[FOM] 225:Better Pi01 Independence

Tue Nov 9 10:47:09 EST 2004

This time we have apparently made a substantive simplification.

We restate posting #224 removing the parameter r (i.e., r = 1).

We have made some cosmetic changes in the way the data is presented, that
makes things flow better. In particular, we give some versions with fewer
letters, as well as with more letters.

We also insert a "nonempty" which we mistakenly omitted in postings #223 and
#224.

My strategy is to let this round of simplifications settle down, at some
risk of an error, and then write up a good rough sketch. If things go as
planned, I will publish a pointer to the rough sketch.

Especially, see PROPOSITION 5 below. This is explicitly arithmetic, and with
the double exponential upper bounds, is explicitly Pi01.

This development of course does not obsolete the BRT development in any way,
shape, or form. However, the explicitly Pi01 statements below are
substantially better than any explicitly Pi01 statements presently coming
out of the BRT development.

##############################################

Let N be the set of all nonnegative integers. Let R containedin N^s+k. Let E
containedin N^s. We write

R<[E] = {y in N^k: (therexists x in N^s)(R(x,y) and max(x) < max(y))}.

R<[E] is read "the upper image of R on E".

We use U. for disjoint union.

All lower case letters represent arbitrary integers >= 1, unless indicated
otherwise.

THEOREM 1. Let R containedin N^2k. There exists a unique A
containedin N^k such that A U. R<[A] = N^k. A contains the origin.

Here is an obvious consequence of Theorem 1.

THEOREM 2. Let R_1,R_2,... containedin N^2k. There exist nonempty
A_1,A_2,... containedin N^k such that every nonempty R_i[A_j] meets every
A_p U. R_p<[A_p]. 

Let q >= 0 and a_1,...,a_q >= 0. We say that R containedin N^s is order
invariant over a_1,...,a_q if and only if for all x,y in N^s, if
(x,a_1,...,a_q),(y,a_1,...,a_q) have the same order type then x in R iff y
in R. 

Let n >= 0. The powers of n are the numbers n^1,n^2,n^3,... .

Let u >= 0. E without u is the set of all elements of E none of whose
coordinates are u.

PROPOSITION 3. Let R_1,R_2,... containedin N^2k each be order invariant over
some k powers of n, where n >> k. There exist nonempty A_1,A_2,...
containedin N^k such that every nonempty R_i[A_j] meets every A_p U.
R_p<[A_p] without n-1.

For finite forms, we start by modifying Theorem 2.

THEOREM 4. Let R_1,...,R_m containedin N^2k. There exist nonempty finite
A_1,...,A_m containedin N^k such that every nonempty R_i[A_j] meets every
A_p U. R_p<[A_p].

PROPOSITION 5. Let R_1,...,R_m containedin N^2k each be order invariant over
some k powers of n, where n >> k. There exist nonempty finite A_1,...,A_m
containedin N^k such that every nonempty R_i[A_j] meets every A_p U.
R_p<[A_p] without n-1.

We now introduce an important new parameter t in Propositions 3,5.

PROPOSITION 6. Let R_1,R_2,... containedin N^2k each be order invariant over
some t powers of n, where n >> k,t. There exist nonempty A_1,A_2,...
containedin N^k such that every nonempty R_i[A_j] meets every A_p U.
R_p<[A_p] without n-1.

PROPOSITION 7. Let R_1,...,R_m containedin N^2k each be order invariant over
some t powers of n, where n >> k,t. There exist nonempty finite A_1,...,A_m
containedin N^k such that every nonempty R_i[A_j] meets every A_p U.
R_p<[A_p] without n-1.

The bounds in Propositions 3,5,6,7 are innocent:

PROPOSITION 3'. Let R_1,R_2,,... containedin N^2k each be order invariant
over some k powers of n, where n >> k. There exist nonempty A_1,A_2,...
containedin N^k such that every nonempty R_i[A_j] meets every A_p U.
R_p<[A_p] without n-1. Furthermore, the >> can be given by a suitable double
exponential expression in k.

PROPOSITION 5'. Let R_1,...,R_m containedin N^2k each be order invariant
over some k powers of n, where n >> k. There exists nonempty finite
A_1,...,A_m containedin N^k such that every nonempty R_i[A_j] meets every
A_p U. R_p<[A_p] without n-1. Furthermore, the >> can be given by a suitable
double exponential expression in k. In addition, the largest coordinate in A
can be bounded above by a suitable double exponential expression in k and
the largest power of n used in the hypothesis.

PROPOSITION 6'. Let R_1,R_2,,... containedin N^2k each be order invariant
over some t powers of n, where n >> k,t. There exist nonempty A_1,A_2,...
containedin N^k such that every nonempty R_i[A_j] meets every A_p U.
R_p<[A_p] without n-1. Furthermore, the >> can be given by a suitable double
exponential expression in k,t.

PROPOSITION 7'. Let R_1,...,R_m containedin N^2k each be order invariant
over some t powers of n, where n >> k,t. There exists nonempty finite
A_1,...,A_m containedin N^k such that every nonempty R_i[A_j] meets every
A_p U. R_p<[A_p] without n-1. Furthermore, the >> can be given by a suitable
double exponential expression in k,t. In addition, the largest coordinate in
A can be bounded above by a suitable double exponential expression in k and
the largest power of n used in the hypothesis.

THEOREM 8. Propositions 3,5,6,7,3',5',6',7' are each provably equivalent,
over RCA0, to the consistency of MAH = ZFC + {there exists an n-Mahlo
cardinal}_n. If we remove "without n-1" then they become provable in RCA0.
In the case of Propositions 5,5',7,7'', we can use EFA instead of RCA0.

THEOREM 9. The level of Mahloness needed to prove any of Propositions
6,7,6',7' is roughly t.

There are other interesting ways to control the strength of these
Propositions. Instead of fixing t, we can set m to be simple functions of n.
This should allow us to get roughly Zermelo set theory and various fragments
thereof. In fact, there should emerge a nice correspondence between
numerical functions and consistency strengths of set theories, presumably
going all the way from fragments of finite set theory = PA up through ZFC
and various fragments, up through levels of the Mahlo cardinal hierarchy.

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

I use www.math.ohio-state.edu/~friedman/ for downloadable manuscripts.
This is the 224th 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 4/22/03  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
200:Advances in Sentential Reflection 12/22/03 11:17PM
201:Algebraic Treatment of First Order Notions 1/11/04 11:26PM
202:Proof(?) of Church's Thesis 1/12/04 2:41PM
203:Proof(?) of Church's Thesis - Restatement 1/13/04 12:23AM
204:Finite Extrapolation 1/18/04 8:18AM
205:First Order Extremal Clauses 1/18/04 2:25PM
206:On foundations of special relativistic kinematics 1 1/21/04 5:50PM
207:On foundations of special relativistic kinematics 2  1/26/04  12:18AM
208:On foundations of special relativistic kinematics 3  1/26/04  12:19AAM
209:Faithful Representation in Set Theory with Atoms 1/31/04 7:18AM
210:Coding in Reverse Mathematics 1  2/2/04  12:47AM
211:Coding in Reverse Mathematics 2  2/4/04  10:52AM
212:On foundations of special relativistic kinematics 4  2/7/04  6:28PM
213:On foundations of special relativistic kinematics 5  2/8/04  9:33PM
214:On foundations of special relativistic kinematics 6  2/14/04 9:43AM
215:Special Relativity Corrections  2/24/04 8:13PM
216:New Pi01 statements  6/6/04  6:33PM
217:New new Pi01 statements  6/13/04  9:59PM
218:Unexpected Pi01 statements  6/13/04  9:40PM
219:Typos in Unexpected Pi01 statements  6/15/04  1:38AM
220:Brand New Corrected Pi01 Statements  9/18/04  4:32AM
221:Pi01 Statements/getting it right  10/7/04  5:56PM
222:Statements/getting it right again  10/9/04  1:32AM
223:Better Pi01 Independence  11/2/04  11:15AM
224:Prettier Pi01 Independence  11/7/04  8:11PM

Harvey Friedman