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ELEMENTAL SENTENTIAL REFLECTION
by
Harvey M. Friedman
Ohio State University
Princeton University
[email protected]
http://www.math.ohiostate.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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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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