[FOM] 200:Advances in Sentential Reflection

[Harvey Friedman]

ADVANCES IN SENTENTIAL REFLECTION
by 
Harvey M. Friedman
12/22/03

We refer the reader to

[1] H.M. Friedman, Sentential Reflection, http://www.mathpreprints.com/

Also see 

[2] H.M. Friedman, Reflection,
http://www.mathpreprints.com/math/Preprint/HarveyFriedman/20031218/1

for a more general discussion.

Recall the following most fundamental form of Sentential Reflection from
[1].

SR(epsilon). If a sentence of L(epsilon) holds in a given category (of
classes), then it holds in some subclass (of the given category).

As stated in [1], SR(epsilon), with no other axioms, is mutually
interpretable with ZFC without the power set axiom (countable set theory),
or, equivalently, mutually interpretable with the (first order theory of)
second order arithmetic, Z_2.

To get mutual interpretability with ZFC, we strengthened SR(epsilon) in [1]
by using what we called "inclusion subclasses".

We say that a class A is an inclusion subclass of the category K if and only
if 

i) A is a subclass of K;
ii) every element of K that is a subclass of an element of A is itself an
element of A. 

SRIS(epsilon). If a given sentence of L(epsilon) holds in a given category,
then it holds in some inclusion subclass.

We proved that SRIS(epsilon), with no other axioms, is mutually
interpretable with ZFC.

But how do we justify the move from "subclasses" to "inclusion subclasses"?
It is possible that one can find a philosophically fundamental reason for
this move. However, we take a different approach here.

In fact, we now make the same crucial "simplicity move" that we have been
making with some frequency in recent years in several contexts in the
foundations of mathematics.

Semiformally, let P be a binary relation between classes. We say that a
class A is a P closed subclass of a category K if and only if

i) A is a subclass of K;
ii) every element of K that is related by P to an element of A is itself an
element of A.

Note that A is an inclusion subclass of K if and only if A is a P closed
subclass of K, where P is given by

P(x,y) if and only if x is a subclass of y.

Formally, let P be any binary relation on classes given by a formula of
L(epsilon) with the two distinct free variables x,y.

We define *sentential reflection for P* to be the scheme

SR(epsilon,P). If a given sentence of L(epsilon) holds in a given category,
then it holds in some P closed subclass.

Thus SRIS(epsilon) is equivalent to SR(epsilon,P), where P is the inclusion
relation.

We have now set things up for a detailed simplicity study.

If I didn't have more pressing commitments right now, I would be working on
the following, with real confidence of success.

*Determine the logical strength, or metamathematical status, of all of the
SR(epsilon,P), where P has at most one quantifier.*

This obviously includes the case where P is inclusion.

There are only finitely many P to consider, since there are only finitely
many formulas in L(epsilon) with a given number of quantifiers, up to
logical equivalence.

For the beginnings of a simplicity study for the truth of sentences of set
theory, see
 
[3] H.M. Friedman, Three-quantifier sentences, Fund. Math., 177 (2003),
213-240. 

It seems "clear" that one can gain a complete understanding of what happens
here in the case where P has at most one quantifier. Of course, one must
also know how to show high logical strength, as in the case where P is the
inclusion relation - a nontrivial task.

Since the actual number of (even) single quantifier P is rather large, we do
want some specific clarifying results about the finitely many schemes that
we get, SR(epsilon,P).

We conjecture (for at most single quantifier P) that all such schemes are
either inconsistent, with small, explicitly constructed, inconsistencies, or
are interpretable in ZFC. This suggests a possible general principle
(discussed by me for years) that

*if a simple scheme isn't obviously inconsistent, then it is consistent, and
in fact interpretable in a well known set theory*

Now what about two quantifier P? This looks like a serious challenge. I
worked with two quantifier formulas with one free variable in [3]. But here
we have to work with two quantifier formulas with two free variables. Ouch!
However, we are not allowing equality, which certainly helps.

In any case, I think I have set things up right for a major simplicity study
for sentential reflection.

*********************************************

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This is the 198th in a series of self contained numbered postings to
FOM covering a wide range of topics in f.o.m. The list of previous
numbered postings #1-149 can be found at
http://www.cs.nyu.edu/pipermail/fom/2003-May/006563.html  in the FOM
archives, 5/8/03 8:46AM. Previous ones counting from #150 are:

150:Finite obstruction/statistics  8:55AM  6/1/02
151:Finite forms by bounding  4:35AM  6/5/02
152:sin  10:35PM  6/8/02
153:Large cardinals as general algebra  1:21PM  6/17/02
154:Orderings on theories  5:28AM  6/25/02
155:A way out  8/13/02  6:56PM
156:Societies  8/13/02  6:56PM
157:Finite Societies  8/13/02  6:56PM
158:Sentential Reflection  3/31/03  12:17AM
159.Elemental Sentential Reflection  3/31/03  12:17AM
160.Similar Subclasses  3/31/03  12:17AM
161:Restrictions and Extensions  3/31/03  12:18AM
162:Two Quantifier Blocks  3/31/03  12:28PM
163:Ouch!  4/20/03  3:08AM
164:Foundations with (almost) no axioms, 4/22/0  5:31PM
165:Incompleteness Reformulated  4/29/03  1:42PM
166:Clean Godel Incompleteness  5/6/03  11:06AM
167:Incompleteness Reformulated/More  5/6/03  11:57AM
168:Incompleteness Reformulated/Again 5/8/03  12:30PM
169:New PA Independence  5:11PM  8:35PM
170:New Borel Independence  5/18/03  11:53PM
171:Coordinate Free Borel Statements  5/22/03  2:27PM
172:Ordered Fields/Countable DST/PD/Large Cardinals  5/34/03  1:55AM
173:Borel/DST/PD  5/25/03  2:11AM
174:Directly Honest Second Incompleteness  6/3/03  1:39PM
175:Maximal Principle/Hilbert's Program  6/8/03  11:59PM
176:Count Arithmetic  6/10/03  8:54AM
177:Strict Reverse Mathematics 1  6/10/03  8:27PM
178:Diophantine Shift Sequences  6/14/03  6:34PM
179:Polynomial Shift Sequences/Correction  6/15/03  2:24PM
180:Provable Functions of PA  6/16/03  12:42AM
181:Strict Reverse Mathematics 2:06/19/03  2:06AM
182:Ideas in Proof Checking 1  6/21/03 10:50PM
183:Ideas in Proof Checking 2  6/22/03  5:48PM
184:Ideas in Proof Checking 3  6/23/03  5:58PM
185:Ideas in Proof Checking 4  6/25/03  3:25AM
186:Grand Unification 1  7/2/03  10:39AM
187:Grand Unification 2 - saving human lives 7/2/03 10:39AM
188:Applications of Hilbert's 10-th 7/6/03  4:43AM
189:Some Model theoretic Pi-0-1 statements  9/25/03  11:04AM
190:Diagrammatic BRT 10/6/03  8:36PM
191:Boolean Roots 10/7/03  11:03 AM
192:Order Invariant Statement 10/27/03 10:05AM
193:Piecewise Linear Statement  11/2/03  4:42PM
194:PL Statement/clarification  11/2/03  8:10PM
195:The axiom of choice  11/3/03  1:11PM
196:Quantifier complexity in set theory  11/6/03  3:18AM
197:PL and primes 11/12/03  7:46AM
198:Strong Thematic Propositions 12/18/03 10:54AM
199:Radical Polynomial Behavior Theorems

Harvey Friedman