[FOM] 229:More Progress in Pi01 Independence

Sat Nov 13 22:41:23 EST 2004

We are still waiting for this round to stabilize, before preparing a rough
sketch.

These developments in no way, shape, or form obsoletes BRT.

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Let N be the set of all nonnegative integers. We will use the lexicographic
ordering of N^k. We also use the sup norm |x| on N^k. We will make use of
the spheres in N^k, under | |.

We will take up the matter of other norms, including the Euclidean norm, at
a later time.

Let R containedin N^2k. We say that A is an antichain for R if and only if A
containedin N^k and for all x,y in E, not R(x,y).

We say that R is strictly dominating if and only if R(x,y) implies |x| <
|y|.

THEOREM 1. For all k >= 1, every strictly dominating R containedin N^2k has
an antichain A containing the k-vectors outside R[A]. Furthermore, A is
unique and contains the origin.

Control over A is hard to obtain:

THEOREM 2. The following is false. For all k >= 1, every strictly dominating
R containedin N^2k has an antichain A containing the k-vectors outside R[A],
where A lies outside the sphere of radius 64^k! -1. We cannot repair this
statement by changing the sphere.

Perhaps we can obtain some control over A if we assume that R is very
concrete:

Let R containedin N^s. We say that R is order invariant if and only if for
all x,y in N^s of the same order type, R(x) iff R(y).

THEOREM 3. The following is false. For all k >= 1, every strictly dominating
order invariant R containedin N^2k has an antichain A containing the
k-vectors outside R[A], where A lies outisde the sphere of radius 64^k! -1.
We cannot repair this statement by changing the sphere in any explicit way
(this can be made rigorous).

We now give up working with all k-vectors outside R[A]:

TEMPLATE. For all k,p >= 1, every strictly dominating order invariant R
containedin N^2k has a finite antichain A containing **all vectors
'generated' by A,p** outside R[A], where A lies outside the sphere of radius
64^k! -1.

Let A containedin N^k and p >= 0. The A,p-minimal vectors are the
lexicographically least elements of the sets

S[{u} x A]

where S containedin N^3k is order invariant and u in {1,2,4,...,2^p}^k, and
we take the least element of the empty set to be 0.

PROPOSITION 4. For all k,p >= 1, every strictly dominating order invariant R
containedin N^2k has a finite antichain A containing the A,p-minimal vectors
outside R[A], where A lies outside the sphere of radius 64^k! -1 in N^k.

In fact, we can use the decrement by 1 of any sufficiently large power of 2
relative to k. Other quantities can also be used.

Note that Proposition 4 is explicitly Pi02. We can write A containedin
[0,b]^k, where b = b(k,p) is an innocent function of k,p, without changing
Proposition 4, so that Proposition 4 becomes explicitly Pi01.

As things stabilize, we will give a reasonable expression for b = b(k,p),
better than double exponential. Also we will use an expression that is more
careful (i.e., lower) than 64^k! -1.

THEOREM 5. Theorems 1-3 are provable in RCA0. Proposition 4 is provably
equivalent, over EFA, to the consistency of MAH = ZFC + {there exists an
n-Mahlo cardinal}_n. If we remove "where A lies outside the sphere of radius
64^k! -1 in N^k", then Proposition 4 become provable in RCA0, as it becomes
a trivial consequence of Theorem 1.

If we set p to be certain simple functions of k, rather than arbitrary, then
we can control the strength of the statement somewhat. We should be able to
get PA and n-th order arithmetic, for various n, as well as significant
fragments of ZFC, ZFC itself, and levels of the Mahlo hierarchy.

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

I use www.math.ohio-state.edu/~friedman/ for downloadable manuscripts.
This is the 229th 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
225:Better Pi01 Independence  11/9/04  10:47AM
226:Nicer Pi01 Independence  11/10/04  10:43AM
227:Progress in Pi01 Independence  11/11/04  11:22PM
228:Further Progress in Pi01 Independence  11/12/04  2:49AM

Harvey Friedman