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Reverse Mathematics: The Playground of Logic
, 2010
"... The general enterprise of calibrating the strength of classical mathematical theorems in terms of the axioms (typically of set existence) needed to prove them was begun by Harvey Friedman in [1971] (see also [1967]). His goals were both philosophical and foundational. What existence assumptions are ..."
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The general enterprise of calibrating the strength of classical mathematical theorems in terms of the axioms (typically of set existence) needed to prove them was begun by Harvey Friedman in [1971] (see also [1967]). His goals were both philosophical and foundational. What existence assumptions are really needed to develop classical mathematics
Reverse mathematics, countable and uncountable: a computational approach
, 2009
"... Reverse mathematics analyzes the complexity of mathematical statements in terms of the strength of axiomatic systems needed to prove them. Its setting is countable mathematics and subsystems of second order arithmetic. We present a similar analysis based on (recursion theoretic) computational comple ..."
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Reverse mathematics analyzes the complexity of mathematical statements in terms of the strength of axiomatic systems needed to prove them. Its setting is countable mathematics and subsystems of second order arithmetic. We present a similar analysis based on (recursion theoretic) computational complexity instead. In the countable case, this view is implicit in many of results in the area. By making it explicit and precise, we provide an alternate approach to this type of analysis for countable mathematics. It may be more intelligible to some mathematicians in that it replaces logic and proof systems with relative computability. In the uncountable case, second order arithmetic and its proof theory is insufficient for the desired analysis. Our computational approach, however, supplies a ready made paradigm for similar analyses. It can be implemented with any appropriate notion of computation on uncountable sets.
A Notion of a Computational Step for Partial Combinatory Algebras
"... Abstract. Working within the general formalism of a partial combinatory algebra (or PCA), we introduce and develop the notion of a step algebra, which enables us to work with individual computational steps, even in very general and abstract computational settings. We show that every partial applicat ..."
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Abstract. Working within the general formalism of a partial combinatory algebra (or PCA), we introduce and develop the notion of a step algebra, which enables us to work with individual computational steps, even in very general and abstract computational settings. We show that every partial applicative structure is the closure of a step algebra obtained by repeated application, and identify conditions under which this closure yields a PCA.
IOS Press
"... Computability in type2 objects with wellbehaved type1 oracles is pnormal ..."
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Computability in type2 objects with wellbehaved type1 oracles is pnormal