[FOM] 350: one dimensional set series
Harvey Friedman
friedman at math.ohio-state.edu
Thu Jul 23 00:11:52 EDT 2009
PREFACE
Let f:N^k into N. We define a nondeterministic notion of f derived
subset of N. Just as in Cantor's famous (deterministic) derived sets,
our derived sets are subsets of the given set. And, as in Cantor, we
iterate the process, and end with a large set.
We prove that for strictly dominating f there exist finite series of
successive f derived sets of any finite length, ending with an
infinite set of odd integers. However, the proof necessarily uses
Mahlo cardinals of all finite orders.
CORRECTION: In #349, http://www.cs.nyu.edu/pipermail/fom/2009-July/013864.html
in section 1, in the definition of set series, I wrote
(E delta RE)-1.
This should be
E-1 delta RE.
ONE DIMENSIONAL SET SERIES AND INCOMPLETENESS
Harvey M. Friedman
July 22, 2009
We let N be the set of all nonnegative integers. We say that f:N^k
into N is upward if and only if for all x_1,...,x_k, f(x_1,...,x_k) >
x_1,...,x_k.
We define fE = {f(x): x in E^k}.
A sum set is a set of the form A+A, where A is a subset of N.
COMPLEMENTATION THEOREM. For all k >= 1 and strictly dominating f:N^k
into N, there exists a unique E contained in N such that E delta fE = N.
Actually, we can use any of the following four trivially equivalent
conclusions:
E delta fE = N
E U. fE = N
E = N\fE
fE = N\E.
Here U. means "disjoint union".
Let f:N^k into N and E contained in N. We say that E^ is f derived
from E if and only if
E^ is the intersection of E with a set A such that A+A is contained in
E delta fE.
A set series for f is a nonempty finite or infinite sequence of
subsets of N, where each set is derived from the preceding set using
f, as above.
THEOREM 2. For all k >= 1 and strictly dominating f:N^k into N, there
is an infinite set series for f consisting of the same infinite set.
Just take E,E,E,..., where E is given by the Complementation Theorem.
The sum sets used will be N = N+N.
PROPOSITION 3. For all k,n >= 1 and strictly dominating f:N^k into N,
there is a set series for f of length n ending with an infinite set of
odd integers.
Here is an explicitly Pi03 form of Proposition 3.
PROPOSITION 4. For all r,t >> k,n,p there exists t such that the
following holds. Let f:[0,t]^k into N be piecewise linear with
coefficients from [0,p]. There is a set series for f of length n,
consisting of subsets of [0,t], ending with the powers of r in [1,t].
Let SMAH = ZFC + {there is a strongly n-Mahlo cardinal}_n. Let SMAH+ =
ZFC + "for all n there is a strongly n-Mahlo cardinal".
THEOREM 5. Propositions 3,4 are provable in SMAH+ but not in any
consistent subsystem of SMAH. In particular, Proposition 3,4 are not
provable in ZFC, assuming ZFC is consistent. ACA proves that
Proposition 3,4 are equivalent to 1-Con(SMAH).
**********************************
I use http://www.math.ohio-state.edu/~friedman/ for downloadable
manuscripts. This is the 350th 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
342: Thematic Pi01 Incompleteness 3 5/23/09 7:48PM
343: Goedel's Second Revisited 1 5/27/09 6:07AM
344: Goedel's Second Revisited 2 6/1/09 9:21PM
345: Thematic Pi01 Incompleteness 4 6/15/09 1:15PM
appears misnumbered as 344.
346: Goedel's Second Revisited 3 6/16/09 11:04PM
347: Goedel's Second Revisited 4 6/20/09 1:25AM
348: Goedel's Second Revisited 5 6/22/09 11:00AM
349: Pi01 Incompleteness/set series 7/20/09 11:21PM
Harvey Friedman
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