[FOM] 342:Thematic Pi01 Incompleteness 3
Harvey Friedman
friedman at math.ohio-state.edu
Sat May 23 16:54:05 EDT 2009
There has been a simplification in the independent statement, pushing
it closer, grammatically, to everyday mathematics. I also changed the
terminology to "positive image terms in A,A',RA".
THEMATIC Pi01 INCOMPLETENESS 3
DRAFT
Harvey M. Friedman
May 23, 2009
We define N to be the set of all nonnegative integers. For any set As
containedin N^k, we write A' = N^k\A. The difference between two sets
A,B is A\B.
Let R containedin N^k x N^k = N^2k. We say that R is upwards if and
only if
for all x,y in N^k, R(x,y) implies max(x) < max(y).
We define RA for A containedin N^k, by
RA = {y in N^k: (therexists x in A)((x,y) in R)}.
THEOREM 1. For all k >= 1 and upwards R containedin N^2k, there exists
A containedin N^k, such that RA = A'. Furthermore, A is unique.
We want to work with very concrete R containedin N^2k.
We say that x,y in N^k are order equivalent if and only if for all 1
<= i,j <= k, x_i < x_j iff y_i < y_j.
We say that R containedin N^k is order invariant if and only if for
all order equivalent x,y in N^k, x in R iff y in R.
We have the following concrete form of Theorem 1. Here EXP means "the
characteristic function is computable in exponential time".
THEOREM 2. For all k >= 1 and upwards order invariant R containedin
N^2k, there exists A containedin N^k such that RA = A'. Furthermore, A
is unique and lies in EXP.
We now consider a family of expressions involving
R,A,A',union,intersection, which we call
*the positive image terms in R,A,A'*.
These are defined inductively as follows.
i. A,A' are positive image terms in R,A,A'.
ii. if s,t are positive image terms in R,A,A', then (s union t), (s
intersect t) are positive image terms in R,A,A'.
iii. if s is a positive image term in R,A,A', then R(s) is a positive
image term in R,A,A'.
Let s be a positive image term in R,A,A'. We define the rank of s to
be the number of occurrences of R in t. We say that s is similar to t
if and only if t is obtained from s by replacing some occurrences of
RA by A', and some occurrences of A' by RA. Here "some" means "none,
some, or all".
In an abuse of notation, if k,R,A have been specified, we think of t
both as a syntactic object and as the subset of N^k obtained by
evaluation.
The following is an obvious weakening of Theorem 2.
THEOREM 3. For all k >= 1 and upwards order invariant R containedin
N^2k, there exists A containedin N^k such that any two similar
positive image terms in R,A,A' are equal. Furthermore, A is unique and
lies in EXP.
Let U,V containedin N^k and S containedin N. We say that U,V agree
over S if and only if
U intersection S^k =
V intersection S^k.
PROPOSITION 4. For all k >= 1 and upwards order invariant R
containedin N^2k, there exists A containedin N^k such that any two
similar positive image terms in R,A,A' of rank at most k agree over some
infinite set disjoint from A+1.
More concreteness:
PROPOSITION 5. For all k >= 1 and upwards order invariant R
containedin N^2k, there exists A containedin N^k such that any two
similar positive image terms in R,A,A' of rank at most k agree over
some infinite geometric progression disjoint from A+1.
More concreteness:
PROPOSITION 6. For all k >= 1 and upwards order invariant R
containedin N^2k, there exists A containedin N^k such that any two
similar positive image terms in R,A,A' of rank at most k agree over
some infinite geometric progression disjoint from A+1, whose second
term is at most (8k)!.
More concreteness:
PROPOSITION 7. For all k,r >= 1 and upwards order invariant R
containedin N^2k, there exists finite A containedin N^k such that any
two similar positive image terms in R,A,A' of rank at most k agree
over some length r+1 geometric progression disjoint from A+1, whose
second term is at most (8k)!.
In Proposition 7, it is easy to see that only numbers from 0 through
(8k)!^r are relevant. Moreover, we can easily adjust the statement so
that only a finite interval of numbers are even mentioned. It is
particularly convenient to use
R containedin [0,r]^2k
A containedin [0,r]
and define order invariance for R relative to the restricted domain
[0,r], and define A' to be [0,r]\A. We thus arrive at the explicitly
Pi01 statement
PROPOSITION 8. For all k,r >= 1 and upwards order invariant R
containedin [0,r]^2k, there exists finite A containedin [0,r]^k such
that any two similar positive image terms in R,A,A' of rank at most k
agree over some length r+1 geometric progression disjoint from A+1,
whose second term is at most (8k)!.
Obviously, the geometric progression will go out of [0,r], but that is
of no consequence.
SMAH = ZFC + {there exists a strongly n-Mahlo cardinal}_n.
SMAH+ = ZFC + "for all n there exists a strongly n-Mahlo cardinal".
THEOREM 9. Propositions 4-8 are provably equivalent to Con(SMAH) over
ACA. They are not provable in any consistent fragment of SMAH. They
are provable in SMAH+ but not in SMAH (the latter using that SMAH is
consistent). These results still hold if we replace "rank at most k"
with "rank at most 8".
We look forward to adjusting the nature of the geometric progression
to get lots of important consistency strengths below SMAH. I.e., what
relationship do we want between the length of the progression and its
second term? Alternatively, replace r by specific expressions in k.
**********************************
I use http://www.math.ohio-state.edu/~friedman/ for downloadable
manuscripts. This is the 342nd 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 #1-249 can be found at
http://www.cs.nyu.edu/pipermail/fom/2005-June/008999.html in the FOM
archives, 6/15/05, 9:18PM. NOTE: The title of #269 has been corrected
from the original.
250. Extreme Cardinals/Pi01 7/31/05 8:34PM
251. Embedding Axioms 8/1/05 10:40AM
252. Pi01 Revisited 10/25/05 10:35PM
253. Pi01 Progress 10/26/05 6:32AM
254. Pi01 Progress/more 11/10/05 4:37AM
255. Controlling Pi01 11/12 5:10PM
256. NAME:finite inclusion theory 11/21/05 2:34AM
257. FIT/more 11/22/05 5:34AM
258. Pi01/Simplification/Restatement 11/27/05 2:12AM
259. Pi01 pointer 11/30/05 10:36AM
260. Pi01/simplification 12/3/05 3:11PM
261. Pi01/nicer 12/5/05 2:26AM
262. Correction/Restatement 12/9/05 10:13AM
263. Pi01/digraphs 1 1/13/06 1:11AM
264. Pi01/digraphs 2 1/27/06 11:34AM
265. Pi01/digraphs 2/more 1/28/06 2:46PM
266. Pi01/digraphs/unifying 2/4/06 5:27AM
267. Pi01/digraphs/progress 2/8/06 2:44AM
268. Finite to Infinite 1 2/22/06 9:01AM
269. Pi01,Pi00/digraphs 2/25/06 3:09AM
270. Finite to Infinite/Restatement 2/25/06 8:25PM
271. Clarification of Smith Article 3/22/06 5:58PM
272. Sigma01/optimal 3/24/06 1:45PM
273: Sigma01/optimal/size 3/28/06 12:57PM
274: Subcubic Graph Numbers 4/1/06 11:23AM
275: Kruskal Theorem/Impredicativity 4/2/06 12:16PM
276: Higman/Kruskal/impredicativity 4/4/06 6:31AM
277: Strict Predicativity 4/5/06 1:58PM
278: Ultra/Strict/Predicativity/Higman 4/8/06 1:33AM
279: Subcubic graph numbers/restated 4/8/06 3:14AN
280: Generating large caridnals/self embedding axioms 5/2/06 4:55AM
281: Linear Self Embedding Axioms 5/5/06 2:32AM
282: Adventures in Pi01 Independence 5/7/06
283: A theory of indiscernibles 5/7/06 6:42PM
284: Godel's Second 5/9/06 10:02AM
285: Godel's Second/more 5/10/06 5:55PM
286: Godel's Second/still more 5/11/06 2:05PM
287: More Pi01 adventures 5/18/06 9:19AM
288: Discrete ordered rings and large cardinals 6/1/06 11:28AM
289: Integer Thresholds in FFF 6/6/06 10:23PM
290: Independently Free Minds/Collectively Random Agents 6/12/06
11:01AM
291: Independently Free Minds/Collectively Random Agents (more) 6/13/06
5:01PM
292: Concept Calculus 1 6/17/06 5:26PM
293: Concept Calculus 2 6/20/06 6:27PM
294: Concept Calculus 3 6/25/06 5:15PM
295: Concept Calculus 4 7/3/06 2:34AM
296: Order Calculus 7/7/06 12:13PM
297: Order Calculus/restatement 7/11/06 12:16PM
298: Concept Calculus 5 7/14/06 5:40AM
299: Order Calculus/simplification 7/23/06 7:38PM
300: Exotic Prefix Theory 9/14/06 7:11AM
301: Exotic Prefix Theory (correction) 9/14/06 6:09PM
302: PA Completeness 10/29/06 2:38AM
303: PA Completeness (restatement) 10/30/06 11:53AM
304: PA Completeness/strategy 11/4/06 10:57AM
305: Proofs of Godel's Second 12/21/06 11:31AM
306: Godel's Second/more 12/23/06 7:39PM
307: Formalized Consistency Problem Solved 1/14/07 6:24PM
308: Large Large Cardinals 7/05/07 5:01AM
309: Thematic PA Incompleteness 10/22/07 10:56AM
310: Thematic PA Incompleteness 2 11/6/07 5:31AM
311: Thematic PA Incompleteness 3 11/8/07 8:35AM
312: Pi01 Incompleteness 11/13/07 3:11PM
313: Pi01 Incompleteness 12/19/07 8:00AM
314: Pi01 Incompleteness/Digraphs 12/22/07 4:12AM
315: Pi01 Incompleteness/Digraphs/#2 1/16/08 7:32AM
316: Shift Theorems 1/24/08 12:36PM
317: Polynomials and PA 1/29/08 10:29PM
318: Polynomials and PA #2 2/4/08 12:07AM
319: Pi01 Incompleteness/Digraphs/#3 2/12/08 9:21PM
320: Pi01 Incompleteness/#4 2/13/08 5:32PM
321: Pi01 Incompleteness/forward imaging 2/19/08 5:09PM
322: Pi01 Incompleteness/forward imaging 2 3/10/08 11:09PM
323: Pi01 Incompleteness/point deletion 3/17/08 2:18PM
324: Existential Comprehension 4/10/08 10:16PM
325: Single Quantifier Comprehension 4/14/08 11:07AM
326: Progress in Pi01 Incompleteness 1 10/22/08 11:58PM
327: Finite Independence/update 1/16/09 7:39PM
328: Polynomial Independence 1 1/16/09 7:39PM
329: Finite Decidability/Templating 1/16/09 7:01PM
330: Templating Pi01/Polynomial 1/17/09 7:25PM
331: Corrected Pi01/Templating 1/20/09 8:50PM
332: Preferred Model 1/22/09 7:28PM
333: Single Quantifier Comprehension/more 1/26/09 4:32PM
334: Progress in Pi01 Incompleteness 2 4/3/09 11:26PM
335: Undecidability/Euclidean geometry 4/27/09 1:12PM
336: Undecidability/Euclidean geometry/2 4/29/09 1:43PM
337: Undecidability/Euclidean geometry/3 5/3/09 6:54PM
338: Undecidability/Euclidean geometry/4 5/5/09 6:38PM
339: Undecidability/Euclidean geometry/5 5/7/09 2:25PM
340: Thematic Pi01 Incompleteness 1 5/13/09 5:56PM
341: Thematic Pi01 Incompleteness 2 5/21/09 7:25PM
Harvey Friedman
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