595 research outputs found
Logic and operator algebras
The most recent wave of applications of logic to operator algebras is a young
and rapidly developing field. This is a snapshot of the current state of the
art.Comment: A minor chang
Leibniz's Principles and Topological Extensions
Three philosophical principles are often quoted in connection with Leibniz:
"objects sharing the same properties are the same object", "everything can
possibly exist, unless it yields contradiction", "the ideal elements correctly
determine the real things". Here we give a precise formulation of these
principles within the framework of the Topological Extensions of [8],
structures that generalize at once compactifications, completions, and
nonstandard extensions. In this topological context, the above Leibniz's
principles appear as a property of separation, a property of compactness, and a
property of analyticity, respectively. Abiding by this interpretation, we
obtain the somehow surprising conclusion that these Leibnz's principles can be
fulfilled in pairs, but not all three together.Comment: 16 page
A topological interpretation of three Leibnizian principles within the functional extensions
Three philosophical principles are often quoted in connection with Leibniz:
"objects sharing the same properties are the same object" (Identity of
indiscernibles), "everything can possibly exist, unless it yields
contradiction" (Possibility as consistency), and "the ideal elements correctly
determine the real things" (Transfer). Here we give a precise
logico-mathematical formulation of these principles within the framework of the
Functional Extensions, mathematical structures that generalize at once
compactifications, completions, and elementary extensions of models. In this
context, the above Leibnizian principles appear as topological or algebraic
properties, namely: a property of separation, a property of compactness, and a
property of directeness, respectively. Abiding by this interpretation, we
obtain the somehow surprising conclusion that these Leibnizian principles may
be fulfilled in pairs, but not all three together.Comment: arXiv admin note: substantial text overlap with arXiv:1012.434
Techniques for approaching the dual Ramsey property in the projective hierarchy
We define the dualizations of objects and concepts which are essential for
investigating the Ramsey property in the first levels of the projective
hierarchy, prove a forcing equivalence theorem for dual Mathias forcing and
dual Laver forcing, and show that the Harrington-Kechris techniques for proving
the Ramsey property from determinacy work in the dualized case as well
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