[FOM] 330: Templating Pi01/Polynomial
Harvey Friedman
friedman at math.ohio-state.edu
Sat Jan 17 17:04:23 EST 2009
TYPO in #328, http://www.cs.nyu.edu/pipermail/fom/2009-January/013287.html
We wrote there
> 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].
The letter n is wrong in both places. This should be
Let x,y in N^k. We write x <=* y if and only if for all 1 <= i <= k,
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[r].
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Here we discuss Templating projects
1. for the Pi01 independence results that were presented in #326, http://www.cs.nyu.edu/pipermail/fom/2008-October/013161.html
2. for the polynomial independence results presented in #328. http://www.cs.nyu.edu/pipermail/fom/2009-January/013287.html
1. TEMPLATING THE Pi01 STATEMENT.
Recall the following from #326:
PROPOSITION 1.6. For all k,r >= 1 and upwards order invariant R
containedin [0,r]^4k, some RRA is a subset of R(A') union A+2 with the
same k dimensional powers of (8k)!.
This is equivalent to the consistency of ZFC with Mahlo cardinalities
of every finite order.
We will take up a slight alternative mentioned at the end of #326.
PROPOSITION 1.6'. For all k,r >= 1 and upwards order invariant R
containedin [1,r]^4k, some RRA is a subset of R(A') union A+1 with the
same k dimensional powers of (8k)!.
For present purposes, we prefer to suppress the quantity (8k)!.
Instead, we will use the following further variant of Proposition 1.6.
PROPOSITION 1.6*. For all r,p >> k and upwards order invariant R
containedin [1,r]^4k, some RRA is a subset of R(A') union A+1 with the
same k dimensional powers of p. The >> can be taken to represent a
double exponential.
We now construct a first template. Write POW(p)^k for the set of all k
dimensional powers of p. Here 1 is the first power of (8k)!.
TEMPLATE 1. For all r,p >> k and upwards order invariant R containedin
[1,r]^4k, there exists A containedin [1,r] such that BOOL(A
+1,R(A'),RRA,POW(p)^k).
In the above Template 1, BOOL(A+1,R(A'),RRA,POW(p)^k) represents any
given Boolean relation between the four sets A+1,R(A'),RRA,POW(p)^k).
A Boolean relation between subsets A_1,...,A_n of E, is the equality
between two expressions involving
A_1,...,A_n,union,intersection,complement, where complementation is
taken with respect to E. In Template 1, the E is [1,r]^k. THere are
2^16 Boolean equations, up to formal equivalence, in four set variables.
I have made a decent start on solving Template 1. It appears to be
difficult, but manageable. One gets an early reduction to 2^14 cases,
still a big number.
Obviously, the list A+1,R(A'),RRA,POW(p)^k) looks ad hoc. More
interesting, and hopefully still within reach of a sustained intense
effort, would be. e.g.,
TEMPLATE 2. For all r,p >> k and upwards order invariant R containedin
[1,r]^4k, there exists A containedin [1,r] such that BOOL(A,A
+1,RA,RRA,R(A'),POW(p)^k).
Obviously, even more interesting would be
TEMPLATE 3. For all r,p >= k and upwards order invariant R containedin
[1,r]^4k, there exists A containedin [1,r] such that BOOL(A,A+1,A
+2,...;RA,RRA,RRRA,...;R(A'),RR(A'),RRR(A')...;POW(p)^k).
2. TEMPLATING THE POLYNOMIAL INDEPENDENCE RESULT.
Recall the following from #328:
THEOREM 5. For every surjective polynomial P:N^k into Z^r, there exist
x <* y such that P(x),P(y) are adjacent.
Here for x,y in N^k,
x <* y iff x not= y, and (forall i in [1,k])(x[i] <= y[i]).
For x,y in N^r,
x,y are adjacent iff x not= y, and y begins with x[2],...,x[r].
We view <* and adjacent, as having variable dimension. Thus, they are
presented in a natural unifying way as follows:
x R y if and only if x,y are distinct vectors from N of the same
length, such that
(forall i in [1,lth(x))), a given order invariant relation holds of
x[i],x[i+1],y[i],y[i+1].
TEMPLATE 1. For every surjective polynomial P:N^k into Z^r, there
exist x R y such that P(x) S P(y). Here R,S are binary relations on
finite sequences from R, presented as above.
I believe that Template 1 is manageable.
There are obviously some easier Templates naturally arise. We can
consider the relations of the forms:
(forall i in (1,lth(x)), a given order invariant relation holds of
x[i],x[i+1],y[i].
The above doesn't cover the comparison of x[n],y[n], but it does cover
all x[i],y[i], 1 <= i < n-1. So it doesn't quite incorporate x <* y -
only a slightly weakened form.
Harvey Friedman
**********************************
I use http://www.math.ohio-state.edu/~friedman/ for downloadable
manuscripts. This is the 330th 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
Harvey Friedman
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