the fact that our "logic" should serve to
partly guide the learning of also those developments of thought.