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Fundamental notions of analysis in subsystems of second-order arithmetic

Summary: Fundamental notions of analysis
in subsystems of second-order arithmetic
Jeremy Avigad
Department of Philosophy
Carnegie Mellon University
Pittsburgh, PA 15213
Ksenija Simic
Department of Mathematics
The University of Arizona
Tucson, AZ 85721
We develop fundamental aspects of the theory of metric, Hilbert, and Banach spaces
in the context of subsystems of second-order arithmetic. In particular, we explore
issues having to do with distances, closed subsets and subspaces, closures, bases,
norms, and projections. We pay close attention to variations that arise when for-
malizing definitions and theorems, and study the relationships between them. For
example, we show that a natural formalization of the mean ergodic theorem can be
proved in ACA0; but even recognizing the theorem's "equivalent" existence asser-
tions as such can also require the full strength of ACA0.


Source: Avigad, Jeremy - Departments of Mathematical Sciences & Philosophy, Carnegie Mellon University


Collections: Multidisciplinary Databases and Resources; Mathematics