4,698 research outputs found
First Order Theories of Some Lattices of Open Sets
We show that the first order theory of the lattice of open sets in some
natural topological spaces is -equivalent to second order arithmetic. We
also show that for many natural computable metric spaces and computable domains
the first order theory of the lattice of effectively open sets is undecidable.
Moreover, for several important spaces (e.g., , , and the
domain ) this theory is -equivalent to first order arithmetic
Normal Numbers and the Borel Hierarchy
We show that the set of absolutely normal numbers is -complete in the Borel hierarchy of subsets of real numbers. Similarly,
the set of absolutely normal numbers is -complete in the effective
Borel hierarchy
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