[FOM] 328: Polynomial Independence 1
Harvey Friedman
friedman at math.ohio-state.edu
Fri Jan 16 19:06:05 EST 2009
We use N for the set of all nonnegative integers, and Z for the set of
all integers.
Let x,y in N^k. We write x <=* y if and only if for all 1 <= i <= n,
x[i] <= y[i]. We write x <* y if and only if x <=* y and x not= y.
We say that x,y in N^r are adjacent if and only if x,y are distinct
and y begins with x[2],...,x[n].
Note that <=*,<*, and adjacency require that all coordinates be
nonnegative.
TEMPLATE. (Given two relations). For every surjective function
(surjective polynomial), there are two related_1 arguments whose
values are related_2.
Not only do we have the above intriguingly basic template, we also
have that the finite form is obtained simply by restricting to
polynomials!
THEOREM 1. For every surjective P:N^k into Z^r, there exist x <* y
such that P(x) <* P(y).
THEOREM 2. For every surjective P:N^k into Z^r, there exist x <* y
such that P(x),P(y) are adjacent.
THEOREM 3. For every surjective polynomial P:N^k into Z^r, there exist
x <* y such that P(x) <* P(y) and P(x),P(y) are adjacent.
THEOREM 4. For every surjective polynomial P:N^k into Z^r, there exist
x <* y such that P(x) <* P(y).
THEOREM 5. For every surjective polynomial P:N^k into Z^r, there exist
x <* y such that P(x),P(y) are adjacent.
THEOREM 6. For every surjective polynomial P:N^k into Z^r, there exist
x <* y such that P(x) <* P(y) and P(x),P(y) are adjacent.
Note that Theorem 1-3 are explicitly Pi11, and Theorems 4-6 are
explicitly Pi03. We give Pi02 forms of Theorems 4-6 using the notion
of controllably surjective.
We say that P:N^k into Z^r is controllably surjective if and only if
there exists a polynomial Q:Z^r into Z such that
(forall x in Z^r)(therexists y in N^k)(P(y) = x and |y| <= Q(x)).
THEOREM 7. For every controllably surjective polynomial P:N^k into
Z^r, there exist x <* y such that P(x) <* P(y).
THEOREM 8. For every controllably surjective polynomial P:N^k into
Z^r, there exist x <* y such that P(x),P(y) are adjacent.
THEOREM 9. For every controllably surjective polynomial P:N^k into
Z^r, there exist x <* y such that P(x) <* P(y) and P(x),P(y) are
adjacent.
ACA' = RCA_0 + "for every x,n, the n-th Turing jump of x exists".
THEOREM A. Theorem 1 is provable in RCA_0. Theorems 2,3 are provably
equivalent to "epsilon_0 is well ordered" over RCA_0.
THEOREM B. Theorem 4 is provable in ISigma_1 but not in PRA. It is
provably equivalent to ISigma_1 over EFA. Theorems 5,6 are provable in
ACA' but not in PA. Theorems 5,6 are provably equivalent to 2-Con(PA)
over EFA.
THEOREM C. Theorem 7 is provable in EFA+ but not in EFA. It is
provably equivalent to EFA+ over EFA. Theorems 8,9 are provable in
ACA' but not in PA. Theorems 8,9 are provably equivalent to 1-Con(PA)
over EFA. They exhibit the well known < epsilon_0 recursive growth
phenomenon.
**********************************
I use http://www.math.ohio-state.edu/~friedman/ for downloadable
manuscripts. This is the 328th 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
Harvey Friedman
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