[FOM] 328: Polynomial Independence 1

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