ElSentRef030303

ElSentRef030303 - 1 ELEMENTAL SENTENTIAL REFLECTION by...

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1 ELEMENTAL SENTENTIAL REFLECTION by Harvey M. Friedman Ohio State University Princeton University friedman@math.ohio-state.edu http://www.math.ohio-state.edu/~friedman/ February 6, 2003 March 3, 2003 Abstract. “Sentential reflection” in the sense of [Fr03] is based on reflecting down from a category of classes. “Elemental sentential reflection” is based on reflecting down from a category of elemental classes. We present various forms of elemental sentential reflection, which are shown to interpret and be interpretable in certain set theories with large cardinal axioms. 1. Introduction. As in [Fr03], we use “class” as a neutral term, without commitment to the developed notions of “set” and “class” that have become standard in set theory and mathematical logic. We use for membership. This framework supports interpretations of sentential reflection that may differ from conventional set theory or class theory. However, we do not pursue this direction here. As in [Fr03], this framework is intended to accommodate objects that are not classes. Such nonclasses are treated as classes with no elements. Thus we are careful not to assume extensionality. In fact, we will not assume any form of extensionality. As in [Fr03], all of our formal theories of classes are in the language L( ), which is the usual classical first order predicate calculus with only the binary relation symbol (no equality). As in [Fr03], we use “category of classes” or just “category” as a neutral term, not specifically related to category theory. They are given by a formula of L( ) with a distinguished free variable, with parameters allowed.
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2 In [Fr03], the following two forms of “sentential reflection” are considered. if a given sentence of L( ) holds in a given category then it holds in a subclass. if a given sentence of L( ) holds in a given category then it holds in an inclusion subclass.
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ElSentRef030303 - 1 ELEMENTAL SENTENTIAL REFLECTION by...

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