Re: Finiteness in Toposes Jan 17 1997 This concerns the possibility , mentioned in my previous message, of two internal toposes of finite objects. The conjecture that there are two natural internal categories of finite objects is partly supported by the fact that there are two natural natural-numbers objects, the usual one N that parameterizes compositional iteration and another semicontinuous one L with the following features: 0) It is a rig, so receives a homomorphism from N and its elementary arithmetic starts out looking very similar. 1) But unlike N it has a least-number-property in the sense that it is inf-complete and better. 2) It can be constructed internally using truth-valued sheaves on N. 3) Hence it also contains a map from (big) omega, which permits (unlike N) the use of the standard method in finite combinatorics where (for example) a binary relation is considered as a matrix which is valued (not only in a rig where 1+1 = 1, but instead) in a rig in which natural numbers are distinct; the resulting generalized characteristic functions are added, multiplied, infed etc. according to the usual methods of arithmetic and analysis and then translated back into the combinatorics of the original finite structures. Of course, in each case one hopes that the answer to a combinatorial problem might turn out decidable, but that shouldnt require us to stay in the bounds of two-valued subsets in the course of a construction. 4) This internally-defined order-complete rig in E has also an external characterization if E is an S-based topos, namely it is the sheaf of germs of S-geometrical morphisms from E to the topos often called S-sets -through-time (I dont think that depends on any presumption that the N in S ,used to parameterize the transitions through time, coincides with its completion in S). In localic or open set terms, there is in S a (T sub zero) space whose points are N, but whose open sets have the usual order on N as their specialization order; continuous functions from any space E to this space are called semi-continuous and there is in E a sheaf of them. 5) The application to the variable linear algebra over algebraic or complex-analytic spaces needs L too, because dimension of a vector space is a semi-continuous function. More precisely, if A is a good module in a ringed topos E, R then for each X and E there should be a map X--> L which is the fiber-wise dimension of X*A. The basic case is perhaps that where E,R is an algebraic affine scheme, and the conceptual problem is to get at what sort of sets contained in A this dimension function is counting (or bounding). One should not expect that equality of dimension will imply isomorphism. This object L has been discussed for 25 years, but I dont know if anyone published the working-out of its properties and role. Bill