580 research outputs found
Interpreting the projective hierarchy in expansions of the real line
We give a criterion when an expansion of the ordered set of real numbers
defines the image of the expansion of the real field by the set of natural
numbers under a semialgebraic injection. In particular, we show that for a
non-quadratic irrational number a, the expansion of the ordered Q(a)-vector
space of real numbers by the set of natural numbers defines multiplication on
the real numbers
Expansions of the real field by open sets: definability versus interpretability
An open set U of the real numbers R is produced such that the expansion
(R,+,x,U) of the real field by U defines a Borel isomorph of (R,+,x,N) but does
not define N. It follows that (R,+,x,U) defines sets in every level of the
projective hierarchy but does not define all projective sets. This result is
elaborated in various ways that involve geometric measure theory and working
over o-minimal expansions of (R,+,x). In particular, there is a Cantor subset K
of R such that for every exponentially bounded o-minimal expansion M of
(R,+,x), every subset of R definable in (M,K) either has interior or is
Hausdorff null.Comment: 14 page
The prospects for mathematical logic in the twenty-first century
The four authors present their speculations about the future developments of
mathematical logic in the twenty-first century. The areas of recursion theory,
proof theory and logic for computer science, model theory, and set theory are
discussed independently.Comment: Association for Symbolic Logi
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