[FOM] 160:Similar Subclasses
Harvey Friedman
friedman at math.ohio-state.edu
Mon Mar 31 00:17:47 EST 2003
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SIMILAR SUBCLASSES
by
Harvey M. Friedman
Ohio State University
March 11, 2003
Abstract. Reflection, in the sense of [Fr03a] and [Fr03b], is based
on the idea that a category of classes has a subclass that is
"similar" to the category. Here we present axiomatizations based on
the idea that a category of classes that does not form a class has
extensionally different subclasses that are "similar". We present two
such similarity principles, which are shown to interpret and be
interpretable in certain set theories with large cardinal axioms.
1. Introduction.
As in [Fr03a], [Fr03b], 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 epsilon
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 [Fr03a], [Fr03b], 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 [Fr03a], [Fr03b], all of our formal theories of classes are in
the language L(epsilon), which is the usual classical first order
predicate calculus with only the binary relation symbol epsilon (no
equality).
As in [Fr03a], [Fr03b], 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(epsilon) with a
distinguished free variable, with parameters allowed.
The Similar Subclass Principle (SSP) asserts the following:
any category not forming a class has two extensionally different
nonempty subclasses which are similar with respect to any two
elements of the first.
We formalize SSP in the expected way by approximation. Let L(epsilon)
be first order predicate calculus with only the binary relation
symbol epsilon (no equality).
SSP consists of the axioms
not(therexists y)(forall x)(x epsilon y iff phi) implies (therexists
y,z)((forall x epsilon y)(phi) & (forall x epsilon z)(phi) &
(therexists x)(x epsilon y) & (therexists x)(x epsilon z) &
(therexists x)(x epsilon y iff x notepsilon z) &
(forallu,v epsilon y)(psi_1 iff psi_1[y/z] & ... & psi_k iff psi_k[y/z]))
where x,y,z,u,v are distinct variables, phi,psi_1,...,psi_k are
formulas of L(epsilon), y,z not in phi, and all free variables in
psi_1,...,psi_k are among y,u,v.
The following definition is used in [Ba75] and [Fr01]. We say that an
ordinal lambda is subtle if and only if
i) lambda is a limit ordinal;
ii) Let C containedin lambda be closed and unbounded, and for each
alpha < lambda let A_alpha containedin alpha be given. There exists
alpha,beta epsilon C, alpha < beta, such that A_alpha = A_beta
intersect alpha.
It is well known that every subtle ordinal is a subtle cardinal (see
[Fr01], p. 3).
We will use the following schematic form of subtlety. SSUB is the
following scheme in the language of ZFC with epsilon,=. Let phi,psi
be formulas, where we view phi as carving out a class on the variable
x, and psi as carving out a binary relation on the variables x,y.
Parameters are allowed in phi,psi.
if phi defines a closed and unbounded class of ordinals and psi
defines a system A_alpha containedin alpha, for all ordinals alpha,
then there exists alpha,beta epsilon C, alpha < beta, where A_alpha =
A_beta intersect alpha.
THEOREM 1. SSP is mutually interpretable with SSUB. SSP is provable
in ZFC + there exists arbitrarily large subtle cardinals. SSP is
provable in SSUB + V = L.
The Strong Similar Subclass Principle (SSP*) asserts the following:
for any category K not forming a class and any class x, there is a
nonempty y containedin K such that for all z epsilon y, y and y\z are
similar with respect to any element of x.
We formalize SSP* in the expected way by approximation, just as we
formalized SSP.
THEOREM 2. ZFC + "there exists a measurable cardinal" is
interpretable in SSP*. SSP* is provable in NBG + "there are
elementary embeddings from V into V with arbitrarily large critical
points".
REFERENCES
[Fr03a] Sentential reflection, abstract.
[Fr03b] Elemental sentential reflection, abstract.
[Ba75] J. Baumgartner, Ineffability properties of cardinals I, in: A.
Hajnal, R. Rado, and V. Sos (eds.), Infinite and Finite Sets,
Colloquia Mathematica Societatis Janos Bolylai, vol. 10 (Amsterdam:
North-Holland, 1975), 109-130.
[Fr01] H. Friedman, Subtle Cardinals and Linear Orderings, Annals of
Pure and Applied Logic 107 (2001) 1-34.
*********************************************
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This is the 160th in a series of self contained postings to FOM covering
a wide range of topics in f.o.m. Previous ones counting from #100 are:
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102: Turing Degrees/2 4/8/01 5:20PM
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125:Disjoint Unions/First Classifications 3/1/02 6:19AM
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127:Combinatorial conditions/BRT 3/11/02 3:34AM
128:Finite BRT/Collapsing Triples 3/11/02 3:34AM
129:Finite BRT/Improvements 3/20/02 12:48AM
130:Finite BRT/More 3/21/02 4:32AM
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156:Societies 8/13/02 6:56PM
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158:Sentential Reflection
159:Elemental Sentential Reflection
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