[FOM] 235:Neatening Affine Pi01 Independence
Harvey Friedman
friedman at math.ohio-state.edu
Sun Nov 28 18:08:05 EST 2004
In this posting, we make some minor changes that improve the flow of the
statement. See Propositions 4,5.
It appears that this round is very close to stabilizing, at which time I
will roll up my sleeves and check everything in detail, and provide a rough
sketch.
These developments in no way, shape, or form obsolete BRT.
##########################################
Let N be the set of all nonnegative integers. For A containedin N and k >=
1, let A^k be the set of all k-tuples from A. Let A^<k be the set of all
tuples from A of nonzero length < k.
For r >= 0, let [r] = {0,...,r}. We say that A,B disagree at z if and only
if A,B are sets and z in A iff z notin B.
Let T:B^k into N, and B,E containedin N^k. We define the upper image of T on
E by
T<[E] = {T(x): x in E and T(x) > max(x)}.
We start with a fixed point theorem.
THEOREM 1. For all T:N^k into N, some A containedin N disagrees with T<[A^k]
at all elements of N.
Theorem 1 has the following obvious finite form.
THEOREM 2. For all T:[p]^k into N, some A containedin [p] disagrees with
T<[A^k] at all elements of [p].
It is easy to see that 0 must lie in A. We can study requirements concerning
membership of positive integers in A.
THEROEM 3. For all T:[p]^k into N, some A containedin [p] including 1 or 2
disagrees with T<[A^k] at all elements of [p].
We note that in Theorem 1, even if k = 1, it is not possible to specify any
positive integer in advance to be included in A, depending only on p.
However, the situation changes if we use rather well behaved T and weaken
"at all elements of N" to integers that are in some sense "generated" by
T,A, and the factorials.
Let k,r >= 1 and E containedin [r]. We write RAF([r]^k,E) for the set of all
restricted affine transformations T:[r]^k into N over E. These are the
affine transformations T:[r]^k into N restricted to the solution set of a
system of linear inequalities, where all coefficients used in the
transformation and the system are from E.
We use cross section notation T_x, where x is a vector of any nonzero finite
length. Note that dom(T_x) depends on the length of x. If x is too long,
then obviously dom(T_x) = emptyset.
We take min(emptyset) = 0. For x in N^s, we define x! = (x_1!,...,x_s!).
PROPOSITION 4. For all T in RAF([p]^k,[k]), some finite A containedin N
including (8k)!! disagrees with T<[A^k] at all min(T_x![A^<k]).
Proposition 4 is easily seen to be Pi01, since we can bound the A and x!.
With a bit of extra work, one can arrive at the following elegant explicitly
Pi01 form, which should be compared with Theorems 2 and 3.
PROPOSITION 5. For all T in RAF([p]^k,[k]), some A containedin [p] including
(8k)!! disagrees with T<[A^k] at all min(T_x![A^<k]) <= p.
As things stabilize, we will sharpen the (8k)!!.
THEOREM 6. Theorems 1-3 are provable in RCA0. Theorems 2,3 are provable in
EFA. Propositions 4,5 is provably equivalent, over ACA, to the consistency
of MAH = ZFC + {there exists an n-Mahlo cardinal}_n. If we remove
"containing (8k)!!" from Proposition 4 or 5, the resulting statement is
provable in EFA.
If we set p to be certain simple functions of k, rather than arbitrary, then
we can control the strength of Proposition 4 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 235th 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
229:More Progress in Pi01 Independence 11/13/04 10:41PM
230:Piecewise Linear Pi01 Independence 11/14/04 9:38PM
231:More Piecewise Linear Pi01 Independence 11/15/04 11:18PM
232:More Piecewise Linear Pi01 Independence/correction 11/16/04 8:57AM
233:Neatening Piecewise Linear Pi01 Independence 11/17/04 12:22AM
234:Affine Pi01 Independence 11/20/04 9:54PM
Harvey Friedman
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