[FOM] 159:Elemental Sentential Reflection
Harvey Friedman
friedman at math.ohio-state.edu
Mon Mar 31 00:17:28 EST 2003
A PDF version of this abstract will be available in a few days on
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ELEMENTAL SENTENTIAL REFLECTION
by
Harvey M. Friedman
Ohio State University
Princeton University
friedman at math.ohio-state.edu
http://www.math.ohio-state.edu/~friedman/
http://www.mathpreprints.com
February 6, 2003
March 3, 2002
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 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 [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(epsilon), which is the usual classical first order predicate
calculus with only the binary relation symbol epsilon (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(epsilon) with a distinguished free variable,
with parameters allowed.
In [Fr03], the following two forms of "sentential reflection" are considered.
if a given sentence of L(epsilon) holds in a given category
then it holds in a subclass.
if a given sentence of L(epsilon) holds in a given category
then it holds in an inclusion subclass.
These are called SR(epsilon) and SRIS(epsilon), respectively. In
[Fr03], they are shown to be mutually interpretable with Z_2 and ZFC,
respectively.
An elemental class is a class which is an element of a class. In some
common formalizations of the usual theory of classes that correspond
to NBG (the von Neumann Bernays Godel theory of classes), the
elemental classes are the sets, and the nonelemental classes are the
proper classes.
In "elemental sentential reflection", we reflect from a category of
elemental classes (i.e., a category all of whose elements are
elemental), to an elemental subclass (of the category of elemental
classes).
ESR(epsilon) and ESRIS(epsilon) are, respectively,
if a given sentence of L(epsilon) holds in a given category of
elemental classes then it holds in an elemental subclass.
if a given sentence of L(epsilon) holds in a given category of
elemental classes then it holds in an elemental inclusion subclass.
Just as in [Fr03] for SR(epsilon) and SRIS(epsilon), these are
formalized as double schemea, with one schematic letter presenting
the category, and one schematic letter presenting the sentence of
L(epsilon).
THEOREM 1.1. ESR(epsilon) is mutually interpretable with Z_2.
ESRIS(epsilon) is mutually interpretable with ZFC.
Thus by merely making the move to elemental sentential reflection, we
do not obtain anything new.
To take full advantage of elemental sentential reflection, we expand
the language L(() in a standard way to support quantification over
subclasses of the domain.
For this purpose, we use the two sorted language L^2(epsilon) with
variables over objects and variables over classes of objects, whose
atomic formulas are
x epsilon y
x epsilon A
where x,y are object variables and A is a class variable.
The object quantifiers are to range over the elements of the given
category or class (the domain), and the class quantifiers are to
range over all subclasses of the given category or class (the domain).
We emphasize that the language of all class theories considered here
will still be L(epsilon), despite the use of the "second order
language" L^2(epsilon).
ESR(epsilon*) is
if a given sentence of L^2(epsilon) holds in a given category of
elemental classes then it holds in an elemental subclass.
THEOREM 1.2. ESR(epsilon*) is mutually interpretable with ZFC +
{there exists a Pi-1-n indescribable cardinal}_n.
We now move to some extremely strong forms of elemental sentential
reflection. We define extensional equality as usual by x == y iff
(forall z)(z epsilon x iff z epsilon y).
Let K be a category or class. We say that x is an extensionally
proper subclass of K if and only if x is a subclass of K and
(therexists y epsilon K)(forall z == y)(z notepsilon x). Here x,y,z
are distinct variables.
ESR(epsilon*)# is
if a given sentence of L^2(epsilon) holds in a given category of
elemental classes not forming an elemental class, then it holds in a
nonelemental extensionally proper subclass.
THEOREM 1.3. ESR(epsilon*)# is interpretable in ZF + "there is a
nontrivial elementary embedding from some L(V(alpha+1)) into
L(V(alpha+1))" and interprets ZF + "there is a nontrivial elementary
embedding from some V(alpha+1) into V(alpha+1)".
It is well known that ZFC + "the existence of a measurable cardinal"
can be interpreted in ZF + "there is a nontrivial elementary
embedding from some V(alpha+1) into V(alpha+1)", or even just ZF +
"there is a nontrivial elementary embedding from some V(alpha) into
some transitive set". Such results are well known as far as the inner
model theory has been developed, which means, e.g., that "a proper
class of Woodin cardinals" can replace "the existence of a measurable
cardinal".
Woodin has many results concerning the relationship between large
cardinals above the reach of inner model theory with and without
choice.
E.g., Woodin has proved the consistency of ZFC + "there is a
nontrivial elementary embedding from some V(alpha+1) into V(alpha+1)"
from NBG + "there is a nontrivial elementary embedding from V into
V". Also Woodin has proved the consistency of ZFC + "there is a
cardinal which is n-huge for all finite n" in ZF + "there is a
nontrivial elementary embedding from some V(alpha+1) into V(alpha+1)".
Hence by Theorem 1.3 and results of Woodin, we see that ZFC + "there
is a cardinal which is n-huge for all finite n" is interpretable in
ESR(epsilon*)#.
The consensus among set theorists is that ZFC + "there is a
nontrivial elementary embedding from some L(V(alpha+1)) into itself"
is consistent.
Finally, consider the following system ESR(epsilon*)##.
if a given sentence of L^2(epsilon) holds in a given category of
elemental classes not forming an elemental class, then it holds in an
elemental subclass and in a nonelemental extensionally proper
subclass.
THEOREM 1.4. ESR(epsilon*)## is mutually interpretable with ZF +
{there exists a Pi-1-n indescribable V(alpha) and a nontrivial
elementary embedding from V(alpha) into V(alpha)}_n. ESR(epsilon*)##
proves the consistency of ZFC + "there is a nontrivial elementary
embedding from some V(alpha+1) into V(alpha+1)".
The last conclusion in Theorem 1.4 uses the above cited results of Woodin.
For the formal presentations of ESR(epsilon*), ESR(epsilon*)#,
ESR(epsilon*)##, let phi be a formula of L(epsilon), and let psi be a
sentence of L^2(epsilon) with no variables in common with phi. Let y
be a variable in phi. We want to define the formula psi[phi,y] of
L(epsilon). For the subclass variables Z in psi, associate object
variables Z', so that the Z' are distinct and do not appear in
phi,psi (and for technical reasons are not x,y). Let psi[phi,y] be
the result of replacing all quantifiers (Qz) in psi by
(Qz|phi[y/z])
and all quantifiers (QZ) in psi by
(QZ'|(forall y epsilon Z')(phi))
and expanding the result to a formula of L(epsilon).
Informally, psi[phi,y] is the (formula expressing the) result of
relativizing all object quantifiers in psi to {y: phi(y)} and all
subclass quantifiers to subclasses of {y: phi(y)}. In particular,
psi[y epsilon x,y] is the (formula expressing the) result of
relativizing all object quantifiers in psi to (the elements of) x,
and all subclass quantifiers to subclasses of x.
ESR(epsilon*) is the formal system in L(epsilon) whose nonlogical axioms are
((forall y)(phi implies E(y)) & psi[phi,y]) implies
(therexists x)(E(x) & (forall y epsilon x)(phi) & psi[y epsilon x,y])
where x,y are distinct variables, phi is a formula of L(epsilon) in
which x is not free, psi is a sentence of L^2(epsilon) with no
variables in common with phi, y is a variable in phi, E(x) is
(therexists y)(x epsilon y), and E(y) is (therexists x)(y epsilon x).
We leave the formalizations of ESR(epsilon*)# and ESR(epsilon*)## to
the reader.
2. ESR(epsilon) and ESRIS(epsilon).
3. ESR(epsilon*).
4. ESR(epsilon*)#.
5. ESR(epsilon*)##.
REFERENCES
[Fr03] Sentential reflection, preprint, January 5, 2003.
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This is the 159th 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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158:Sentential Reflection
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