Abstract
Let K = hR; i be a closed ordered dierential eld, in the sense
of Singer [20], and C its eld of constants. In this note, we prove that, for
sets denable in the pair M= hR;Ci, the -dimension from [5] and the large
dimension from [11] coincide. As an application, we characterize the denable
sets in K that are internal to C as those sets that are denable in M and
have -dimension 0. We further show that, for sets denable in K, having
-dimension 0 does not generally imply co-analyzability in C (in contrast to
the case of transseries). We also point out that the coincidence of dimensions
also holds in the context of dierentially closed elds and in the context of
transseries.
of Singer [20], and C its eld of constants. In this note, we prove that, for
sets denable in the pair M= hR;Ci, the -dimension from [5] and the large
dimension from [11] coincide. As an application, we characterize the denable
sets in K that are internal to C as those sets that are denable in M and
have -dimension 0. We further show that, for sets denable in K, having
-dimension 0 does not generally imply co-analyzability in C (in contrast to
the case of transseries). We also point out that the coincidence of dimensions
also holds in the context of dierentially closed elds and in the context of
transseries.
| Original language | English |
|---|---|
| Journal | Notre Dame Journal of Formal Logic |
| Publication status | Accepted/In press - 9 Oct 2020 |