The answer to the general question is surely "no": the sentences which hold in Sh(X + X) are exactly those which hold in Sh(X). Whether one can separate Sh(R) from Sh([0,\infty)) in this way is a more interesting question. Does "Brouwer's continuity theorem" hold in Sh([0,\infty))? The proof that I know for Sh(R) doesn't work over [0,\infty), but that may be because it's not the best proof. Peter Johnstone On Thu, 5 Mar 2009, Thomas Streicher wrote:
Bob Lubarsky has recently asked me a question I could answer only partially. He is not reading this mailing list and so I forward his question (since I am also interested in it). The question is whether for nonisomorphic spaces X and Y one can always find a formula in higher order arithmetic or in the language of set theory which holds in one of the toposes Sh(X), Sh(Y) but not in the other. More concretely he asked about the folowing 4 spaces
R (reals) R_{\geq 0} (nonnegative reals)
Q (rationals) Q_{\geq 0} (nonnegative rationals)
AC_N holds for sheaves over spaces in the second line but not for sheaves over the spaces in the first line. But I couldn't tell him how to logically separate R and R_{\geq 0} or Q and Q_{\geq 0}.
The background of Bob's question is his work on "forcing with settling down" (http://www.math.fau.edu/lubarsky/forcing with settling.pdf) providing a model for CZF without Fullness but Exponentiation where, moreover, the Dedekind reals are not a set but a proper class.
Thomas