Three problems concerning ideals of differentiable functions

Abstract

In this paper, we study the validity of the following two statements in the internal logic of the toposes of Synthetic Differential Geometry: 1. (1) The integral of f is non-negative if f is non-negative; 2. (2) If f=0 in the set of non-negative reals, and f=0 in the set of non-negative reals, then f=0. We find statements (1) and (2) to be true in the toposes considered. We also prove that 3. (3) For n greater than two, the arrow tn from the line to itself is not a stable effective epic. This answers a question raised by Quê-Moerdijk-Reyes.

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