[FOM] 318:Polynomials and PA #2

Harvey Friedman friedman at math.ohio-state.edu
Mon Feb 4 00:07:57 EST 2008

```We correct three typos in the statement of Theorem B from #317. We also add
two additional statements. We use N for the set of all nonnegative integers.

Let P:Z^k into Z^r be a polynomial and x in Z^r. We write |P^-1(x)| for the
Euclidean distance from 0 in Z^k to the pre image of x. I.e., the minimum
Euclidean norm of the y with P(y) = x. If P^-1(x) is empty, then we take
|P^-1(x)| to be infinity.

THEOREM A. For all k >= 1 and polynomials P:Z^k into Z^k, there are distinct
positive integers x_1,...,x_k+2 such that |P^-1(x_1,...,x_k)| <=
|P^-1(x_2,...,x_k+1)| <= |P^-1(x_3,...,x_k+2)|.

THEOREM B. For all k,r,t >= 1 and polynomials P:Z^k into Z^r, there exist
distinct positive integers x_1,...,x_r+t-1 such that |P^-1(x_1,...,x_r)| <=
... <= |P^-1(x_t,...,x_r+t-1)|.

THEOREM C. For all k >= 1 and polynomials P:Z^k into Z^k, there are distinct
positive integers x_1,...,x_k+2 such that
|P^-1(x_2,...,x_k+1)|-|P^-1(x_1,...,x_k)| in 2N.

THEOREM D. For all k,r,t >= 1 and polynomials P:Z^k into Z^r, there are
distinct positive integers x_1,...,x_r+1 such that
|P^-1(x_2,...,x_r+1)|-|P^-1(x_1,...,x_r)| in tN.

THEOREM 1. Theorems A,B,C,D are provable in ACA but not in ACA_0. In
particular, they are not provable in PA. In each of these Theorems, the
mapping from P's into the least possible x's eventually dominates all
recursive functions.

THEOREM 2. Theorems A,B,C,D are provably equivalent, over EFA = exponential
function arithmetic, to the 2-consistency of PA. I.e.,"every 2
quantifier sentence provable in PA is true".

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

manuscripts. This is the 318th 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

Harvey Friedman

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